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Big Ideas Math Book Answer Key Grade 5 Chapter 10 Divide Fractions

Cross Check the Solutions from our Big Ideas Math Answers Grade 5 Chapter 10 Divide Fractions and understand the areas you are facing difficulty. Score higher grades in your exams and refer to Big Ideas Math Book Solution Key Grade 5 Chapter 10 Divide Fractions to have strong command over fractions.

Lesson: 1 Interpret Fractions as Division

• Lesson 10.1 Interpret Fractions as Division
• Interpret Fractions as Division Homework & Practice 10.1

Lesson: 2 Mixed Numbers as Quotients

• Lesson 10.2 Mixed Numbers as Quotients
• Mixed Numbers as Quotients Homework & Practice 10.2

Lesson: 3 Divide Whole Numbers by Unit Fractions

• Lesson 10.3 Divide Whole Numbers by Unit Fractions
• Divide Whole Numbers by Unit Fractions Homework & Practice 10.3

Lesson: 4 Divide Unit Fractions by Whole Numbers

• Lesson 10.4 Divide Unit Fractions by Whole Numbers
• Divide Unit Fractions by Whole Numbers Homework & Practice 10.4

Lesson: 5 Problem Solving: Fraction Division

• Lesson 10.5 Problem Solving: Fraction Division
• Problem Solving: Fraction Division Homework &  10.5

Chapter: 10 – Divide Fractions

• Divide Fractions Performance Task 10
• Divide Fractions Activity
• Divide Fractions Chapter Practice 10

Lesson 10.1 Interpret Fractions as Division

Explore and Grow`

You share 4 sheets of construction paper equally among 8 people. Write a division expression that represents the situation. What fraction of a sheet of paper does each person get? Use a model to support your answer?
Answer:
The division expression that represents the fraction of a sheet of paper does each person get is:
4 ÷ 8 = \(\frac{1}{2}\)

Explanation:
It is given that you have 4 sheets of construction paper equally among 8 people.
Hence,
The division expression that represents the fraction of a sheet of paper is:
( The number of sheets of construction paper ) ÷ ( The number of people )
= 4 ÷ 8
= \(\frac{1}{2}\)
Hence, from the above,
We can conclude that the fraction of a sheet of paper does each person get is: \(\frac{1}{2}\)

Structure
How can you check your answer using multiplication?
Answer: We can check the answer by using the partial products method or by using the simplification method.

Think and Grow: Divide Whole Numbers
You can use models to divide whole numbers that have a fraction as the quotient.
Answer: 
From the above model,
The number of colored parts is: 4
The total number of parts are: 8
So,
The fraction of the colored part out of the total number of parts = 4 ÷ 8
= \(\frac{4}{8}\) = \(\frac{1}{2}\)
In \(\frac{1}{2}\),
1 represents the quotient
2 represents the remainder
Example
Find 2 ÷ 3.
One Way: Use a tape diagram. Show 2 wholes. Divide each whole into 3 equal parts.

Another Way: Use an area model. Show 2 wholes. Divide each whole into 3 equal parts. Then separate the parts into 3 equal groups.

Show and Grow

Divide. Use a model to help

Question 1.
2 ÷ 4 =0.5
Answer:
From the above model,
The number of colored parts is: 2
The number of total parts is: 4
So,
The fraction of the colored parts out of the total number of parts = 2 ÷ 4
= \(\frac{2}{4}\)
= \(\frac{1}{2}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.5

Question 2.
1 ÷ 3 = 0.33
Answer:

From the above model,
The number of colored parts is: 1
The number of total parts is: 3
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 3
= \(\frac{1}{3}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.33

Apply and Grow: Practice

Divide. Use a model to help.

Question 3.
1 ÷ 8 =0.018
Answer:
From the above model,
The number of colored parts is: 1
The number of total parts is: 8
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 8
= \(\frac{1}{8}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.018

Question 4.
1 ÷ 4 =0.25
Answer:
From the above model,
The number of colored parts is: 1
The number of total parts is: 4
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 4
= \(\frac{1}{4}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.25

Question 5.
2 ÷ 6 =0.33
Answer:
From the above model,
The number of colored parts is: 2
The number of total parts is: 6
So,
The fraction of the colored parts out of the total number of parts = 2 ÷ 6
= \(\frac{2}{6}\)
= \(\frac{1}{3}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.33

Question 6.
2 ÷ 5 = 0.4
Answer:
From the above model,
The number of colored parts is: 2
The number of total parts is: 5
So,
The fraction of the colored parts out of the total number of parts = 2 ÷ 5
= \(\frac{2}{5}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.4

Question 7.
3 ÷ 7 = 0.42
Answer: 
From the above model,
The number of colored parts is: 3
The number of total parts is: 7
So,
The fraction of the colored parts out of the total number of parts = 3 ÷ 7
= \(\frac{3}{7}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.42

Question 8.
5 ÷ 6 = 0.83
Answer:
From the above model,
The number of colored parts is: 5
The number of total parts is: 6
So,
The fraction of the colored parts out of the total number of parts = 5 ÷ 6
= \(\frac{5}{6}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.83

Question 9.
How many 6s are in 1?
Answer: There are six \(\frac{1}{6}\)s in 1

Explanation:
The number of 6s in 1 can be obtained by dividing 1 into 6 equal parts.
So,
The figure obtained will be like;

From the above model,
The number of colored parts is: 1
The number of total parts is: 6
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 6
= \(\frac{1}{6}\)
Hence, from the above,
We can conclude that there are six 6s in 1

Question 10.
How many 10s are in 9?
Answer: There are 9 \(\frac{9}{10}\)s in 9

Explanation:
The model for the number of 10s in 9 are:

From the above model,
The number of colored parts is: 9
The number of total parts is: 10
So,
The fraction of the colored parts out of the total number of parts = 9 ÷ 10
= \(\frac{9}{10}\)
Hence, from the above,
We can conclude that there are nine 9s in 10

Question 11.
Number Sense
For which equations does k = 8?

Answer: Let the equations named A), B), C), and D)
So,
The four equations are:

From the above equations,
The value ‘k’ must be in the numerator.
So,
In A), the value of the numerator is: 3
In B), the value of the numerator is: k
In C), the value of the numerator is: 2
In D) the value of the numerator is: 8
So,
From the above numerator values,
We can say that “k=8” holds good for Equation B)

Question 12.
Writing
Write and solve a real-life problem for 7 ÷ 12.
Answer:
From the above model,
The number of colored parts is: 7
The number of total parts is: 12
So,
The fraction of the colored parts out of the total number of parts = 7 ÷ 12
= \(\frac{7}{12}\)
Hence,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.58

Think and Grow: Modeling Real Life

Example
Three fruit bars are shared equally among 4 friends. What fraction of a fruit bar does each friend get?

Divide 3 by 4 to find what fraction of a fruit bar each friend gets.
Use an area model to find 3 ÷ 4. Show 3 whole fruit bars. Divide each fruit bar into 4 equal parts. Then separate the parts into 4 equal groups.

Show and Grow

Question 13.
You cut a 5-foot streamer into 6 pieces of equal size. What is the length of each piece in feet? in inches?
Answer: The length of each piece in feet is: \(\frac{5}{6}\)

Explanation:
It is given that you cut a 5-foot streamer into 6 equal pieces of equal size.
So,
The model representing the 6 equal pieces of the 5-foot streamer is:

From the above model,
We can see that each part in the model represents \(\frac{5}{6}\) of each part.
Hence, from the above,
We can conclude that the length of each piece of a 5-foot streamer in feet is: \(\frac{5}{6}\)

Question 14.
Four circular lemon slices are shared equally among 8 glasses of water. What fraction of a lemon slice does each glass get?

Answer: The fraction of a lemon slice does each glass get is: \(\frac{1}{2}\)

Explanation:
It is given that the four circular lemon slices are shared equally among 8 glasses of water.
So,
The model representing the portion that each glass get is:

From the above model,
We can say that each part represents \(\frac{1}{2}\) part
Hence, from the above,
We can conclude that the portion of a lemon slice does glass get is: \(\frac{1}{2}\)

Question 15.
You cut a 5-foot streamer into 6 pieces of equal size. What is the length of each piece in feet? in inches?
Answer: The length of each piece in feet is: \(\frac{5}{6}\)

Explanation:
It is given that you cut a 5-foot streamer into 6 equal pieces of equal size.
So,
The model representing the 6 equal pieces of the 5-foot streamer is:

From the above model,
We can see that each part in the model represents \(\frac{5}{6}\) of each part.
Hence, from the above,
We can conclude that the length of each piece of a 5-foot streamer in feet is: \(\frac{5}{6}\)

Question 16.
DIG DEEPER!
A fruit drink is made using \(\frac{7}{4}\) quarts of orange juice and \(\frac{5}{4}\) quarts of pineapple juice. The drink is shared equally among 12 guests. What fraction of a quart does each guest get?
Answer: The fraction of a quart does each guest get is: \(\frac{1}{4}\)

Explanation:
It is given that a fruit drink is made using \(\frac{7}{4}\) quarts of orange juice and \(\frac{5}{4}\) quarts of pineapple juice.
So,
The total amount of fruit juice= \(\frac{7}{4}\) + \(\frac{5}{4}\)
= \(\frac{ 7 + 5}{4}\)
= \(\frac{12}{4}\)
It is also given that the drink is shared equally among 12 guests
So,
The fraction of a quart does each gust get = \(\frac{12}{4}\) ÷ 12
= \(\frac{12}{4}\) ÷ \(\frac{12}{1}\)
= \(\frac{12}{4}\) × \(\frac{1}{12}\)
= \(\frac{1}{4}\)
Hence, from the above,
We can conclude that the fraction of a quart does each person get is: \(\frac{1}{4}\)

Interpret Fractions as Division Homework & Practice 10.1

Divide. Use a model to help.

Question 1.
1 ÷ 6 =0.16
Answer:

From the above model,
The number of colored parts is: 1
The number of total parts is: 6
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 6
= \(\frac{1}{6}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.16

Question 2.
1 ÷ 7 =0.14
Answer:
From the above model,
The number of colored parts is: 1
The number of total parts is: 7
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 7
= \(\frac{1}{7}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.14

Question 3.
1 ÷ 5 = 0.20
Answer:
From the above model,
The number of colored parts is: 1
The number of total parts is: 5
So,
The fraction of the colored parts out of the total number of parts = 1 ÷ 5
= \(\frac{1}{5}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.20

Question 4.
3 ÷ 4 = 0.75
Answer:
From the above model,
The number of colored parts is: 3
The number of total parts is: 4
So,
The fraction of the colored parts out of the total number of parts = 3 ÷ 4
= \(\frac{3}{4}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.75

Question 5.
6 ÷ 7 = 0.85
Answer:
From the above model,
The number of colored parts is: 6
The number of total parts is: 7
So,
The fraction of the colored parts out of the total number of parts = 6 ÷ 7
= \(\frac{6}{7}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.85

Question 6.
5 ÷ 9 = 0.55
Answer:
From the above model,
The number of colored parts is: 5
The number of total parts is: 9
So,
The fraction of the colored parts out of the total number of parts = 5 ÷ 9
= \(\frac{5}{9}\)
So,
The fraction of the colored parts out of the total number of parts in the decimal form is: 0.55

Question 7.
YOU BE THE TEACHER
Your friend says \(\frac{5}{12}\) is equivalent to 12 ÷ 5. Is your friend correct? Explain.
Answer: No, your friend s not correct.

Explanation:
The given fraction is: \(\frac{5}{12}\)
From the given fraction,
The numerator is: 5
The denominator is: 12
We can write a fraction in the following form:
Fraction = \(\frac{Numerator}{Denominator}\)
So,
\(\frac{5}{12}\) is equivalent to 5 ÷ 12
But, according to your friend,
\(\frac{5}{12}\) is equivalent to 12 ÷ 5
Hence, from the above,
we can conclude that your friend is not correct.

Question 8.
Writing
Explain how fractions and division are related.

Question 9.
Structure
Write a division equation represented by the model.

Answer:
The division equation represented by the model is: 1 ÷ 4

Explanation:
The given model is:

From the given model,
The number of shaded parts is: 1
The total number of parts are: 4
So,
The division equation can be represented as:
Division equation = (The number of shaded parts) ÷ ( The total number of parts )
= 1 ÷ 4
= \(\frac{1}{4}\)

Question 10.
Number Sense
Eight friends share multiple vegetable pizzas, and each gets \(\frac{3}{8}\) of a pizza. How many pizzas do they share?
Answer: The total number of pizzas the eight friends shared are: 3 pizzas

Explanation:
It is given that the eight friends share multiple vegetable pizzas and each gets \(\frac{3}{8}\) of a pizza.
So,
The total number of pizzas shared by the eight friends = \(\frac{3}{8}\) × 8
= \(\frac{3}{8}\) × \(\frac{8}{1}\)
= \(\frac{3 × 8}{8 × 1}\)
= \(\frac{3}{1}\)
= 3
Hence, from the above,
We can conclude that the total number of pizzas shared by the eight friends is: 3 pizzas

Question 11.
Modeling Real Life
Seven friends each run an equal part of a 5-kilometer relay race. What fraction of a kilometer does each friend complete?
Answer: The fraction of a kilometer does each friend complete is: \(\frac{5}{7}\) kilometer

Explanation:
It is given that there are seven friends each run an equal part of a 5-kilometer relay race.
So,
The fraction that each friend run = \(\frac{The total distance} {The number of friends}\)
= \(\frac{5}{7}\)
Hence, from the above,
We can conclude that the fraction of a kilometer does each friend complete is: \(\frac{5}{7}\) kilometer

Question 12.
Modeling Real Life
A group of friends equally share 3 bags of pretzels. Each friend gets \(\frac{3}{5}\) of a bag of pretzels. How many friends are in the group?
Answer: The total number of friends in the group are: 5

Explanation:
It is given that a group of friends equally share 3 bags of pretzels and each friend gets \(\frac{3}{5}\) of a bag of pretzels.
So,
The total number of friends = \(\frac{The total number of bags}{The amount each friend gets}\)
= \(\frac{3}{1}\) × \(\frac{5}{3}\)
= \(\frac{5}{1}\)
= 5
Hence, from the above,
We can conclude that the total number of friends are: 5

Review & Refresh

Multiply.

Question 13.

Answer: 9 × \(\frac{2}{3}\) = 6

Explanation:
The given fractions are: \(\frac{9}{1}\) and \(\frac{2}{3}\)
So,
\(\frac{9}{1}\) × \(\frac{2}{3}\)
= \(\frac{9 × 2}{1 × 3}\)
= \(\frac{6}{1}\)
= 6
Hence,
9 × \(\frac{2}{3}\) = 6

Question 14.

Answer: 5 × \(\frac{7}{10}\) = \(\frac{7}{2}\)

Explanation:
The given fractions are: \(\frac{5}{1}\) and \(\frac{7}{10}\)
So,
\(\frac{5}{1}\) × \(\frac{7}{10}\)
= \(\frac{5 × 7}{1 × 10}\)
= \(\frac{7}{2}\)
Hence,
5 × \(\frac{7}{10}\) = \(\frac{7}{2}\)

Question 15.

Answer: 3 × \(\frac{5}{12}\) = \(\frac{5}{4}\)

Explanation:
The given fractions are: \(\frac{3}{1}\) and \(\frac{5}{12}\)
So,
\(\frac{3}{1}\) × \(\frac{5}{12}\)
= \(\frac{3 × 5}{1 × 12}\)
= \(\frac{5}{4}\)
Hence,
3 × \(\frac{5}{12}\) = \(\frac{5}{4}\)

Lesson 10.2 Mixed Numbers as Quotients

Explore and Grow

You share 6 sheets of construction paper equally among 4 people. Write a division expression that represents the situation. How much paper does each person get? Use a model to support your answer.
Answer:
The division expression representing the situation is: 6 ÷ 4

Explanation:
It s given that you have shared 6 sheets of construction paper equally among 4 people
So,
The division equation representing the sharing of construction papers is: 6 ÷ 4
Now,
6 ÷ 4 = \(\frac{6}{4}\)
So,
The above equation represents that 4 is divided into 6 parts.
So,
The model representing the situation is:

From the above model,
We can say that the amount of does each person get is: 1\(\frac{1}{2}\) or 1.5 or \(\frac{3}{2}\)

Precision
Does each person get less than or more than 1 sheet of paper? Use the dividend and divisor to explain why your answer makes sense.
Answer:
From the above problem,
We can say that each person gets more than 1 paper.
So,
The division equation of the above problem is: 6 ÷ 4
The equivalent form of 6 ÷ 4 is: \(\frac{6}{4}\)
Now,
The simplest form of \(\frac{6}{4}\) is: \(\frac{3}{2}\) ( The simplest form is the division of the numerator and the denominator with the common multiple if we can divide)
The mixed form of \(\frac{3}{2}\) is: 1\(\frac{1}{2}\)

Think and Grow: Divide Whole Numbers

You can use models to divide whole numbers that have a mixed number as the quotient.
Example
Find 3 ÷ 2.
One Way:
Use a tape diagram. Show 3 wholes. Divide each whole into 2 equal parts.

Another Way: Use an area model. Show 3 wholes. Divide each whole into 2 equal parts. Then separate the parts into 2 equal groups.

Show and Grow

Divide. Use a model to help

Question 1.
5 ÷ 3 = ___
Answer: 5 ÷ 3 = 1\(\frac{2}{3}\)

Explanation;
The given division equation is: 5 ÷ 3
The model representing the division equation is:

From the above model,
5 ÷ 3 = 3 ÷ 3
= 1 R 2
Hence,
We can say that each part is divided into 1\(\frac{2}{3}\) or \(\frac{5}{3}\)

Question 2.
7 ÷ 2 = ___

Answer: 7 ÷ 2 = 3\(\frac{1}{2}\)

Explanation;
The given division equation is: 7 ÷ 2
The model representing the division equation is:

From the above model,
7 ÷ 2 = 6 ÷ 2
= 3 R 1
Hence,
We can say that each part is divided into 3\(\frac{1}{2}\) or \(\frac{7}{2}\) or 3.5

Apply and Grow: Practice

Divide. Use a model to help.

Question 3.
12 ÷ 7 = ___

Answer: 12 ÷ 7 = 1\(\frac{5}{7}\)

Explanation;
The given division equation is: 12 ÷ 7
The model representing the division equation is:

From the above model,
12 ÷ 7 = 7 ÷ 7
= 1 R 5
Hence,
We can say that each part is divided into 1\(\frac{5}{7}\) or \(\frac{12}{7}\)

Question 4.
25 ÷ 20 = ___

Answer: 25 ÷ 20 = 1\(\frac{5}{20}\) = \(\frac{5}{4}\)

Explanation;
The given division equation is: 25 ÷ 20
The model representing the division equation is:

From the above model,
25 ÷ 20 = 20 ÷ 20
= 1 R 5
Hence,
We can say that each part is divided into 1\(\frac{5}{20}\) or \(\frac{5}{4}\)

Question 5.
15 ÷ 4 = ___

Answer: 15 ÷ 4 = 3\(\frac{3}{4}\)

Explanation;
The given division equation is: 15 ÷ 4
The model representing the division equation is:

From the above model,
15 ÷ 4 = 12 ÷ 4
= 3 R 3
Hence,
We can say that each part is divided into 3\(\frac{3}{4}\) or \(\frac{15}{4}\)

Question 6.
13 ÷ 6 = ___

Answer: 13 ÷ 6 = 2\(\frac{1}{6}\)

Explanation;
The given division equation is: 13÷ 6
The model representing the division equation is:

From the above model,
13 ÷ 6 = 12 ÷ 6
= 2 R 1
Hence,
We can say that each part is divided into 2\(\frac{1}{6}\) or \(\frac{13}{6}\)

Question 7.
16 ÷ 8 = ___

Answer: 16 ÷ 8 = 2

Explanation;
The given division equation is: 16÷ 8
The model representing the division equation is:

From the above model,
16 ÷ 8
= 2 R 0
Hence,
We can say that each part is divided into 2 equal parts

Question 8.
92 ÷ 50 = ___

Answer: 92 ÷ 50 = 1\(\frac{21}{25}\)

Explanation;
The given division equation is: 92÷ 50
So,
92 ÷ 50 = 50 ÷ 50
= 1 R 42
Hence,
We can say that each part is divided into 1\(\frac{42}{50}\) or 1\(\frac{21}{25}\)

Question 9.
How many 3s are in 7?
Answer: The number of 3 in 7 are: \(\frac{7}{3}\) or 2\(\frac{1}{3}\)

Explanation:
The division equation is: 7 ÷ 3
So,
The model for the given division equation is:

From the above model,
7 ÷ 3 = 6 ÷ 3
= 2 R 1
Hence, from the above,
We can conclude that there are 2\(\frac{1}{3}\) 3s in 7

Question 10.
How many 6s are in 21?
Answer: The number of 6s in 21 are: \(\frac{21}{6}\) or 3\(\frac{3}{6}\)

Explanation:
The division equation is: 21 ÷ 6
So,
The model for the given division equation is:

From the above model,
21 ÷ 6 = 18 ÷ 6
= 3 R 3
Hence, from the above,
We can conclude that there are 3\(\frac{3}{6}\) 3s in 21

Question 11.
YOU BE THE TEACHER
Your friend says that \(\frac{35}{6}\) is equivalent to 35 ÷ 6. Is your friend correct? Explain.
Answer: Yes, your friend is correct

Explanation:
It is given that \(\frac{35}{6}\)
We know that,
The decimal equation can be converted into a fraction as \(\frac{Numerator}{Denominator}\)
So,
\(\frac{35}{6}\) = 35 ÷ 6
Hence, from the above,
We can conclude that your friend is correct

Question 12.
Writing
Write and solve a real-life problem for 24 ÷ 5.
Answer: 24 ÷ 5 = 4\(\frac{4}{5}\)

Explanation;
The given division equation is: 24÷ 5
The model for the above division equation is:

From the above model,
24 ÷ 5 = 20 ÷ 5
= 4 R 4
Hence,
We can say that each part is divided into 4\(\frac{4}{5}\)

Think and Grow: Modeling Real Life

Example
You share 7 bales of hay equally among 3 horse stalls. How many whole bales are in each stall? What fractional amount of a bale is in each stall?
Divide 7 by 3 to find how many bales of hay are in each stall. Use an area model to help.

Show and Grow

Question 13.
Six muffins are shared equally among 4 friends. How many whole muffins does each friend get? What fractional amount of a muffin does each friend get?
Answer: Each friend will get 1 muffin and 2 muffins are leftovers
The fractional part of a muffin does each friend get is: \(\frac{1}{2}\)

Explanation:
It is given that there are six muffins are shared equally among 4 friends.
So,
The number of muffins each friend get = 6 ÷ 4
= 4 ÷ 4
= 1 R 2
Hence, from the above,
We can conclude that each friend gets 1 muffin each and the fraction of each muffin get is: \(\frac{1}{2}\)

Question 14.
A cyclist bikes 44 miles in 5 days. She bikes the same distance each day. Does she bike more than 8\(\frac{1}{2}\) miles each day? Explain.
Answer: She bikes more than 8\(\frac{1}{2}\) miles each day.

Explanation:
It is given that a cyclist bikes 44 miles in 5 days.
So,
The distance that she bikes each day = 44 ÷ 5
So,
44 ÷ 5 = 40 ÷ 5
= 8 R 4
= 8\(\frac{4}{5}\) miles
But, it is given that she bikes 8\(\frac{1}{2}\) miles each day
Hence, from the above,
We can conclude that she bikes more than 8\(\frac{1}{2}\) miles each day.

Question 15.
DIG DEEPER!
At Table A, 4 students share 7 packs of clay equally. At Table B, 5 students share 8 packs of clay equally. At which table does each student get a greater amount of clay? Explain.

Answer: At Table A, each student gets a greater amount of clay.

Explanation:
It is given that at Table A, 4 students share 7 packs of clay equally.
So,
The representation of clay at table A is: \(\frac{7}{4}\)
It is also given that at Table B, 5 students share 8 packs of clay equally.
So,
The representation of clay at table B is: \(\frac{8}{5}\)
So,
For comparison, equate the denominators.
So,
Multiply the first fraction at table A by \(\frac{5}{5}\) and the fraction at table B by \(\frac{4}{4}\)
So,
\(\frac{7}{4}\) × \(\frac{5}{5}\)
= \(\frac{35}{20}\)
So,
\(\frac{8}{5}\) × \(\frac{4}{4}\)
= \(\frac{32}{20}\)
Hence, from the above,
We can conclude that at table A, the students will get more amount of clay.

Mixed Numbers as Quotients Homework & Practice 10.2

Divide. Use a model to help.

Question 1.
5 ÷ 2 = ___
Answer: 5 ÷ 2 = 2\(\frac{1}{2}\)

Explanation;
The given division equation is: 5 ÷ 2
The model representing the division equation is:

From the above model,
5 ÷ 2 = 4 ÷ 2
= 2 R 1
Hence,
We can say that 5 ÷ 2 = 2\(\frac{1}{2}\) or 2.5 or \(\frac{5}{2}\)

Question 2.
10 ÷ 7 = ___
Answer: 10 ÷ 7 = 1\(\frac{3}{7}\) = \(\frac{10}{7}\)

Explanation;
The given division equation is: 10 ÷ 7
The model representing the division equation is:

From the above model,
10 ÷ 7 = 7 ÷ 7
= 1 R 3
Hence,
We can say that 10 ÷ 7 = 1\(\frac{3}{7}\) or \(\frac{10}{7}\)

Question 3.
3 ÷ 9 = ___
Answer: 3 ÷ 9 = \(\frac{1}{3}\)

Explanation;
The given division equation is: 3 ÷ 9
The model representing the division equation is:

From the above model,
3 and 9 are the multiples of 3.
So,
3 ÷ 9 = \(\frac{1}{3}\)
Hence,
We can say that 3 ÷ 9 = \(\frac{1}{3}\)

Question 4.
11 ÷ 4 = ___
Answer: 11 ÷ 4 = 2\(\frac{3}{4}\)

Explanation;
The given division equation is: 11 ÷ 4
The model representing the division equation is:

From the above model,
11 ÷ 4 = 8 ÷ 4
= 2 R 3
Hence,
We can say that 11 ÷ 4 = \(\frac{11}{4}\) or 2\(\frac{3}{4}\)

Question 5.
13 ÷ 6 = ___
Answer: 13 ÷ 6 = 2\(\frac{1}{6}\)

Explanation;
The given division equation is: 13 ÷ 6
The model representing the division equation is:

From the above model,
13 ÷ 6 = 12 ÷ 6
= 2 R 1
Hence,
We can say that 13 ÷ 6 = \(\frac{13}{6}\) or 2\(\frac{1}{6}\)

Question 6.
45 ÷ 8 = ___
Answer: 45 ÷ 8 = 5\(\frac{5}{8}\)

Explanation;
The given division equation is: 45 ÷ 8
The model representing the division equation is:

From the above model,
45 ÷ 8 = 40 ÷ 8
= 5 R 5
Hence,
We can say that 45 ÷ 8 = \(\frac{45}{8}\) or 5\(\frac{5}{8}\)

Question 7.
Number Sense
Between which two whole numbers is the quotient of 74 and 9?
Answer: The quotient of 74 and 9 is between 8 and 9

Explanation:
The given two numbers are 7 and 9
So,
By using the partial quotients method,
74 ÷ 9= 72 ÷ 9
= 8 R 2
So,
74 ÷ 9 = \(\frac{74}{9}\) or 8\(\frac{2}{9}\) or 8.3
Hence, from the above,
We can conclude that the quotient of 74 and 9 is between 8 and 9

Question 8.
Reasoning
Three friends want to share 22 baseball cards. For this situation, why does the quotient 7 R1 make more sense than the quotient 7\(\frac{1}{3}\)?
Answer:
It is given that three friends want to share 22 baseball cards.
So,
We have to find the number of baseball cards each friend possesses.
So,
It is sufficient to write the number of baseball cards possessed by each friend in the remainder form rather than the fraction form.
So,
The number of baseball cards possessed by each friend = \(\frac{The total number of baseball cards}{The number of friends}\)
= 22 ÷ 3
= 21 ÷ 3
= 7 R 1
Hence, from the above,
We can conclude that the remainder form is sufficient to find the number of baseball cars possessed by each friend rather than the fraction form.

Question 9.
DIG DEEPER!
Is \(\frac{2}{5}\) × 3 equivalent to 2 × 3 ÷ 5? Explain.
Answer: Yes, \(\frac{2}{5}\) × 3 equivalent to 2 × 3 ÷ 5

Explanation:
The given fraction and the number is: \(\frac{2}{5}\) and 3
So,
\(\frac{2}{5}\) × 3 = \(\frac{2}{5}\) × \(\frac{3}{1}\)
= \(\frac{2 × 3}{5}\)
= 2 × 3 ÷ 5
Hence, from the above,
We can conclude that \(\frac{2}{5}\) × 3 equivalent to 2 × 3 ÷ 5

Question 10.
Modeling Real Life
A bag of 4 balls weighs 6 pounds. Each ball weighs the same amount. What is the weight of each ball?
Answer: The weight of each ball is: \(\frac{3}{2}\) pounds or 1.5 pounds

Explanation:
It is given that a bag of 4 balls weighs 6 pounds
So,
The weight of each ball = \(\frac{The total weight of the balls}{The number of balls}\)
= 6 ÷ 4
Since 6 and 4 are the multiples of 2, divide the two numbers by 2
So,
6 ÷ 4 = 3 ÷ 2
So,
3 ÷ 2 = 2 ÷ 2
= 1 R 1
= 1\(\frac{1}{2}\) pounds
Hence, from the above,
We can conclude that the weight of each ball is: 1\(\frac{1}{2}\) pounds or 1.5 pounds

Question 11.
Modeling Real Life
Zookeepers order 600 pounds of bamboo for the pandas. The bamboo lasts 7 days. How many whole pounds of bamboo do the pandas eat each day? What fractional amount of a pound do the pandas eat each day?

Answer:
The amount of bamboos the pandas eat each day is around 85 pounds
The amount of bamboos the pandas eat each day in the fraction form is: 85\(\frac{5}{7}\)

Explanation:
It is given that zookeepers order 600 pounds of bamboo for the pandas and the bamboos last 7 days for the pandas
So,
The number of bamboos the pandas eat each day = 600 ÷ 7
So,
By using the partial quotients method,
600 ÷ 7 = ( 560 + 35 ) ÷ 7
= ( 560 ÷ 7 ) + ( 35 ÷ 7 )
= 80 + 5
= 85 R 5
Hence, from the above,
We can conclude that
The amount of bamboos the pandas eat each day is around 85 pounds
The amount of bamboos the pandas eat each day in the fraction form is: 85\(\frac{5}{7}\)

Question 12.
Modeling Real Life
A plumber has 20 feet of piping. He cuts the piping into 6 equal pieces. Is each piece greater than, less than, or equal to 3\(\frac{1}{2}\) feet?
Answer: Each piece is less than 3\(\frac{1}{2}\) feet

Explanation:
It is given that a plumber has 20 feet of piping and he cuts the piping into 6 equal pieces.
So,
The length of each piece = 20 ÷ 6
By using the partial quotients method,
20 ÷ 6 = 18 ÷ 6
= 3 R 2
So,
20 ÷ 6 = 3\(\frac{2}{6}\)
Now,
3\(\frac{1}{2}\) = \(\frac{7}{2}\)
3\(\frac{2}{6}\) = \(\frac{20}{6}\)
For comparison, we have to equate whether the denominators or the numerators.
So,
Multiply 3\(\frac{1}{2}\) with \(\frac{3}{3}\)
So,
3\(\frac{1}{2}\) = \(\frac{21}{6}\)
Hence, from the above,
We can conclude that each piece is less than 3\(\frac{1}{2}\) feet

Review & Refresh

Add.

Question 13.

Answer: \(\frac{2}{9}\) + \(\frac{2}{3}\) = \(\frac{8}{9}\)

Explanation:
The two given fractions are: \(\frac{2}{9}\) and \(\frac{2}{3}\)
So, in addition, we have to make either the numerators or the denominators equal
So,
Multiply \(\frac{2}{3}\)  with \(\frac{3}{3}\)
So,
\(\frac{2}{3}\)  = \(\frac{6}{9}\)
Hence, from the above,
We can conclude that \(\frac{2}{9}\) + \(\frac{2}{3}\) = \(\frac{8}{9}\)

Question 14.

Answer: \(\frac{1}{10}\) + \(\frac{3}{4}\) = \(\frac{34}{40}\)

Explanation:
The two given fractions are: \(\frac{1}{10}\) and \(\frac{3}{4}\)
So, in addition, we have to make either the numerators or the denominators equal
So,
Multiply \(\frac{1}{10}\)  with \(\frac{4}{4}\)
Multiply \(\frac{3}{4}\)  with \(\frac{10}{10}\)
So,
\(\frac{1}{10}\)  = \(\frac{4}{40}\)
\(\frac{3}{4}\)  = \(\frac{30}{40}\)
Hence, from the above,
We can conclude that \(\frac{1}{10}\) + \(\frac{3}{4}\) = \(\frac{34}{40}\)

Question 15.

Answer: \(\frac{3}{5}\) + \(\frac{5}{6}\) + \(\frac{1}{5}\) = \(\frac{49}{30}\)

Explanation:
The three given fractions are: \(\frac{3}{5}\) , \(\frac{5}{6}\) and \(\frac{1}{5}\)
So, in addition, we have to make either the numerators or the denominators equal
So,
Multiply \(\frac{3}{5}\)  with \(\frac{6}{6}\)
Multiply \(\frac{5}{6}\)  with \(\frac{5}{5}\)
Multiply \(\frac{1}{5}\)  with \(\frac{6}{6}\)
So,
\(\frac{3}{5}\)  = \(\frac{18}{30}\)
\(\frac{5}{6}\)  = \(\frac{25}{30}\)
\(\frac{1}{5}\)  = \(\frac{6}{30}\)
Hence, from the above,
We can conclude that \(\frac{3}{5}\) + \(\frac{5}{6}\) +\(\frac{1}{5}\)  = \(\frac{49}{30}\)

Lesson 10.3 Divide Whole Numbers by Unit Fractions

Explore and Grow

Write a real-life problem that can be represented by 6 ÷ \(\frac{1}{2}\)?
Answer:
Suppose, we have an apple and there are 6 children and we are giving each child half of the piece.
So,
Each child receives 6 ÷ \(\frac{1}{2}\) piece of the apple

What is the solution to the problem? Use a model to support your answer?
Answer:
The above problem is the division of an apple among the six children
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
The amount each child receive from an apple = 6 ÷ \(\frac{1}{2}\)
= 6 × \(\frac{2}{1}\)
= \(\frac{6}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 6 × 2}{1 × 1}\)
= 12

Structure
How can you check your answer using multiplication?
Answer:
We can check the answer using multiplication by the two rules regarding division and multiplication. They are:
A) a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
B) a= \(\frac{a}{1}\)

Think and Grow: Divide Whole Numbers by Unit Fractions

You can use models to divide whole numbers by unit fractions.
Example
Find 4 ÷ \(\frac{1}{3}\)
One Way:
Use a tape diagram to find how many \(\frac{1}{3}\)s are in 4. There are 4 wholes.
Divide each whole into 3 equal parts. Each part is \(\frac{1}{3}\).

Because there are 3 one-thirds in 1 whole, there are
4 × 3 equal parts = 12 one-thirds in 4 wholes.

Show and Grow

Divide. Use a model to help

Question 1.

Answer: 3 ÷ \(\frac{1}{2}\) = 6

Explanation:
The given numbers are: 3 and \(\frac{1}{2}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
3 ÷ \(\frac{1}{2}\)  = 3 × \(\frac{2}{1}\)
= \(\frac{3}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 3 × 2}{1 × 1}\)
= 6
Hence,
3÷ \(\frac{1}{2}\) = 6

Question 2.

Answer: 2 ÷ \(\frac{1}{5}\) = 10

Explanation:
The given numbers are: 2 and \(\frac{1}{5}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
2 ÷ \(\frac{1}{5}\)  = 2 × \(\frac{5}{1}\)
= \(\frac{5}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 5 × 2}{1 × 1}\)
= 10
Hence,
2÷ \(\frac{1}{5}\) = 10

Apply and Grow: Practice

Divide. Use a model to help.

Question 3.

Answer: 1 ÷ \(\frac{1}{3}\) = 3

Explanation:
The given numbers are: 1 and \(\frac{1}{3}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
1 ÷ \(\frac{1}{3}\)  = 1 × \(\frac{3}{1}\)
= \(\frac{3}{1}\) × \(\frac{1}{1}\)
= \(\frac{ 3 × 1}{1 × 1}\)
= 3
Hence,
1÷ \(\frac{1}{3}\) = 3

Question 4.

Answer: 3 ÷ \(\frac{1}{5}\) = 15

Explanation:
The given numbers are: 3 and \(\frac{1}{5}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
3 ÷ \(\frac{1}{5}\)  = 3 × \(\frac{5}{1}\)
= \(\frac{3}{1}\) × \(\frac{5}{1}\)
= \(\frac{ 3 × 5}{1 × 1}\)
= 15
Hence,
3÷ \(\frac{1}{5}\) = 15

Question 5.

Answer: 5 ÷ \(\frac{1}{3}\) = 15

Explanation:
The given numbers are: 5 and \(\frac{1}{3}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
5 ÷ \(\frac{1}{3}\)  = 5 × \(\frac{3}{1}\)
= \(\frac{3}{1}\) × \(\frac{5}{1}\)
= \(\frac{ 3 × 5}{1 × 1}\)
= 15
Hence,
5÷ \(\frac{1}{3}\) = 15

Question 6.

Answer: 4 ÷ \(\frac{1}{4}\) = 16

Explanation:
The given numbers are: 4 and \(\frac{1}{4}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
4 ÷ \(\frac{1}{4}\)  = 4 × \(\frac{4}{1}\)
= \(\frac{4}{1}\) × \(\frac{4}{1}\)
= \(\frac{ 4 × 4}{1 × 1}\)
= 16
Hence,
4÷ \(\frac{1}{4}\) = 16

Question 7.

Answer: 7 ÷ \(\frac{1}{2}\) = 14

Explanation:
The given numbers are: 7 and \(\frac{1}{2}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
7 ÷ \(\frac{1}{2}\)  = 7 × \(\frac{2}{1}\)
= \(\frac{7}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 7 × 2}{1 × 1}\)
= 14
Hence,
7÷ \(\frac{1}{2}\) = 14

Question 8.

Answer: 2 ÷ \(\frac{1}{7}\) = 14

Explanation:
The given numbers are: 2 and \(\frac{1}{7}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
2 ÷ \(\frac{1}{7}\)  = 2 × \(\frac{7}{1}\)
= \(\frac{2}{1}\) × \(\frac{7}{1}\)
= \(\frac{ 7 × 2}{1 × 1}\)
= 14
Hence,
2÷ \(\frac{1}{7}\) = 14

Question 9.
How many \(\frac{1}{4}\)s are in 5?
Answer: There are 20 \(\frac{1}{4}\)s in 5

Explanation:
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
Now,
We have to find the number of \(\frac{1}{4}\)s in 5
So,
5 ÷ \(\frac{1}{4}\)  = 5 × \(\frac{4}{1}\)
= \(\frac{5}{1}\) × \(\frac{4}{1}\)
= \(\frac{ 5 × 4}{1 × 1}\)
= 20
Hence, from the above,
We can conclude that there are 20 \(\frac{1}{4}\)s in 5.

Question 10.
How many \(\frac{1}{6}\)s are in 2?
Answer: There are 12 \(\frac{1}{6}\)s in 2

Explanation:
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
Now,
We have to find the number of \(\frac{1}{6}\)s in 2
So,
2 ÷ \(\frac{1}{6}\)  = 2 × \(\frac{6}{1}\)
= \(\frac{2}{1}\) × \(\frac{6}{1}\)
= \(\frac{ 2 × 6}{1 × 1}\)
= 12
Hence, from the above,
We can conclude that there are 12 \(\frac{1}{6}\)s in 2.

Question 11.
YOU BE THE TEACHER
Newton finds 6 ÷ \(\frac{1}{3}\). Is he correct? Explain.

Answer: No, Newton is not correct

Explanation:
The given division equation is: 6 ÷ \(\frac{1}{3}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
6 ÷ \(\frac{1}{3}\)  = 6 × \(\frac{3}{1}\)
= \(\frac{6}{1}\) × \(\frac{3}{1}\)
= \(\frac{ 3 × 6}{1 × 1}\)
= 18
But, according to Newton,
6 ÷ \(\frac{1}{3}\) = 2
Hence, from the above,
We can conclude that Newton is not correct.

Question 12.
Writing
Write and solve a real-life problem for 4 ÷ \(\frac{1}{2}\).
Answer:
Suppose we have 4 bags of wheat and we have to distribute the 4 bags by dividing each bag of wheat in half
So,
Each person receives 4 ÷ \(\frac{1}{2}\) bag of wheat
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
4 ÷ \(\frac{1}{2}\)  = 4 × \(\frac{2}{1}\)
= \(\frac{4}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 4 × 2}{1 × 1}\)
= 8
Hence, from the above,
We can conclude that there are 8 bags of wheat when divide the 4 bags of wheat in half.

Think and Grow: Modeling Real Life

Example
A chef makes 3 cups of salsa. A serving of salsa is \(\frac{1}{8}\) cup. How many servings does the chef make?
To find the number of servings, find the number of \(\frac{1}{8}\) cups in 3 cups.
Use an area model to find 3 ÷ \(\frac{1}{8}\). Divide each cup into 8 equal parts.

Show and Grow

Question 13.
A litter of kittens weighs a total of 2 pounds. Each newborn kitten weighs \(\frac{1}{4}\) pound. How many kittens are in the litter?
Answer: The number of kittens in the litter are: 8 kittens

Explanation:
It is given that a litter of kittens weighs a total of 2 pounds and each newborn kitten weighs \(\frac{1}{4}\) pound.
So,
The number of kittens in the litter = \(\frac{The total weight of litter}{The weight of each newborn kitten}\)
= 2 ÷ \(\frac{1}{4}\)
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
2 ÷ \(\frac{1}{4}\)  = 2 × \(\frac{4}{1}\)
= \(\frac{4}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 4 × 2}{1 × 1}\)
= 8
Hence, from the above,
We can conclude that the number of kittens in the litter are: 8 kittens

Question 14.
You put signs on a walking trail that is 7 miles long. You put a sign at the start and at the end of the trail. You also put a sign every \(\frac{1}{10}\) mile. How many signs do you put on the trail?
Answer: The total number of signs you put on the trail is: 72

Explanation:
It is given that you put signs on a walking trail that is 7 miles long and you put a sign at the start and at the end of the trail.
It is also given that you put a sign every \(\frac{1}{10}\) mile.
So,
The total number of signs you put on the trail = The sign at the start of the trail + The sign at the end of the trail + The total number of signs for \(\frac{1}{10}\) mile
Now,
The total number of signs for \(\frac{1}{10}\) mile = 7 ÷ \(\frac{1}{10}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
7 ÷ \(\frac{1}{10}\)  = 7 × \(\frac{10}{1}\)
= \(\frac{7}{1}\) × \(\frac{10}{1}\)
= \(\frac{ 7 × 10}{1 × 1}\)
= 70
So,
The total number of signs you put on the trail = 1 + 1 + 70
= 72
hence, from the above,
We can conclude that there are 72 signs that you put on the trail

Question 15.
DIG DEEPER!
You have 2 boards that are each 8 feet long. You cut \(\frac{1}{2}\)– foot pieces to make square picture frames. How many picture frames can you make?
Answer: The number of picture frames you can make is: 32

Explanation:
It is given that you have 2 boards that are each 8 feet long.
So,
The total length of 2 boards = 2 × 8 = 16 feet
It is also given that you cut \(\frac{1}{2}\)– foot pieces to make square picture frames.
So,
The total number of picture frames = \(\frac{The total length of 2 boards}{The length of each square frame}\)
= 16 ÷ \(\frac{1}{2}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
16 ÷ \(\frac{1}{2}\)  = 16 × \(\frac{2}{1}\)
= \(\frac{16}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 16 × 2}{1 × 1}\)
= 32
Hence, from the above,
We can conclude that we can make 32 picture frames.

Divide Whole Numbers by Unit Fractions Homework & Practice 10.3

Divide. Use a model to help.

Question 1.

Answer: 1 ÷ \(\frac{1}{9}\) = 9

Explanation:
The given numbers are: 1 and \(\frac{1}{9}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
1 ÷ \(\frac{1}{9}\)  = 1 × \(\frac{9}{1}\)
= \(\frac{1}{1}\) × \(\frac{9}{1}\)
= \(\frac{ 1 × 9}{1 × 1}\)
= 9
Hence,
1÷ \(\frac{1}{9}\) = 9

Question 2.

Answer: 2 ÷ \(\frac{1}{3}\) = 6

Explanation:
The given numbers are: 2 and \(\frac{1}{3}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
2 ÷ \(\frac{1}{3}\)  = 2 × \(\frac{3}{1}\)
= \(\frac{3}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 3 × 2}{1 × 1}\)
= 6
Hence,
2÷ \(\frac{1}{3}\) = 6

Question 3.

Answer: 5 ÷ \(\frac{1}{2}\) = 10

Explanation:
The given numbers are: 5 and \(\frac{1}{2}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
5 ÷ \(\frac{1}{2}\)  = 5 × \(\frac{2}{1}\)
= \(\frac{5}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 5 × 2}{1 × 1}\)
= 10
Hence,
5÷ \(\frac{1}{2}\) = 10

Divide. Use a model to help.

Question 4.

Answer: 9 ÷ \(\frac{1}{4}\) = 36

Explanation:
The given numbers are: 9 and \(\frac{1}{4}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
9 ÷ \(\frac{1}{4}\)  = 9 × \(\frac{4}{1}\)
= \(\frac{9}{1}\) × \(\frac{4}{1}\)
= \(\frac{ 9 × 4}{1 × 1}\)
= 36
Hence,
9÷ \(\frac{1}{4}\) = 36

Question 5.

Answer: 7 ÷ \(\frac{1}{3}\) = 21

Explanation:
The given numbers are: 7 and \(\frac{1}{3}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
7 ÷ \(\frac{1}{3}\)  = 7 × \(\frac{3}{1}\)
= \(\frac{3}{1}\) × \(\frac{7}{1}\)
= \(\frac{ 3 × 7}{1 × 1}\)
= 21
Hence,
7÷ \(\frac{1}{3}\) = 21

Question 6.

Answer: 8 ÷ \(\frac{1}{5}\) = 40

Explanation:
The given numbers are: 8 and \(\frac{1}{5}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
8 ÷ \(\frac{1}{5}\)  = 8 × \(\frac{5}{1}\)
= \(\frac{8}{1}\) × \(\frac{5}{1}\)
= \(\frac{ 8 × 5}{1 × 1}\)
= 40
Hence,
8÷ \(\frac{1}{5}\) = 40

Question 7.
Number Sense
Explain how you can check your answer for Exercise 6.
Answer:
We can check the answer for exercise 6 by using the below model:

From the above model,
Each part represents \(\frac{8}{5}\)
So,
The total value of the 5 parts is: \(\frac{40}{5}\)
Hence,
In the above way, we can say that we check the answer

Question 8.
YOU BE THE TEACHER
Descartes finds 5 ÷ \(\frac{1}{4}\). Is he correct? Explain.

Answer: Yes, he is correct

Explanation:
We can write 5 as \(\frac{20}{4}\) or \(\frac{5}{1}\)
But, we only take \(\frac{20}{4}\) because the divided number given is 4
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{20}{4}\) ÷ \(\frac{1}{4}\)
= \(\frac{20}{4}\) × \(\frac{4}{1}\)
= \(\frac{ 20 × 4}{4 × 1}\)
= 20
Hence, from the above,
We can conclude that Descartes is correct.

Question 9.
Modeling Real Life
You need \(\frac{1}{2}\) pound of clay to make a pinch pot. How many pinch pots can you make with 12 pounds of clay?

Answer: You can make 24 pinch pots with 12 pounds of clay

Explanation:
It is given that you need \(\frac{1}{2}\) pound of clay to make a pinch pot.
It is also given that you have 12 pounds of clay
So,
The number of pinch pots you can make by using 12 pounds of clay = \(\frac{The total amount of clay}{The amount of clay used to make each pinch pot}\)
= 12 ÷ \(\frac{1}{2}\)
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
12 ÷ \(\frac{1}{2}\)
= \(\frac{12}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 12 × 2}{4 × 1}\)
= 24
Hence, from the above,
We can conclude that we can make 24 pinch pots by using 12 pounds of clay.

Question 10.
Modeling Real Life
Your art teacher has 5 yards of yellow string and 4 yards of green string. She cuts both colors \(\frac{1}{3}\)-yard pieces to hang of string into student artwork. How many pieces of student artwork can she hang?
Answer: The number of pieces of student artwork she can hang is: 27

Explanation:
It is given that your art teacher has 5 yards of yellow string and 4 yards of green string.
So,
The total number of yards of string = 5 + 4 = 9 yards of string
It is also given that she cuts both colors \(\frac{1}{3}\)-yard pieces to hang of string into student artwork.
So,
The number of pieces of student artwork she can hang = \(\frac{The total number of yards of strings}{The length of each yard f string}\)
= 9 ÷ \(\frac{1}{3}\)
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
9 ÷ \(\frac{1}{3}\)
= \(\frac{9}{1}\) × \(\frac{3}{1}\)
= \(\frac{ 9 × 3}{1 × 1}\)
= 27
Hence, from the above,
We can conclude that there are 27 pieces of student artwork that she can hang.

Review & Refresh

Question 11.

Answer: \(\frac{2}{5}\) × \(\frac{3}{4}\) = \(\frac{6}{20}\)

Explanation:
The given fractions are: \(\frac{3}{4}\) and \(\frac{2}{5}\)
So,
\(\frac{2}{5}\) × \(\frac{3}{4}\)
= \(\frac{2 × 3}{5 × 4}\)
= \(\frac{6}{20}\)
Hence,
\(\frac{2}{5}\) × \(\frac{3}{4}\) = \(\frac{6}{20}\)

Question 12.

Answer: \(\frac{1}{8}\) × \(\frac{5}{8}\) = \(\frac{5}{64}\)

Explanation:
The given fractions are: \(\frac{1}{8}\) and \(\frac{5}{8}\)
So,
\(\frac{1}{8}\) × \(\frac{5}{8}\)
= \(\frac{1 × 5}{8 × 8}\)
= \(\frac{5}{64}\)
Hence,
\(\frac{1}{8}\) × \(\frac{5}{8}\) = \(\frac{5}{64}\)

Question 13.

Answer: \(\frac{4}{9}\) × \(\frac{2}{7}\) = \(\frac{8}{63}\)

Explanation:
The given fractions are: \(\frac{4}{9}\) and \(\frac{2}{7}\)
So,
\(\frac{4}{9}\) × \(\frac{2}{7}\)
= \(\frac{2 × 4}{7 × 9}\)
= \(\frac{8}{63}\)
Hence,
\(\frac{4}{9}\) × \(\frac{2}{7}\) = \(\frac{8}{63}\)

Lesson 10.4 Divide Unit Fractions by Whole Numbers

Write a real-life problem that can be represented by \(\frac{1}{2}\) ÷ 3?
Answer:
Suppose we have 3 people and those 3 people each has to share \(\frac{1}{2}\) of the apple

What is the solution to the problem? Use a model to support your answer?
Answer:
The above problem is: We have to share \(\frac{1}{2}\) each for the 3 people
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{2}\) ÷ 3
= \(\frac{1}{2}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{2 × 3}\)
= \(\frac{1}{6}\)
Hence,
\(\frac{1}{6}\) is the solution to the above problem.

Precision
Is the answer greater than or less than 1? Explain?
Answer: The answer is less than 1

Explanation:
The answer for the problem is: \(\frac{1}{6}\)
So,
For the comparison of \(\frac{1}{6}\) with 1, we have to see whether the numerators or the denominators are equal or not
So, in this case, the numerators are equal
So, compare the denominators
So,
1 < 6
Hence, from the above,
We can conclude that \(\frac{1}{6}\) is less than 1

Think and Grow: Divide Unit Fractions by Whole Numbers

You can use models to divide unit fractions by whole numbers.

Show and Grow

Divide. Use a model to help.

Question 1.

Answer: \(\frac{1}{4}\) ÷ 2 = \(\frac{1}{8}\)

Explanation:
The given numbers are: \(\frac{1}{4}\) and 2
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{4}\) ÷ 2
= \(\frac{1}{4}\) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{2 × 4}\)
= \(\frac{1}{8}\)
Hence,
\(\frac{1}{4}\) ÷ 2 = \(\frac{1}{8}\)

Question 2.

Answer: \(\frac{1}{2}\) ÷ 5 = \(\frac{1}{10}\)

Explanation:
The given numbers are: \(\frac{1}{2}\) and 5
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{2}\) ÷ 5
= \(\frac{1}{2}\) × \(\frac{1}{5}\)
= \(\frac{ 1 × 1}{2 × 5}\)
= \(\frac{1}{10}\)
Hence,
\(\frac{1}{2}\) ÷ 5 = \(\frac{1}{10}\)

Apply and Grow: Practice

Divide. Use a model to help.

Question 3.

Answer: \(\frac{1}{5}\) ÷ 3 = \(\frac{1}{15}\)

Explanation:
The given numbers are: \(\frac{1}{5}\) and 3
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{5}\) ÷ 3
= \(\frac{1}{5}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{5 × 3}\)
= \(\frac{1}{15}\)
Hence,
\(\frac{1}{5}\) ÷ 3 = \(\frac{1}{15}\)

Question 4.

Answer: \(\frac{1}{6}\) ÷ 2 = \(\frac{1}{12}\)

Explanation:
The given numbers are: \(\frac{1}{6}\) and 2
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{6}\) ÷ 2
= \(\frac{1}{6}\) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{2 × 6}\)
= \(\frac{1}{12}\)
Hence,
\(\frac{1}{6}\) ÷ 2 = \(\frac{1}{12}\)

Question 5.

Answer: \(\frac{1}{3}\) ÷ 5 = \(\frac{1}{15}\)

Explanation:
The given numbers are: \(\frac{1}{3}\) and 5
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{3}\) ÷ 5
= \(\frac{1}{3}\) × \(\frac{1}{5}\)
= \(\frac{ 1 × 1}{3 × 5}\)
= \(\frac{1}{15}\)
Hence,
\(\frac{1}{3}\) ÷ 5 = \(\frac{1}{15}\)

Question 6.

Answer: \(\frac{1}{5}\) ÷ 4 = \(\frac{1}{20}\)

Explanation:
The given numbers are: \(\frac{1}{5}\) and 4
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{5}\) ÷ 4
= \(\frac{1}{5}\) × \(\frac{1}{4}\)
= \(\frac{ 1 × 1}{5 × 4}\)
= \(\frac{1}{20}\)
Hence,
\(\frac{1}{5}\) ÷ 4 = \(\frac{1}{20}\)

Question 7.

Answer: \(\frac{1}{3}\) ÷ 3 = \(\frac{1}{9}\)

Explanation:
The given numbers are: \(\frac{1}{3}\) and 3
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{3}\) ÷ 3
= \(\frac{1}{3}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{3 × 3}\)
= \(\frac{1}{9}\)
Hence,
\(\frac{1}{3}\) ÷ 3 = \(\frac{1}{9}\)

Question 8.

Answer: \(\frac{1}{8}\) ÷ 2 = \(\frac{1}{16}\)

Explanation:
The given numbers are: \(\frac{1}{8}\) and 2
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{8}\) ÷ 2
= \(\frac{1}{8}\) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{2 × 8}\)
= \(\frac{1}{16}\)
Hence,
\(\frac{1}{8}\) ÷ 2 = \(\frac{1}{16}\)

Question 9.
How many 6s are in \(\frac{1}{2}\)?
Answer: There are \(\frac{1}{12}\) 6s in \(\frac{1}{2}\)

Explanation:
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
Now,
We have to find the number of 6s in \(\frac{1}{2}\)
So,
\(\frac{1}{2}\) ÷ 6
= \(\frac{1}{2}\) × \(\frac{1}{6}\)
= \(\frac{ 1 × 1}{2 × 6}\)
= \(\frac{1}{12}\)
Hence, from the above,
We can conclude that there are \(\frac{1}{12}\) 6s in \(\frac{1}{2}\)

Question 10.
How many 2s are in \(\frac{1}{3}\) ?
Answer: There are \(\frac{1}{6}\) 2s in \(\frac{1}{3}\)

Explanation:
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
Now,
We have to find the number of 2s in \(\frac{1}{3}\)
So,
\(\frac{1}{3}\) ÷ 2
= \(\frac{1}{3}\) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{2 × 3}\)
= \(\frac{1}{6}\)
Hence, from the above,
We can conclude that there are \(\frac{1}{6}\) 2s in \(\frac{1}{2}\)

Question 11.
Writing
Write and solve a real-life problem for

Answer:
Suppose a box has 7 chocolates. We have to divide these seven chocolates into further \(\frac{1}{2}\) parts so that the chocolates can be distributed to more people
So,
The each part of chocolate we can get = \(\frac{1}{2}\) ÷ 7
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{2}\) ÷ 7
= \(\frac{1}{2}\) × \(\frac{1}{7}\)
= \(\frac{ 1 × 1}{2 × 7}\)
= \(\frac{1}{14}\)
Hence, from the above,
We can conclude that we can get \(\frac{1}{14}\) part of each chocolate.

Question 12.
Reasoning
Complete the statements.

Think and Grow: Modeling Real Life

You melt \(\frac{1}{4}\) quart of soap. You pour the soap into 4 of the same-sized molds. What fraction of a quart of soap does each mold hold?
You are dividing \(\frac{1}{4}\) quart into 4 equal parts, so you need to find \(\frac{1}{4}\) ÷ 4.

Show and Grow

Question 13.
You buy \(\frac{1}{2}\) pound of grapes. You equally divide the grapes into 2 bags. What fraction of a pound of grapes do you put into each bag?
Answer: The fraction of a pound of grapes you put into each bag is: \(\frac{1}{8}\) pound

Explanation:
It is given that you buy \(\frac{1}{2}\) pound of grapes.
It is also given that you equally divide the grapes into 2 bags.
So,
The number of grapes in each bag = \(\frac{1}{2}\) ÷ 2
Now,
The fraction of pound of grapes you put into each bag = \(\frac{The number of grapes in each bag}{2}\)
=  ( \(\frac{1}{2}\) ÷ 2 ) ÷ 2
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
( \(\frac{1}{2}\) ÷ 2 ) ÷ 2
= ( \(\frac{1}{2}\) × \(\frac{1}{2}\) ) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{2 × 2}\) × \(\frac{1}{2}\)
= \(\frac{1}{4}\) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{2 × 4}\)
= \(\frac{1}{8}\)
Hence, from the above
We can conclude that the fraction of pound of grapes in each bag is: \(\frac{1}{8}\) pound

Question 14.
You have \(\frac{1}{8}\) cup of red sand, \(\frac{1}{4}\) cup of blue sand, and \(\frac{1}{2}\) cup of white sand. You equally divide the sand into 3 containers. What fraction of a cup of sand do you pour into each container?
Answer: The fraction of a cup of sand you pour into each container is: \(\frac{7}{24}\)

Explanation:
It is given that you have \(\frac{1}{8}\) cup of red sand, \(\frac{1}{4}\) cup of blue sand, and \(\frac{1}{2}\) cup of white sand.
So,
The total amount of sand = \(\frac{1}{8}\) cup of red sand + \(\frac{1}{4}\) cup of blue sand + \(\frac{1}{2}\) cup of white sand
In addition, we have to see either the numerators are equal or the denominators are equal.
If the numerators are equal we have to ake the denominators also equal.
So,
\(\frac{1}{4}\) is multplied by \(\frac{2}{2}\)
\(\frac{1}{2}\) is multiplied by \(\frac{4}{4}\)
So,
\(\frac{1}{4}\) = \(\frac{2}{8}\)
\(\frac{1}{2}\) = \(\frac{4}{8}\)
So,
\(\frac{1}{8}\) + \(\frac{2}{8}\) + \(\frac{4}{8}\) = \(\frac{7}{8}\)
It is also given that all the sand is equally distributed into 3 containers
So,
The amount of sand in each container = \(\frac{7}{8}\) ÷ 3
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{7}{8}\) ÷ 3
= \(\frac{7}{8}\) × \(\frac{1}{3}\)
= \(\frac{ 7 × 1}{8 × 3}\)
= \(\frac{7}{24}\)
Hence, from the above,
We can conclude that the amount of sand in each container is: \(\frac{7}{24}\) cup.

Question 15.
DIG DEEPER!
You, your friend, and your cousin share \(\frac{1}{2}\) of a vegetable pizza and \(\frac{1}{4}\) of a cheese share pizza. The pizzas are the same size. What fraction of a pizza do you get in all?

Divide. Use a model to help
Answer: The fraction of a pizza you got is: \(\frac{3}{12}\)

Explanation:
It is given that you, your friend, and your cousin share \(\frac{1}{2}\) of a vegetable pizza and \(\frac{1}{4}\) of a cheese share pizza.
So,
The total amount of pizza = \(\frac{1}{2}\) of a vegetable pizza + \(\frac{1}{4}\) of a cheese share pizza
In addition, we have to see either the numerators are equal or the denominators are equal.
If the numerators are equal we have to ake the denominators also equal.
So,
\(\frac{1}{2}\) is multplied by \(\frac{2}{2}\)
So,
\(\frac{1}{2}\) = \(\frac{2}{4}\)
So,
\(\frac{2}{4}\) + \(\frac{1}{4}\) = \(\frac{3}{4}\)
So,
The fraction of pizza each get = \(\frac{The total amount of pizza}{3}\)
= \(\frac{3}{4}\) ÷ 3
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{3}{4}\) ÷ 3
= \(\frac{3}{4}\) × \(\frac{1}{3}\)
= \(\frac{ 3 × 1}{4 × 3}\)
= \(\frac{3}{12}\)
Hence, from the above,
We can conclude that the fraction of pizza each get is: \(\frac{3}{12}\)

Divide Unit Fractions by Whole Numbers Homework & Practice 10.4

Question 1.

Answer: \(\frac{1}{3}\) ÷ 4 = \(\frac{1}{12}\)

Explanation:
The given numbers are: \(\frac{1}{3}\) and 4
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{3}\) ÷ 4
= \(\frac{1}{3}\) × \(\frac{1}{4}\)
= \(\frac{ 1 × 1}{3 × 4}\)
= \(\frac{1}{12}\)
Hence,
\(\frac{1}{3}\) ÷ 4 = \(\frac{1}{12}\)

Question 2.

Answer: \(\frac{1}{6}\) ÷ 3 = \(\frac{1}{18}\)

Explanation:
The given numbers are: \(\frac{1}{6}\) and 3
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{6}\) ÷ 3
= \(\frac{1}{6}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{6 × 3}\)
= \(\frac{1}{18}\)
Hence,
\(\frac{1}{6}\) ÷ 3 = \(\frac{1}{18}\)

Question 3.

Answer: \(\frac{1}{4}\) ÷ 5 = \(\frac{1}{20}\)

Explanation:
The given numbers are: \(\frac{1}{4}\) and 5
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{4}\) ÷ 5
= \(\frac{1}{4}\) × \(\frac{1}{5}\)
= \(\frac{ 1 × 1}{5 × 4}\)
= \(\frac{1}{20}\)
Hence,
\(\frac{1}{4}\) ÷ 5 = \(\frac{1}{20}\)

Divide. Use a model to help.

Question 4.

Answer: \(\frac{1}{5}\) ÷ 9 = \(\frac{1}{45}\)

Explanation:
The given numbers are: \(\frac{1}{5}\) and 9
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{5}\) ÷ 9
= \(\frac{1}{5}\) × \(\frac{1}{9}\)
= \(\frac{ 1 × 1}{5 × 9}\)
= \(\frac{1}{45}\)
Hence,
\(\frac{1}{5}\) ÷ 9 = \(\frac{1}{45}\)

Question 5.

Answer: \(\frac{1}{8}\) ÷ 6 = \(\frac{1}{48}\)

Explanation:
The given numbers are: \(\frac{1}{8}\) and 6
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{8}\) ÷ 6
= \(\frac{1}{8}\) × \(\frac{1}{6}\)
= \(\frac{ 1 × 1}{8 × 6}\)
= \(\frac{1}{48}\)
Hence,
\(\frac{1}{8}\) ÷ 6 = \(\frac{1}{48}\)

Question 6.

Answer: \(\frac{1}{7}\) ÷ 4 = \(\frac{1}{28}\)

Explanation:
The given numbers are: \(\frac{1}{7}\) and 4
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{7}\) ÷ 4
= \(\frac{1}{7}\) × \(\frac{1}{4}\)
= \(\frac{ 1 × 1}{7 × 4}\)
= \(\frac{1}{28}\)
Hence,
\(\frac{1}{7}\) ÷ 4 = \(\frac{1}{28}\)

Question 7.
YOU BE THE TEACHER
Your friend divides \(\frac{1}{3}\) by 7 to get \(\frac{1}{21}\). He checks his answer by multiplying \(\frac{1}{21}\) × \(\frac{1}{3}\). Does your friend check his answer correctly? Explain.
Answer: No, your friend does not check his answer correctly

Explanation:
It is given that your friend divides \(\frac{1}{3}\) by 7 to get \(\frac{1}{21}\).
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{3}\) ÷ 7
= \(\frac{1}{3}\) × \(\frac{1}{7}\)
= \(\frac{ 1 × 1}{7 × 3}\)
= \(\frac{1}{21}\)
It is also given that your friend checks his answer by multiplying \(\frac{1}{21}\) × \(\frac{1}{3}\).
Now,
\(\frac{1}{21}\) × \(\frac{1}{3}\)
= \(\frac{1 × 1}{21 × 3}\)
= \(\frac{1}{63}\)
But, your friend wanted to check whether \(\frac{1}{21}\) × \(\frac{1}{3}\) = \(\frac{1}{7}\)
But, the value becomes \(\frac{1}{63}\)
Hence, from the above,
We can conclude that your friend does not check the answer correctly.

Question 8.
Logic
Find the missing numbers.

Question 9.
Modeling Real Life
You win tickets that you can exchange for prizes. You exchange \(\frac{1}{5}\) of your tickets and then divide them equally among 3 prizes. What fraction of your tickets do you spend on each prize?
Answer: The fraction of your tickets you spend on each prize is: \(\frac{1}{15}\)

Explanation:
It is given that you win tickets that you can exchange for prizes.
It is also given that you exchange \(\frac{1}{5}\) of your tickets and then divide them equally among 3 prizes
So,
The fraction of the tickets spent on each prize = \(\frac{The value of Exchange}{The number of prizes}\)
= \(\frac{1}{5}\) ÷ 3
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{5}\) ÷ 3
= \(\frac{1}{5}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{5 × 3}\)
= \(\frac{1}{15}\)
Hence, from the above,
We can conlude that the fraction of tickets you spend on each prize is: \(\frac{1}{15}\)

Question 10.
DIG DEEPER!
You have \(\frac{1}{8}\) gallon of melted crayon wax. You pour the wax equally into 8 different molds to make new crayons. What fraction of a cup of melted wax is in each mold? Think: 1 gallon is 16 cups.

Answer: The fraction of a cup of melted wax in each mold is: \(\frac{1}{4}\)

Explanation:
It is given that you have \(\frac{1}{8}\) gallon of melted crayon wax.
It is also given that you pour the wax equally into 8 different molds to make new crayons.
So,
The fraction of melted crayon wax in each mold in gallons = \(\frac{The total amount of melted crayon wax }{The number of molds}\)
= \(\frac{1}{8}\) ÷ 8
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{8}\) ÷ 8
= \(\frac{1}{8}\) × \(\frac{1}{8}\)
= \(\frac{ 1 × 1}{8 × 8}\)
= \(\frac{1}{64}\) gallons
But, it is given that
1 gallon = 16 cups
So,
The total number of cups that the melted crayon wax contained = \(\frac{1}{64}\) × \(\frac{16}{1}\)
= \(\frac{1 × 16 }{64 × 1}\)
= \(\frac{1}{4}\)
Hence, from the above,
We can conclude that there are \(\frac{1}{4}\) cups of melted crayon wax in each mold.

Review & Refresh

Question 11.
0.9 ÷ 0.1 = ___
Answer: 0.9 ÷ 0.1 = 9

Explanation:
The given decimal numbers are: 0.9 and 0.1
The representation of the decimal numbers in the fraction form is: \(\frac{9}{10}\) and \(\frac{1}{10}\)
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{9}{10}\) ÷ \(\frac{1}{10}\)  = \(\frac{9}{10}\) × \(\frac{10}{1}\)
= \(\frac{ 9 × 10}{10 × 1}\)
= 9
Hence, 0.9 ÷ 0.1 = 9

Question 12.
38.6 ÷ 100 = ___

Answer: 38.6 ÷ 100 = 0.386

Explanation:
The given numbers are: 38.6 and 100
The representation of the numbers in the fraction form is: \(\frac{386}{10}\) and \(\frac{100}{1}\)
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{386}{10}\) ÷ \(\frac{100}{1}\)  = \(\frac{386}{10}\) × \(\frac{1}{100}\)
= \(\frac{ 386 × 1}{100 × 10}\)
= \(\frac{386}{1000}\)
= 0.386
Hence, 38.6 ÷ 100 = 0.386

Question 13.
2.57 ÷ 0.01 = ___
Answer: 2.57 ÷ 0.01 = 257

Explanation:
The given decimal numbers are: 2.57 and 0.01
The representation of the decimal numbers in the fraction form is: \(\frac{257}{100}\) and \(\frac{1}{100}\)
Now,
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{257}{100}\) ÷ \(\frac{1}{100}\)  = \(\frac{257}{100}\) × \(\frac{100}{1}\)
= \(\frac{ 257 × 100}{100 × 1}\)
= 257
Hence, 2.57 ÷ 0.01 = 257

Lesson 10.5 Problem Solving: Fraction Division

Explore and Grow

You want to make a \(\frac{1}{3}\) batch of the recipe. How you can use division to find the amount of each ingredient you need?
Answer:
It is given that you want to make a \(\frac{1}{3}\) batch of the recipe.
So,
From \(\frac{1}{3}\),
1 represents a batch of the recipe
3 represents the total number of ingredients in a batch
So,
The amount of each ingredient you need = \(\frac{The amount of the batch of the recipe }{The total number of ingredients}\)
= \(\frac{1}{3}\) ÷ 3
Now,
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{3}\) ÷ 3
= \(\frac{1}{3}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{3 × 3}\)
= \(\frac{1}{9}\)
Hence, from the above,
We can conclude that the amount of each ingredient you need is: \(\frac{1}{9}\)

Reasoning
Without calculating, explain how you can tell whether you need more than or less than 1 tablespoon of olive oil?
Answer: You need less than 1 tablespoon of olive oil

Explanation:
From the above problem,
The amount of each ingredient is: \(\frac{1}{9}\)
Since the amount of each ingredient is less than 1, you need less than 1 tablespoon of olive oil

Think and Grow: Problem Solving: Fraction Division

Example
You have 4 cups of yellow paint and 3 cups of blue paint. How many batches of green paint can you make?

Understand the Problem

What do you know?

• You have 4 cups of yellow paint and 3 cups of blue paint.
• One batch of green paint is made of \(\frac{1}{2}\) cup of yellow and \(\frac{1}{3}\) cup of blue.

What do you need to find?

• You need to find how many batches of green paint you can make.

Make a Plan
How will you solve?

• Find how many batches are possible from yellow, and how many from blue.
• Choose the lesser number of batches.

Solve

So, you can make 8 batches of green paint.

Show and Grow

Question 1.
In the example, explain why you choose the fewer number of batches.
Answer: In the above example, the yellow paint has the less number of batches as the amount of each batch of yellow paint-filled is more than the batch of green paint
Hence,
We choose the fewer number of batches of yellow paint

Apply and Grow: Practice

Understand the problem. What do you know? What do you need to find? Explain.

Question 2.
A landowner donates 3 acres of land to a city. The mayor of the city uses 1 acre of the land for a playground and the rest of the land for community garden plots. Each garden plot is \(\frac{1}{3}\) acre. How many plots are there?
Understand the problem. Then make a plan. How will you solve it? Explain?
Answer: The number of plots in the community is: 6

Explanation:
It is given that a landowner donates 3 acres of land to a city and the mayor of the city uses 1 acre of the land for a playground and the rest of the land for community garden plots.
So,
The portion of the land used for community garden plots is: 2 acres
It is also given that each garden plot is \(\frac{1}{3}\) acre.
So,
The number of plots = \(\frac{The portion of the land used for community garden plots}{The area of each garden plot}\)
= 2 ÷ \(\frac{1}{3}\)
= 2 × \(\frac{3}{1}\)
= \(\frac{2}{1}\) × \(\frac{3}{1}\)
= 6
Hence, from the above,
We can conclude that there are 6 plots

Question 3.
A craftsman uses \(\frac{3}{4}\) gallon of paint to paint 4 identical dressers. He uses the same amount of paint on each dresser. How much paint does he use to paint 7 of the same dressers?

Answer: The paint used by the craftsman to paint 7 of the same dressers is: \(\frac{21}{16}\)

Explanation:
It is given that a craftsman uses \(\frac{3}{4}\) gallon of paint to paint 4 identical dressers.
So,
The paint used to paint each dresser = \(\frac{3}{4}\) ÷ 4
= \(\frac{3}{4}\) × \(\frac{1}{4}\)
= \(\frac{3}{16}\) gallon
So,
The amount of paint used to paint the 7 identical dressers = \(\frac{The paint used to paint each dresser}{1}\) × 7
= \(\frac{3}{16}\) × \(\frac{7}{1}\)
= \(\frac{3 × 7}{16 × 1}\)
= \(\frac{21}{16}\) gallon
Hence, from the above,
We can conclude that the paint used to paint 7 identical dressers is: \(\frac{21}{16}\) gallon

Question 4.
An airplane travels 125 miles in \(\frac{1}{4}\) hour. It travels the same number of miles each hour. How many miles does the plane travel in 5 hours?
Answer: The number of miles the plane travel in 5 hours is: 2,500 miles

Explanation:
It is given that an airplane travels 125 miles in \(\frac{1}{4}\) hour
So,
The number of miles traveled by plane in 1 hour = 125 ÷ \(\frac{1}{4}\)
= 125 × \(\frac{4}{1}\)
= 125 × 4
= 500 miles
So,
The number of miles traveled by plane in 5 hours = ( The number of miles traveled by plane in 1 hour ) × 5
= 500 × 5
= 2,500 miles
Hence, from the above,
We can conclude that the number of miles traveled by plane in 5 hours is: 2,500 miles

Question 5.
You make bows for gifts using \(\frac{2}{3}\) yard of ribbon for each bow. You have 4 feet of red ribbon and 5 feet of green ribbon. How many bows can you make?
Answer: The number of bows you can make is: 2 bows

Explanation:
It is given that you make bows for gifts using \(\frac{2}{3}\) yard of ribbon for each bow.
It is also given that you have 4 feet of red ribbon and 5 feet of green ribbon
So,
The total length of ribbon = 5 + 4 = 9 feet
we know that,
1 foot = \(\frac{1}{3}\) yards
So,
9 feet = 9 × \(\frac{1}{3}\) yards
= \(\frac{9}{1}\) yards × \(\frac{1}{3}\) yards
= 3 yards
So,
The number of bows you can make = \(\frac{2}{3}\) yards × 3
= 2 bows
Hence, from the above,
We can conclude that the number of bows we can make is: 2

Question 6.
A landscaper buys 1 gallon of plant fertilizer. He uses \(\frac{1}{5}\) of the fertilizer, and then divides the rest into 3 smaller bottles. How many gallons does he put into each bottle?
Answer: The number of gallons he put into each bottle is: \(\frac{4}{15}\)

Explanation:
It is given that a landscaper buys 1 gallon of plant fertilizer and he uses \(\frac{1}{5}\) of the fertilizer
So,
The remaining amount of the fertilizer = 1 – \(\frac{1}{5}\)
= \(\frac{4}{5}\) gallons
It is also given that he divided the remaining amount of fertilizer into 3 smaller bottles.
So,
The amount of fertilizer put into each bottle = \(\frac{The remaining amount of the fertilizer}{The total number of bottles}\)
= \(\frac{4}{5}\) ÷ 3
= \(\frac{4}{5}\) × \(\frac{1}{3}\)
= \(\frac{4 × 1}{5 × 3}\)
= \(\frac{4}{15}\) gallons
hence, from the above,
We can conclude that the amount of remaining fertilizer put into each bottle is: \(\frac{4}{15}\) gallons

Think and Grow: Modeling Real Life

Example
A sponsor donates $0.10 to a charity for every \(\frac{1}{4}\) kilometer of the triathlon an athlete completes. The athlete completes the entire triathlon. How much money does the sponsor donate?

Think: What do you know? What do you need to find? How will you solve?
Write and solve an equation.
Add 1.9, 90, and 21.1 to find how many kilometers the athlete completes.
Divide the sum by \(\frac{1}{4}\) to find how many \(\frac{1}{4}\) kilometers the athlete completes.
Multiply the quotient by $0.10 to find how much money the sponsor donates.

Let m represent the total amount of money donated.

Show and Grow

Question 7.
You earn $5 for every \(\frac{1}{2}\) hour you do yard work. How much money do you earn in 1 week?

Answer: The amount you earn in 1 week is: $700

Explanation:
The given table is:

From the above table,
The total amount of time = 5\(\frac{1}{2}\) + 3 + 1\(\frac{1}{2}\)
= \(\frac{11}{2}\) + 3 + \(\frac{3}{2}\)
= \(\frac{11 + 3}{2}\) + 3
= 7 + 3
= 10 hours
It is given that you earn $5 for every \(\frac{1}{2}\) hour you do yard work
So,
The amount earned in 10 hours in a day = 10 ÷\(\frac{1}{2}\) × 5 ( Since we have the time in hours but the money earned is given in half an hour basis )
= 20 × 5
= $100
We know that 1 week = 7 days
So,
The amount earned in 1 week = 100 × 7 = $700
hence, from the above,
We can conclude that we can earn $700 in a week

Problem Solving: Fraction Division Homework &  10.5

Understand the problem. Then make a plan. How will you solve? Explain.

Question 1.
A train travels 75 miles in \(\frac{1}{2}\) hour. How many miles does the train travel in 8 hours?
Answer: The number of miles the train travel in  hours is: 1,200 miles

Explanation:
It is given that a train travels 75 miles in \(\frac{1}{2}\) hour.
So,
The number of miles the train travel in 1 hour = 75 ÷ \(\frac{1}{2}\)
= 75 × 2
= 150 miles
So,
The number of miles the train travel in 8 hours = The number of miles traveled by train in 1 hour × 8
= 150 × 8
= 1,200 miles
Hence, from the above,
We can conclude that the train travels 1,200 miles in 8 hours.

Question 2.
You need \(\frac{2}{3}\) yard of fabric to create a headband. You have 12 feet of blue fabric and 4 feet of yellow fabric. How many headbands can you make with all of the fabric?
Answer: The number of headbands you can make with all of the fabric is: 8 headbands

Explanation:
It is given that you need \(\frac{2}{3}\) yard of fabric to create a headband.
It is also given that you have 12 feet of blue fabric and 4 feet of yellow fabric.
So,
The total length of the fabric = 12 + 4 = 16 feet
We know that
1 foot = \(\frac{1}{3}\) yards
So,
16 feet = \(\frac{16}{3}\) yards
So,
The number of headbands you can create with all the fabric = \(\frac{The total length of the fabric}{The length of each fabric}\)
= \(\frac{16}{3}\) ÷ \(\frac{2}{3}\)
= \(\frac{16}{3}\) × \(\frac{3}{2}\)
= \(\frac{16 × 3}{3 × 2}\)
= 8 headbands
Hence, from the above,
We can conclude that we can create 8 headbands with all the fabric.

Question 3.
An art teacher has 8 gallons of paint. Her class uses \(\frac{3}{4}\) of the paint. The teacher divides the rest of the paint into 4 bottles. How much paint is in each bottle?
Answer: The amount of paint in each bottle is: \(\frac{1}{2}\)

Explanation:
It is given that an art teacher has 8 gallons of paint and her class uses \(\frac{3}{4}\) of the paint.
So,
The remaining amount of paint = \(\frac{1}{4}\) × 8
= \(\frac{1}{4}\) × \(\frac{8}{1}\)
=\(\frac{1 × 8}{4 × 1}\)
= 2 gallons
It is also given that the remaining amount of the paint divided into 4 bottles by the teacher
So,
The amount of paint present in each bottle = 2 ÷ 4
= \(\frac{1}{2}\) gallons
Hence, from the above,
We can conclude that the amout of paint present in each bottle is: \(\frac{1}{2}\) gallons

Question 4.
You mix 3\(\frac{1}{4}\) cups of frozen strawberries and 4\(\frac{1}{2}\) cups of frozen blueberries in a bowl. A smoothie requires \(\frac{1}{2}\) cup of your berry mix. How many smoothies can you make?
Answer: The number of smoothies you can make is:

Explanation:
It is given that you mix 3\(\frac{1}{4}\) cups of frozen strawberries and 4\(\frac{1}{2}\) cups of frozen blueberries in a bowl.
So,
The amount of berry mix = 3\(\frac{1}{4}\) cups of frozen strawberries + 4\(\frac{1}{2}\) cups of frozen blueberries
= 3\(\frac{1}{4}\) + 4\(\frac{1}{2}\)
= \(\frac{13}{4}\) + \(\frac{9}{2}\)
In addition, equate the denominators
So,
Multiply \(\frac{9}{2}\) with \(\frac{2}{2}\)
So,
\(\frac{9}{2}\) = \(\frac{18}{4}\)
So,
The amount of berry mix = \(\frac{13}{4}\) + \(\frac{18}{4}\)
= \(\frac{31}{4}\)
Now,
It is also given that the smoothie requires \(\frac{1}{2}\) cup of your berry mix.
So,
The number of smoothies = \(\frac{31}{4}\) ÷ \(\frac{1}{2}\)
= \(\frac{31}{4}\) × \(\frac{2}{1}\)
= \(\frac{31 × 2}{4 × 1}\)
= \(\frac{31}{2}\)
Hence, from the above,
We can conclude that the number of smoothies we can make are: \(\frac{31}{2}\)

Question 5.
Modeling Real Life
A sponsor donates $0.10 for every \(\frac{1}{4}\) dollar donated at the locations shown. How much money does the sponsor donate?

Answer: The amount of money the sponsor donates is: $40.4

Explanation:
The given table is:

From the above table,
The total amount of money collected = 25.25 + 12.50 + 63.25
= $101
It is given that a sponsor donates $0.10 for every \(\frac{1}{4}\) dollar
So,
the total amount of donated = The total amount of money collected ÷ \(\frac{1}{4}\) × 0.10
= 101 ÷ \(\frac{1}{4}\) × 0.10
= 101 × 4 × 0.10
= 04 × 0.10
= $40.4
Hence, from the above,
We can conclude that the amount of money donated by a sponsor is: $40.4

Question 6.
DIG DEEPER!
A nurse earns $16 for every \(\frac{1}{2}\) hour at work. How much money does she earn in 5 days?

Answer: The money she earns in 5 days is: $1,280

Explanation:
The given table is:

From the above table,
The total amount of time = 6\(\frac{3}{4}\) + 1 + \(\frac{1}{4}\)
= \(\frac{27}{4}\) + 1 + \(\frac{1}{4}\)
= \(\frac{27 + 1}{4}\) + 1
= 7 + 1
= 8 hours
It is given that a nurse earns $16 for every \(\frac{1}{2}\) hour at work.
So,
The money she earned for 1 hour = 16 ÷ \(\frac{1}{2}\)
= 16 × 2 = $32
So,
The money earned for 8 hours = The money earned in 1 hour × 8
= 32 × 8 = $256
So,
The money earned in 5 days = The money earned in 1 day × 5
= 256 × 5 = $1,280
hence, from the above,
we can conclude that she can earn $1,280 in 5 days.

Review & Refresh

Find the quotient. Then check your answer.

Question 7.

Answer:  186 ÷ 12 = 1 R 4

Explanation:
Let 185.88 be rounded to 186
So,
By using the partial quotients method,
186 ÷ 12 = ( 120 + 36 + 24 ) ÷ 12
= ( 120 ÷ 12 ) + ( 36 ÷ 12 ) + ( 24 ÷ 12 )
= 10 + 3 + 2
= 17 R 4
Hence, 186 ÷ 12 = 17 R 4

Question 8.

Answer: 74 ÷ 24 = 3 R 2

Explanation:
Let 74.4 be rounded to 74
So,
By using the partial quotients method,
74 ÷ 24 = 72 ÷ 24
= 3 R 2
Hence, 74 ÷ 24 = 3 R 2

Question 9.

Answer: 42 ÷ 46 = 0.9

Explanation:
Let 42.32 be rounded to 42
So,
By using the partial quotients method,
42 ÷ 46 = 0.9
Hence,
42 ÷ 46 = 0.9

Divide Fractions Performance Task 10

Your city has a robotics competition. Each team makes a robot that travels through a maze. The time each robot spends in the maze is used to find the team’s score.
1. One-third of the students in your grade participate in the competition. The number of participating students is divided into 12 teams.

a. What fraction of the total number of students in your grade is on each team?
Answer: The fraction of the total number of students in your grade is: \(\frac{1}{36}\)

Explanation:
It is given that there are \(\frac{1}{3}\) of the students in your grade are participating in the competition.
It is also given that the participating students are divided into 12 teams.
So,
The fraction of the total number of students in each team = \(\frac{The number of participating students}{Th total number of teams}\)
= \(\frac{1}{3}\) ÷ 12
= \(\frac{1}{3}\) ÷ \(\frac{12}{1}\)
= \(\frac{1}{3}\) × \(\frac{1}{12}\)
= \(\frac{1 × 1}{12 × 3}\)
= \(\frac{1}{36}\)
Hence, from the above,
We can conclude that the fraction of students that are in each team is: \(\frac{1}{36}\)

b. There are 3 students on each team. How many students are in your grade?
Answer: The number of students in your grade is: 36

Explanation:
It is given that the number of students is divided into 12 teams
It is also given that there are 3 students on each team
So,
The total number of students = The number of teams × The number of students in each team
= 12 × 3 = 36 students
Hence, from the above,
We can conclude that there are 36 students in your grade

Question 2.
The maze for the competition is shown.
a. Write the length of the maze in feet.

Answer:
The given maze for the competition is:

From the above maze,
The total length of the maze is: 8 feet 6 inches
We know that,
1 foot = 12 inches
Hence,
1 inch = \(\frac{1}{12}\) feet
So,
6 inches = 6 × \(\frac{1}{12}\)
= \(\frac{1}{12}\) × \(\frac{6}{1}\)
= \(\frac{1}{2}\) feet
So,
The total length of the maze in feet = 8 feet + \(\frac{1}{2}\) feet
= 8.5 feet or 8\(\frac{1}{2}\) feet

b. The length of the maze is divided into 6 equal sections. What is the length of each section of the maze?
Answer: The length of each section of the maze is: \(\frac{17}{12}\) feet

Explanation:
From the above Exercise,
The total length of the maze in feet is: 8.5 feet or 8\(\frac{1}{2}\) feet
It is given that the length of the maze is divided into 6 equal sections
So,
The length of each section of the maze = 8\(\frac{1}{2}\) ÷ 6
= 8\(\frac{1}{2}\) ÷ \(\frac{6}{1}\)
= 8\(\frac{1}{2}\) × \(\frac{1}{6}\)
= \(\frac{17}{2}\) × \(\frac{1}{6}\)
= \(\frac{17}{12}\) feet
Hence, from the above,
We can conclude that the length of each section in a maze is: \(\frac{17}{12}\) feet

Question 3.
Each team has 200 seconds to complete the maze. The rules require judges to use the expression (200 – x) ÷ \(\frac{1}{5}\), where x is the total number of seconds, to find a team’s total score.
a. Your robot completes the maze in 3 minutes 5 seconds. How many points does your team earn?
Answer: The number of points your team earn is: 75 points

Explanation:
It is given that each team has 200 seconds to complete the maze and the rules require judges to use the expression (200 – x) ÷ \(\frac{1}{5}\) where x is the total number of seconds
It is also given that your robot completes the maze in 3 minutes 5 seconds
We know that,
1 minute = 60 seconds
So,
The time taken by the robot to complete the maze in seconds = ( 3 × 60 ) + 5
= 185 seconds
So,
x= 185
So,
200 – x = 200 – 185 = 15
So,
The number of points the team earned = ( 200 – x ) ÷ \(\frac{1}{5}\)
= 15 ÷ \(\frac{1}{5}\)
= 15 × 5
= 75 points
Hence, from the above,
We can conclude that the number of points earned by the team is: 75 points

b. Do you think the team with the most points or the fewest points wins? Use an example to justify your answer.
Answer: The team with the most points wins the competition because
Reason:
Suppose team A takes 2 minutes and team B takes 3 minutes to complete the competition
So,
The time is taken by team A in seconds = 120 seconds
So,
x= 120
So,
200 – x = 200 – 120 = 80
Now,
The time is taken by team B in seconds = 180 seconds
So,
x= 180
So,
200 – x = 200 – 180 = 20
Now,
The number of points earned by team A = 80 ÷ \(\frac{1}{5}\)
= 400 points
The number of points earned by team B = 20 ÷ \(\frac{1}{5}\)
= 100 points
Hence, from the above,
We can conclude that the team with more points wins the competition

Divide Fractions Activity

Fraction Connection: Division

Directions:

1. Players take turns rolling three dice.
2. On your turn, evaluate the expression indicated by your roll and cover the answer.
3. The first player to get four in a row, horizontally, vertically, or diagonally, wins!

Divide Fractions Chapter Practice 10

10.1 Interpret Fractions as Division

Divide. Use a model to help.

Question 1.
1 ÷ 2 = ___
Answer: 1 ÷ 2 = \(\frac{1}{2}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
Hence,
1 ÷ 2 = \(\frac{1}{2}\)

Question 2.
3 ÷ 10 = __
Answer: 3 ÷ 10 = \(\frac{3}{10}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
Hence,
3 ÷ 10 = \(\frac{3}{10}\)

Question 3.
4 ÷ 7 = __
Answer: 4 ÷ 7 = \(\frac{4}{7}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
Hence,
4 ÷ 7 = \(\frac{4}{7}\)

Question 4.
11 ÷ 15 = ___
Answer: 11 ÷ 15 = \(\frac{11}{15}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
Hence,
11 ÷ 15 = \(\frac{11}{15}\)

Question 5.
8 ÷ 9 = ___
Answer: 8 ÷ 9 = \(\frac{8}{9}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
Hence,
8 ÷ 9 = \(\frac{8}{9}\)

Question 6.
13 ÷ 20 = ___
Answer: 13 ÷ 20 = \(\frac{13}{20}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
Hence,
13 ÷ 20 = \(\frac{13}{20}\)

Question 7.
Modeling Real Life
Nine friends equally share 12 apples. What fraction of an apple does each friend get?
Answer: The fraction of an apple each friend get is: \(\frac{9}{12}\)

Explanation:
It is given that nine friends equally share 12 apples.
So,
The fraction of an apple each friend get = \(\frac{The number of friends}{The number of apples}\)
= \(\frac{9}{12}\)
Hence, from the above,
We can conclude that the fraction of an apple each friend get is: \(\frac{9}{12}\)

10.2 Mixed Numbers as Quotients

Divide. Use a model to help

Question 8.
8 ÷ 3 = ___

Answer: 8 ÷ 3 = 2\(\frac{2}{3}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
a\(\frac{b}{c}\) = \(\frac{ac + b}{c}\)
\(\frac{a}{b}\) = Remainder\(\frac{Quotient}{Divisor}\)
So,
8 ÷ 3 = \(\frac{8}{3}\)
By using the partial quotients method,
8 ÷ 3 = 6 ÷ 3
= 2 R 2
Hence,
8 ÷ 3 = 2\(\frac{2}{3}\)

Question 9.
6 ÷ 5 = ___
Answer: 6 ÷ 5 = 1\(\frac{1}{5}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
a\(\frac{b}{c}\) = \(\frac{ac + b}{c}\)
\(\frac{a}{b}\) = Remainder\(\frac{Quotient}{Divisor}\)
So,
6 ÷ 5 = \(\frac{6}{5}\)
By using the partial quotients method,
6 ÷ 5 = 5 ÷ 5
= 1 R 1
Hence,
6 ÷ 5 = 1\(\frac{1}{5}\)

Question 10.

10 ÷ 4 = ___
Answer: 10 ÷ 4 = 2\(\frac{2}{4}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
a\(\frac{b}{c}\) = \(\frac{ac + b}{c}\)
\(\frac{a}{b}\) = Remainder\(\frac{Quotient}{Divisor}\)
So,
10 ÷ 4 = \(\frac{10}{4}\)
By using the partial quotients method,
10 ÷ 4 = 8 ÷ 4
= 2 R 2
Hence,
10 ÷ 4 = 2\(\frac{2}{4}\)

Question 11.

20 ÷ 11 = __
Answer: 20 ÷ 11 = 1\(\frac{9}{11}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
a\(\frac{b}{c}\) = \(\frac{ac + b}{c}\)
\(\frac{a}{b}\) = Remainder\(\frac{Quotient}{Divisor}\)
So,
20 ÷ 11 = \(\frac{20}{11}\)
By using the partial quotients method,
20 ÷ 11 = 11 ÷ 11
= 1 R 9
Hence,
20 ÷ 11 = 1\(\frac{9}{11}\)

Question 12.

25 ÷ 2 = ___
Answer: 25 ÷ 2 = 12\(\frac{1}{2}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
a\(\frac{b}{c}\) = \(\frac{ac + b}{c}\)
\(\frac{a}{b}\) = Remainder\(\frac{Quotient}{Divisor}\)
So,
25 ÷ 2 = \(\frac{25}{2}\)
By using the partial quotients method,
25 ÷ 2 = 24 ÷ 2
= 12 R 1
Hence,
25 ÷ 2 = 12\(\frac{1}{2}\)

Question 13.

64 ÷ 9 = ___
Answer: 64 ÷ 9 = 7\(\frac{1}{9}\)

Explanation:
We know that,
a ÷ b = \(\frac{a}{b}\)
a\(\frac{b}{c}\) = \(\frac{ac + b}{c}\)
\(\frac{a}{b}\) = Remainder\(\frac{Quotient}{Divisor}\)
So,
64 ÷ 9 = \(\frac{64}{9}\)
By using the partial quotients method,
64 ÷ 9 = 63 ÷ 9
= 7 R 1
Hence,
64 ÷ 9 = 7\(\frac{1}{9}\)

10.3 Divide Whole Numbers by Unit Fractions

Divide. Use a model to help.

Question 14.

Answer: 4 ÷ \(\frac{1}{2}\) = 8

Explanation:
The given numbers are: 4 and \(\frac{1}{2}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
4 ÷ \(\frac{1}{2}\)  = 4 × \(\frac{2}{1}\)
= \(\frac{4}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 2 × 4}{1 × 1}\)
= 8
Hence,
4÷ \(\frac{1}{2}\) = 8

Question 15.

Answer: 6 ÷ \(\frac{1}{5}\) = 30

Explanation:
The given numbers are: 6 and \(\frac{1}{5}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
6 ÷ \(\frac{1}{5}\)  = 6 × \(\frac{5}{1}\)
= \(\frac{6}{1}\) × \(\frac{5}{1}\)
= \(\frac{ 6 × 5}{1 × 1}\)
= 30
Hence,
6÷ \(\frac{1}{5}\) = 30

Question 16.

Answer: 7 ÷ \(\frac{1}{4}\) = 28

Explanation:
The given numbers are: 7 and \(\frac{1}{4}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
7 ÷ \(\frac{1}{4}\)  = 7 × \(\frac{4}{1}\)
= \(\frac{7}{1}\) × \(\frac{4}{1}\)
= \(\frac{ 7 × 4}{1 × 1}\)
= 28
Hence,
7÷ \(\frac{1}{4}\) = 36

Question 17.

Answer: 8 ÷ \(\frac{1}{3}\) = 24

Explanation:
The given numbers are: 8 and \(\frac{1}{3}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
8 ÷ \(\frac{1}{3}\)  = 8 × \(\frac{3}{1}\)
= \(\frac{8}{1}\) × \(\frac{3}{1}\)
= \(\frac{ 8 × 3}{1 × 1}\)
= 24
Hence,
8÷ \(\frac{1}{3}\) = 24

Question 18.

Answer: 9 ÷ \(\frac{1}{2}\) = 18

Explanation:
The given numbers are: 9 and \(\frac{1}{2}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
9 ÷ \(\frac{1}{2}\)  = 9 × \(\frac{2}{1}\)
= \(\frac{9}{1}\) × \(\frac{2}{1}\)
= \(\frac{ 9 × 2}{1 × 1}\)
= 18
Hence,
9÷ \(\frac{1}{2}\) = 18

Question 19.

Answer: 2 ÷ \(\frac{1}{10}\) = 20

Explanation:
The given numbers are: 2 and \(\frac{1}{10}\)
We know that,
a ÷ \(\frac{a}{b}\) = a × \(\frac{b}{a}\)
a= \(\frac{a}{1}\)
So,
2 ÷ \(\frac{1}{10}\)  = 2 × \(\frac{10}{1}\)
= \(\frac{2}{1}\) × \(\frac{10}{1}\)
= \(\frac{ 2 × 10}{1 × 1}\)
= 20
Hence,
2÷ \(\frac{1}{10}\) = 20

10.4 Divide Unit Fractions by Whole Numbers

Divide. Use a model to help.

Question 20.

Answer: \(\frac{1}{7}\) ÷ 2 = \(\frac{1}{14}\)

Explanation:
The given numbers are: \(\frac{1}{7}\) and 2
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{7}\) ÷ 2
= \(\frac{1}{7}\) × \(\frac{1}{2}\)
= \(\frac{ 1 × 1}{7 × 2}\)
= \(\frac{1}{14}\)
Hence,
\(\frac{1}{7}\) ÷ 2 = \(\frac{1}{14}\)

Question 21.

Answer: \(\frac{1}{2}\) ÷ 9 = \(\frac{1}{18}\)

Explanation:
The given numbers are: \(\frac{1}{2}\) and 9
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{2}\) ÷ 9
= \(\frac{1}{2}\) × \(\frac{1}{9}\)
= \(\frac{ 1 × 1}{2 × 9}\)
= \(\frac{1}{18}\)
Hence,
\(\frac{1}{2}\) ÷ 9 = \(\frac{1}{18}\)

Question 22.

Answer: \(\frac{1}{3}\) ÷ 7 = \(\frac{1}{21}\)

Explanation:
The given numbers are: \(\frac{1}{3}\) and 7
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{3}\) ÷ 7
= \(\frac{1}{3}\) × \(\frac{1}{7}\)
= \(\frac{ 1 × 1}{7 × 3}\)
= \(\frac{1}{21}\)
Hence,
\(\frac{1}{3}\) ÷ 7 = \(\frac{1}{21}\)

Question 23.

Answer: \(\frac{1}{6}\) ÷ 5 = \(\frac{1}{30}\)

Explanation:
The given numbers are: \(\frac{1}{6}\) and 5
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{6}\) ÷ 5
= \(\frac{1}{6}\) × \(\frac{1}{5}\)
= \(\frac{ 1 × 1}{6 × 5}\)
= \(\frac{1}{30}\)
Hence,
\(\frac{1}{6}\) ÷ 5 = \(\frac{1}{30}\)

Question 24.

Answer: \(\frac{1}{7}\) ÷ 3 = \(\frac{1}{21}\)

Explanation:
The given numbers are: \(\frac{1}{7}\) and 3
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{7}\) ÷ 3
= \(\frac{1}{7}\) × \(\frac{1}{3}\)
= \(\frac{ 1 × 1}{7 × 3}\)
= \(\frac{1}{21}\)
Hence,
\(\frac{1}{7}\) ÷ 3 = \(\frac{1}{21}\)

Question 25.

Answer: \(\frac{1}{8}\) ÷ 4 = \(\frac{1}{32}\)

Explanation:
The given numbers are: \(\frac{1}{8}\) and 4
We know that,
\(\frac{a}{b}\) ÷ a  = \(\frac{a}{b}\) × \(\frac{1}{a}\)
a= \(\frac{a}{1}\)
So,
\(\frac{1}{8}\) ÷ 4
= \(\frac{1}{8}\) × \(\frac{1}{4}\)
= \(\frac{ 1 × 1}{8 × 4}\)
= \(\frac{1}{32}\)
Hence,
\(\frac{1}{8}\) ÷ 4 = \(\frac{1}{32}\)

10.5 Problem Solving: Fraction Division

Question 26.
A mechanic buys 1 gallon of oil. She uses \(\frac{1}{6}\) of the oil, and then divides the rest into 4 smaller bottles. How much does she put into each bottle?

Answer: The amount of oil she put into each bottle is: \(\frac{5}{24}\)

Explanation:
It is given that a mechanic buys 1 gallon of oil and she uses \(\frac{1}{6}\) of the oil
So,
the remaining part of the oil = 1 –  \(\frac{1}{6}\)
= \(\frac{5}{6}\)
It is also given that she divides the rest of the oil into 4 smaller bottles.
So,
The amount of oil in each bottle = \(\frac{The remaining part of the oil}{The number of bottles}\)
= \(\frac{5}{6}\) ÷ 4
= \(\frac{5}{6}\) × \(\frac{1}{4}\)
= \(\frac{5 × 1}{6 × 4}\)
= \(\frac{5}{24}\)
Hence, from the above,
We can conclude that the amount of oil in each bottle is: \(\frac{5}{24}\)

Conclusion:

Make use of the quick links and try to solve the problems in a simple manner. Redefine your true self with the BIM Answer Key for Grade 5 curated by subject experts. Test your knowledge by solving the questions which are given at the end of the chapter.

Big Ideas Math Answers Grade 5 Chapter 14 Classify Two-Dimensional Shapes pdf is available here. So, the students who are in search of Big Ideas Math Book 5th Grade Answer Key Chapter 14 Classify Two-Dimensional Shapes can get them on this page. Go through the Big Ideas Math Answers Grade 5 Chapter 14 Classify Two-Dimensional Shapes and improve your performance skills. Test your skills by solving the Performance Task, Activity, Chapter Practice which is provided at the end of the chapter.

Big Ideas Math Book 5th Grade Answer Key Chapter 14 Classify Two-Dimensional Shapes

Big Ideas Math Answers 5th Grade Chapter 14 Classify Two-Dimensional Shapes are very easy to understand. You can quickly grasp the mathematical practices with the help of Big Ideas Math Answer Key Grade 5 Chapter 14 Classify Two-Dimensional Shapes. The topics covered in this chapter are Classify Triangles, Quadrilaterals, Relate Quadrilaterals. Just hit the below-provided links and Download Big Ideas Math Book 5th Grade Answer Key Chapter 14 Classify Two-Dimensional Shapes.

Lesson: 1 Classify Triangles

• Lesson 14.1 Classify Triangles
• Classify Triangles Homework & Practice 14.1

Lesson: 2 Classify Quadrilaterals

• Lesson 14.2 Classify Quadrilaterals
• Classify Quadrilaterals Homework & Practice 14.2

Lesson: 3 Relate Quadrilaterals

• Lesson 14.3 Relate Quadrilaterals
• Relate Quadrilaterals Homework & Practice 14.3

Classify Two-Dimensional Shapes

• Classify Two-Dimensional Shapes Performance Task 14
• Classify Two-Dimensional Shapes Activity
• Classify Two-Dimensional Shapes Chapter Practice 14
• Classify Two-Dimensional Shapes Cumulative Practice 1-14
• Classify Two-Dimensional Shapes Steam Performance Task 1-14

Lesson 14.1 Classify Triangles

Explore and Grow

Draw and label a triangle for each description. If a triangle cannot be drawn, explain why.

Precision
Draw a triangle that meets two of the descriptions above.

Answer:

Think and Grow: Classify Triangles

Key Idea
Triangles can be classified by their sides.

An equilateral triangle has three sides with the same length.

An isosceles triangle has two sides with the same length.

A scalene triangle has no sides with the same length.

Key Idea
Triangles can be classified by their angles.

An acute triangle has three acute angles.

An obtuse triangle has one obtuse angle.

A right triangle has one right angle.

An equiangular triangle has three angles with the same measure.
Example
Classify the triangle by its angles and its sides.

The triangle has one ___ angle
and ___ sides with the same length.
So, it is a ___ triangle.

Answer:
The triangle has one right angle
and no sides with the same length.
So, it is a right triangle.

Show and Grow

Classify the triangle by its angles and its sides

Question 1.

Answer:  Equilateral triangle.

Explanation: An equilateral triangle has three sides of the same length.

Question 2.

Answer: Isosceles triangle

Explanation: An Isosceles triangle has two sides of the same size. Two of its angle also measure equal.

Question 3.

Answer:  Scalene triangle.

Explanation: A Scalene triangle has no sides are congruent (same size)

Apply and Grow: Practice

Classify the triangle by its angles and its sides.

Question 4.

Answer:  Right triangle.

Explanation: In a triangle one of the angle is a right angle (90 Deg ) called as Right triangle.

Question 5.

Answer: Isosceles triangle.

Explanation: An Isosceles triangle has two sides of the same size. Two of its angle also measure equal.

Question 6.

Answer: Equiangular triangle

Explanation: In an equilateral triangle, all the lengths of the sides are equal. In such a case, each of the interior angles will have a measure of 60 degrees. Since the angles of an equilateral triangle are the same, it is also known as an equiangular triangle. The figure given below illustrates an equilateral triangle.

Question 7.

Answer:  Isosceles triangle.

Explanation: An Isosceles triangle has two sides of the same length. Two of its angle also measure equal.

Question 8.

Answer:  Scalene triangle.

Explanation: A Scalene triangle has no sides are congruent (Same size) and angles also all different.

Question 9.

Answer:  Right triangle.

Explanation: In a triangle one of the angle is a right angle (90 deg ) called a Right triangle.

Question 10.
A triangular sign has a 40° angle, a 55° angle, and an 85° angle. None of its sides have the same length. Classify the triangle by its angles and its sides.

Answer:  Scalene triangle.

Explanation: A Scalene triangle has no sides that are congruent (Same size) and angles also all different.

Question 11.
YOU BE THE TEACHER
Your friend says the triangle is an acute triangle because it has two acute angles. Is your friend correct? Explain.

Answer:  Above is no acute triangle and it is called a scalene triangle.

Explanation: A Scalene triangle has only no sides that are congruent (Same size) and angles also all different. So it is called a scalene triangle.

Question 12.
DIG DEEPER!
Draw one triangle for each category. Which is the appropriate category for an equiangular triangle? Explain your reasoning.

Answer:

From the figure, we can say that acute triangles have the same length. So, the first triangle is the equiangular triangle.

Think and Grow: Modeling Real Life

Example
A bridge contains several identical triangles. Classify each triangle by its angles and its sides. What is the length of the bridge?

Each triangle has ___ angles with the same measure and ___ sides with the same length.
So, each triangle is ___ and ___.
The side lengths of 6 identical triangles meet to form the length of the bridge. So, multiply the side length by 6 to find the length of the bridge.
27 × 6 = ___
So, the bridge is ___ long.

Answer:
Each triangle has 3 angles with the same measure and 3 sides with the same length.
The side lengths of 6 identical triangles meet to form the length of the bridge. So, multiply the side length by 6 to find the length of the bridge.
27 × 6 = 162
So, the bridge is 162 ft long.

Show and Grow

Question 13.
The window is made using identical triangular panes of glass. Classify each triangle by its angles and its sides. What is the height of the window?

Answer:

The length of the two sides of the triangle is the same.
18 in + 18 in = 36 inches
Thus the height of the window is 36 inches

Question 14.
DIG DEEPER!
You connect four triangular pieces of fabric to make the kite. Classify the triangles by their angles and their sides. Use a ruler and a protractor to verify your answer.

Answer:
The name of the blue triangle is isosceles right angle triangle.
The two sides of the triangle are the same.
The name of the red triangle is isosceles right-angle triangle.
The two sides of the triangle are the same.
The name of the green triangle is isosceles right angle triangle.
The two sides of the triangle are the same.
The name of the yellow triangle is isosceles right angle triangle.
The two sides of the triangle are the same.

Classify Triangles Homework & Practice 14.1

Classify the triangle by its angles and its sides.

Question 1.

Answer:  Scalene triangle.

Explanation: A Scalene triangle has only no sides that are congruent (Same size) and angles also all different. So it is called a scalene triangle

Question 2.

Answer: Isosceles triangle.

Explanation: An Isosceles triangle has two sides of the same size. Two of its angle also measure equal.

Question 3.

Answer: Scalene triangle.

Explanation: A Scalene triangle has no sides that are congruent (Same size) and angles also all different.

Question 4.

Answer: Isosceles triangle.

Explanation: An Isosceles triangle has two sides of the same length. Two of its angle also measure equal.

Question 5.

Answer:  Scalene triangle.

Explanation: A Scalene triangle has no sides that are congruent (Same size) and angles also all different.

Question 6.

Answer:  Equiangular triangle.

Explanation:
In an equilateral triangle, all the lengths of the sides are equal. In such a case, each of the interior angles will have a measure of 60 degrees. Since the angles of an equilateral triangle are the same, it is also known as an equiangular triangle. The figure given below illustrates an equilateral triangle.

Question 7.
A triangular race flag has two 65° angles and a 50° angle. Two of its sides have the same length. Classify the triangle by its angles and its sides.

Answer: Isosceles triangle.

Explanation:  An Isosceles triangle has two sides of the same size. Two of its angle also measure equal.

Question 8.
A triangular measuring tool has a 90° angle and no sides of the same length. Classify the triangle by its angles and its sides.

Answer: Right triangle.

Explanation: In a triangle one of the angle is a right angle (90 Deg ) called a Right triangle.

Question 9.

Structure

Draw a triangle with vertices A(2, 2), B(2, 6), and C(6, 2) in the coordinate plane. Classify the triangle by its angles and its sides. Explain your reasoning.

Answer:

Question 10.

YOU BE THE TEACHER

Your friend says that both Newton and Descartes are correct. Is your friend correct? Explain.

Answer: Yes

Explanation:  An acute triangle is a triangle in which each angle is an acute angle. Any triangle which is not acute is either a right triangle or an obtuse triangle. All acute triangle angles are less than 90 degrees. For example, an equilateral triangle is always acute, since all angles (which are 60) are all less than 90.

Question 11.
DIG DEEPER!
The sum of all the angle measures in a triangle is 180°. A triangle has a 34° angle and a 26° angle. Is the triangle acute, right, or obtuse? Explain.

Answer: Scalene triangle.

Explanation: A Scalene triangle has no sides that are congruent (Same size) and angles also all different.

Question 12.
Modeling Real Life
A designer creates the logo using identical triangles. Classify each triangle by its angles and its sides. What is the perimeter of the logo?

Answer: Equiangular triangle

Explanation: In an equilateral triangle, all the lengths of the sides are equal. In such a case, each of the interior angles will have a measure of 60 degrees. Since the angles of an equilateral triangle are the same, it is also known as an equiangular triangle. The figure given below illustrates an equilateral triangle.

Question 13.
DIG DEEPER!
The window is made using identical triangular panes of glass. Classify each triangle by its angles and its sides. What are the perimeter and the area of the window? Explain your reasoning.

Answer: Right triangle.

Explanation: A right triangle is a triangle in which one of the angles is 90 degrees. In a right triangle, the side opposite to the right angle (90-degree angle) will be the longest side and is called the hypotenuse. You may come across triangle types with combined names like right isosceles triangle and such, but this only implies that the triangle has two equal sides with one of the interior angles being 90 degrees. The figure given below illustrates a right triangle

Review & Refresh

Question 14.

Answer : \(\frac{1}{4}\) =0.25

Explanation: 2 divides by 8  with 1/4 times, So the answer is 1/4.

Question 15.

Answer : \(\frac{15}{4}\) = 3.75

Explanation: 15 divides by 4 with \(\frac{15}{4}\) times, So the answer is \(\frac{15}{4}\) or 3.75.

Question 16.

Answer : \(\frac{15}{12}\) = \(\frac{1}{4}\) = 1.25

Explanation: 15 divides by 12 with \(\frac{1}{4}\) times, So the answer is \(\frac{1}{4}\).

Lesson 14.2 Classify Quadrilaterals

Explore and Grow

Draw and label a quadrilateral for each description. If a quadrilateral cannot be drawn, explain why

Precision
Draw a quadrilateral that meets three of the descriptions above.

Answer:

Think and Grow: Classify Quadrilaterals

Key Idea
Quadrilaterals can be classified by their angles and their sides.
A trapezoid is a quadrilateral that has exactly one pair of parallel sides.

A parallelogram is a quadrilateral that has two pairs of parallel sides. Opposite sides have the same length.

A rectangle is a parallelogram that has four right angles.

A rhombus is a parallelogram that has four sides with the same length.

A square is a parallelogram that has four right angles and four sides with the same length.

Example
Classify the quadrilateral in as many ways as possible.

Answer:

Show and Grow

Classify the quadrilateral in as many ways as possible.

Question 1.

Answer:  Square

Explanation: A square is a parallelogram that has four right angles and four sides with the same length.

Question 2.

Answer: Trapezoid 

Explanation: A trapezoid is a quadrilateral that has exactly one pair of parallel sides

Apply and Grow: Practice

Classify the quadrilateral in as many ways as possible.

Question 3.

Answer: Parallelogram

Explanation: A parallelogram is a quadrilateral that has two pairs of parallel sides. The opposite sides have the same length.

Question 4.

Answer:  Square

Explanation: A square is a parallelogram that has four right angles and four sides with the same length.

Question 5.

Answer: Rectangle

Explanation: A rectangle is a parallelogram that has four right angles and diagonals are congruent.

Question 6.

Answer: Rhombus

Explanation: A rhombus is a parallelogram with four congruent sides and A rhombus has all the properties of a parallelogram. The diagonals intersect at right angles.

Question 7.
A sign has the shape of a quadrilateral that has two pairs of parallel sides, four sides with the same length, and no right angles

Answer: Parallelogram

Explanation:
Assume that a quadrilateral has parallel sides or equal sides unless that is stated. A parallelogram has two parallel pairs of opposite sides. A rectangle has two pairs of opposite sides parallel, and four right angles.

Question 8.
A tabletop has the shape of a quadrilateral with exactly one pair of parallel sides.

Answer: A trapezoid is a quadrilateral that has exactly one pair of parallel sides. A parallelogram is a quadrilateral that has two pairs of parallel sides.

Question 9.
YOU BE THE TEACHER
Your friend says that a quadrilateral with at least two right angles must be a parallelogram. Is your friend correct? Explain.

Explanation: A trapezoid is only required to have two parallel sides. However, a trapezoid could have one of the sides connecting the two parallel sides perpendicular to the parallel sides which would yield two right angles.

Question 10.
Which One Doesn’t Belong? Which cannot set of lengths be the side lengths of a parallelogram?

Answer: 9 yd, 5 yd, 5 yd, 3 yd

Explanation: A parallelogram is a quadrilateral that has two pairs of parallel sides. Opposite sides have the same length, So the above one is not a parallelogram.

Think and Grow: Modeling Real Life

Example
The dashed line shows how you cut the bottom of a rectangular door so it opens more easily. Classify the new shape of the door.

Draw the new shape of the door.

Answer:
The original shape of the door was a rectangle, so it had one pairs of parallel sides. The new shape of the door has exactly one pair of parallel sides. So, the new shape of the door is a trapezoid.

Show and Grow

Question 11.
DIG DEEPER!
The dashed line shows how you cut the corner of the trapezoidal piece of fabric. The line you cut is parallel to the opposite side. Classify the new shape of the four-sided piece of fabric.

Answer: Parallelogram

Explanation: A parallelogram is a quadrilateral that has two pairs of parallel sides. The opposite sides have the same length.

Question 12.
A farmer encloses a section of land using the four pieces of fencing. Name all of the four-sided shapes that the farmer can enclose with the fencing.

Answer: Parallelogram

Explanation: A parallelogram is a quadrilateral that has two pairs of parallel sides. Opposite sides have the same length, So four-sided shapes of fencing look like Parallelogram.

Classify Quadrilaterals Homework & Practice 14.2

Classify the quadrilateral in as many ways as possible.

Question 1.

Answer: Trapezoid 

Explanation: A trapezoid is a quadrilateral that has exactly one pair of parallel sides

Question 2.

Answer: Trapezoid

Explanation: A Trapezoid is a quadrilateral with exactly one pair of parallel sides. (There may be some confusion about this word depending on which country you’re in. In India and Britain, they say trapezium; in America, trapezium usually means a quadrilateral with no parallel sides.)

Question 3.

Answer: Rectangle

Explanation: A rectangle is a parallelogram that has four right angles and diagonals are congruent.

Question 4.

Answer: Square

Explanation: The diagonals of a square bisect each other and meet at 90°. The diagonals of a square bisect its angles. The opposite sides of a square are both parallel and equal in length. All four angles of a square are equal (each being 360°/4 = 90°, a right angle).

Question 5.
A name tag has the shape of a quadrilateral that has two pairs of parallel sides and four right angles. Opposite sides are the same length, but not all four sides are the same length.

Answer: Rectangle

Explanation: A rectangle is a parallelogram that has four right angles and diagonals are congruent.

Question 6.
A napkin has the shape of a quadrilateral that has two pairs of parallel sides, four sides with the same length, and four right angles.

Answer: Square

Explanation: A square is a parallelogram that has four right angles and four sides of the same length.

Question 7.
Reasoning
Can you draw a quadrilateral that is not a square, but has four right angles? Explain.

Answer: A rectangle is a parallelogram that has four right angles

Question 8.
Structure
Plot two more points in the coordinate plane to form a square. What two points can you plot to form a parallelogram? What two points can you plot to form a trapezoid? Do not use the same pair of points twice.

Answer:

Question 9.
DIG DEEPER!
Which quadrilateral can be classified as a parallelogram, and rectangle, square, rhombus? Explain.

Answer: Square

Explanation: A square can be defined as a rhombus which is also a rectangle – in other words, a parallelogram with four congruent sides and four right angles. A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Question 10.
Modeling Real Life
The dashed line shows how you fold the flap of the envelope so it closes. Classify the new shape of the envelope.

Answer: Rectangle

Explanation: A rectangle is a parallelogram that has four right angles and diagonals are congruent.

Question 11.
DIG DEEPER!
A construction worker tapes off a section of land using the four pieces of caution tape. Name all of the possible shapes that the worker can enclose with the tape.

Answer: Trapezoid

Explanation: A trapezoid is a quadrilateral that has exactly one pair of parallel sides.

Question 12.

Answer: \(\frac{1}{2}\)

Explanation: \(\frac{2}{3}\) –\(\frac{1}{6}\) equal to \(\frac{1}{2}\).

Question 13.

Answer: 0.112

Explanation: \(\frac{1}{2}\) is equal to 0.5 and 7/18 is equal to 0.3888.So subtraction from 0.5 to 0.3888 is 0.112.

Question 14.

Answer: 0.289

Explanation: \(\frac{2}{5}\) is equal to 0.4 and 1/9 is equal to 0.111,So subtraction from 0.4 to 0.111is 0.289.

Lesson 14.3 Relate Quadrilaterals

Explore and Grow

Label the Venn diagram to show the relationships among quadrilaterals. The first one has been done for you.

Reasoning
Explain how you decided where to place each quadrilateral.

Think and Grow: Relate Quadrilaterals

Key Idea
The Venn diagram shows the relationships among quadrilaterals.

Example
Tell whether the statement is true or false.
All rhombuses are rectangles.
Rhombuses do not always have four right angles.
So, the statement is ___.

Answer: So, the statement is true.

Example
Tell whether the statement is true or false.
All rectangles are parallelograms.
All rectangles have two pairs of parallel sides.
So, the statement is ___.

Show and Grow

Tell whether the statement is true or false. Explain.

Question 1.
Some rhombuses are squares.

Answer: true

Explanation: A rhombus is a quadrilateral (plane figure, closed shape, four sides) with four equal-length sides and opposite sides parallel to each other. All squares are rhombuses, but not all rhombuses are squares. The opposite interior angles of rhombuses are congruent.

Question 2.
All parallelograms are rectangles.

Answer: False

Explanation: A rectangle is a parallelogram with four right angles, so all rectangles are also parallelograms and quadrilaterals. On the other hand, not all quadrilaterals and parallelograms are rectangles. A rectangle has all the properties of a parallelogram

Apply and Grow: Practice

Tell whether the statement is true or false. Explain.

Question 3.
All rectangles are squares.

Answer: False

Explanation: All squares are rectangles, but not all rectangles are squares.

Question 4.
Some parallelograms are trapezoids.

Answer: True

Explanation: A trapezoid has one pair of parallel sides and a parallelogram has two pairs of parallel sides. So a parallelogram is also a trapezoid.

Question 5.
Some rhombuses are rectangles.

Answer: False

Explanation: A rhombus is defined as a parallelogram with four equal sides. Is a rhombus always a rectangle? No, because a rhombus does not have to have 4 right angles. Kites have two pairs of adjacent sides that are equal.

Question 6.
All trapezoids are quadrilaterals.

Answer: True

Explanation: Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides. The correct answer is that all trapezoids are quadrilaterals. Trapezoids are four-sided polygons, so they are all quadrilaterals

Question 7.
All squares are rhombuses.

Answer: True

Explanation: All squares are rhombuses, but not all rhombuses are squares. The opposite interior angles of rhombuses are congruent

Question 8.
Some trapezoids are squares.

Answer: True

Explanation: A trapezoid is a quadrilateral with at least one pair of parallel sides. In a square, there are always two pairs of parallel sides, so every square is also a trapezoid. Conversely, only some trapezoids are squares

Question 9.
Reasoning
Use the word cards to complete the graphic organizer.

Answer: The first box to be filled with Square, 3d box to be filled with Rectangle,4th box to be filled with trapezoid and final box to be filled with Quadrilateral.

Explanation: A Square can be defined as a rhombus which is also a rectangle, in other words, a parallelogram with four congruent sides and four right angles. A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Question 10.
Reasoning
All rectangles are parallelograms. Are all parallelograms rectangles? Explain.

Answer: True

Explanation: A rectangle is considered a special case of a parallelogram because, A parallelogram is a quadrilateral with 2 pairs of opposite, equal and parallel sides. A rectangle is a quadrilateral with 2 pairs of opposite, equal and parallel sides but also forms right angles between adjacent sides.

Question 11.
Precision
Newton says the figure is a square. Descartes says the figure is a parallelogram. Your friend says the figure is a rhombus. Are all three correct? Explain.

Answer: No

Explanation:  A square has two pairs of parallel sides, four right angles, and all four sides are equal. It is also a rectangle and a parallelogram. A rhombus is defined as a parallelogram with four equal sides. No, because a rhombus does not have to have 4 right angles.

Think and Grow: Modeling Real Life

Example
You use toothpicks to create several parallelograms. You notice that opposite angles of parallelograms have the same measure. For what other quadrilaterals is this also true?

Parallelograms have the property that opposite angles have the same measure. Subcategories of parallelograms must also have this property.
___, ___, and ___ are subcategories of parallelograms.

So, ___, ____, and ____ also have opposite angles with the same measure.

Answer: Rectangle, Rhombus and Square are subcategories of parallelograms.

Show and Grow

Question 12.
You use pencils to create several rhombuses. You notice that diagonals of rhombuses are perpendicular and divide each other into two equal parts. For what other quadrilateral is this also true? Explain your reasoning.

Answer: Square, Parallelogram, Rhombus are perpendicular and divided into the equal parts.

Question 13.
DIG DEEPER!
You place two identical parallelograms side by side. What can you conclude about the measures of adjacent angles in a parallelogram? For what other quadrilaterals is this also true? Explain your reasoning.

Answer:
The adjacent angles of the parallelogram is supplementary.
Opposite angles of the parallelogram are equal.

Relate Quadrilaterals Homework & Practice 14.3

Tell whether the statement is true or false. Explain.

Question 1.
All trapezoids are parallelograms.

Answer: False

All rectangles are parallelograms.

Answer: True

Question 3.
All squares are quadrilaterals.

Answer: True

Some quadrilaterals are trapezoids.

Answer: True

Question 5.
Some parallelograms are rectangles.

Answer: True

Question 6.
All squares are rectangles and rhombuses.

Answer: False

Question 7.
YOU BE THE TEACHER
Newton says he can draw a quadrilateral that is not a trapezoid and not a parallelogram. Is Newton correct? Explain.

Answer: False

Question 8.
Writing
Explain why a parallelogram is not a trapezoid.

Explanation: a square is a quadrilateral, a parallelogram, a rectangle, and a rhombus Is a trapezoid a parallelogram? No, because a trapezoid has only one pair of parallel sides.

Reasoning
Write always, sometimes, or never to make the statement true? Explain.

Question 9.
A rhombus is ___ a square.

Answer: A rhombus is  some times a square

Explanation: A rhombus is a square. This is sometimes true. Â It is true when a rhombus has 4 right angles. It is not true when a rhombus does not have any right angles.

Question 10.
A trapezoid is __ a rectangle.

Answer: A trapezoid is sometimes a rectangle.

Explanation: A rectangle has one pair of parallel sides.

Question 11.
A parallelogram is ___ a quadrilateral.

Answer: A parallelogram is always a quadrilateral.

Explanation: A parallelogram must have 4 sides, so they must always be quadrilaterals.

Question 12.
DIG DEEPER!
A quadrilateral has exactly three sides that have the same length. Why can the figure not be a rectangle?

Explanation: A rectangle is a parallelogram that has four right angles. opposite sides are in the same length, so the above one is not a rectangle.

Question 13.

Modeling Real Life
You fold the rectangular piece of paper. You notice that the line segments connecting the halfway points of opposite sides are perpendicular. For what other quadrilateral is this also true?

Explanation: A rectangle is a parallelogram that has four right angles. opposite sides are in the same length, so the above one is not a rectangle.

Question 14.
DIG DEEPER!
You tear off the four corners of the square and arrange them to form a circle. You notice that the sum of the angle measures of a square is equal to 360°. For what other quadrilaterals is this also true?

Answer: The sum of the angles in a parallelogram are 360°

Review & Refresh

Question 15.
5 pt = ___ c

Answer : 5 pt = 10 c

Explanation:
Convert from pints to cups.
1 pint = 2 cups
5 pints = 5 × 2 cups
5 pints = 10 cups

Question 17.
20 qt = ___ c

Answer : 20 qt = 80 c

Explanation:
Convert from quarts to cups.
1 quart = 4 cups
20 qt = 20 × 4 cups = 80 cups

Classify Two-Dimensional Shapes Performance Task 14

A homeowner wants to install solar panels on her roof to generate electricity for her house. A solar panel is 65 inches long and 39 inches wide.

Question 1.
a. The shape of the panel has 4 right angles. Sketch and classify the shape of the solar panel.

Answer :

Explanation: A rectangle is a parallelogram that has four right angles. So the shape of the solar panel is a rectangle.

b. There are 60 identical solar cells in a solar panel, arranged in an array. Ten cells meet to form the length of the panel, and six cells meet to form the width. Classify the shape of each solar cell. Explain your reasoning.

Answer: A rectangle is a parallelogram that has four right angles. So the shape of the solar panel is a rectangle.

Question 2.
The home owner measures three sections of her roof.

a. Classify the shape of each section in as many ways as possible.

Answer:  Right triangle.

Explanation: In a triangle one of the angle is a right angle (90 Deg ) called as Right triangle

Answer: Rectangle

Explanation: Rectangle is a parallelogram with four right angles, so all rectangles are also parallelograms and quadrilaterals. On the other hand, not all quadrilaterals and parallelograms are rectangles.

Answer: Isosceles trapezoid

Explanation: An isosceles trapezoid is a trapezoid whose non-parallel sides are congruent.

b. About how many solar panels can fit on the measured sections of the roof? Explain your reasoning.

Question 3.
One solar panel can produce about 30 kilowatt-hours of electricity each month. The homeowner uses her electric bills to determine that she uses about 1,200 kilowatt-hours of electricity each month.

a. How many solar panels should the homeowner install on her roof?

Answer: 40 Solar panels

Explanation: 40 Solar panels X 30 kilowatt-hours of electricity each month per one solar panel equal to 1,200 kilowatt-hours of electricity per month, So the answer is 40 solar panels.

b. Will all of the solar panels fit on the measured sections of the roof? Explain.

Classify Two-Dimensional Shapes Activity

Quadrilateral Lineup

Directions:

1. Players take turns spinning the spinner.
2. On your turn, cover a quadrilateral that matches your spin.
3. If you land on, Lose a turn, then do not cover a quadrilateral.
4. The first player to get four in a row twice, horizontally, vertically, Recor diagonally, wins!

Classify Two-Dimensional Shapes Chapter Practice 14

14.1 Classify Triangles

Classify the triangle by its angles and its sides.

Question 1.

Answer: Scalene triangle.
Explanation: A Scalene triangle has no sides that are congruent (Same size) and angles also all different.

Question 2.

Answer: Right triangle.
Explanation: In a triangle one of the angle is a right angle (90 deg ) called as Right triangle.

Question 3.

Answer: Isosceles triangle.

Explanation: An Isosceles triangle has two sides of the same length. Two of its angle also measure equal.

Question 4.

Answer: Equiangular triangle

Explanation: In an equilateral triangle, all the lengths of the sides are equal. In such a case, each of the interior angles

Question 5.

Answer: Right triangle.

Explanation: In a triangle one of the angle is a right angle (90° ) called as Right triangle.

Question 6.

Answer: Scalene triangle.

Explanation: A Scalene triangle has no sides that are congruent (Same size) and angles also all different.

14.2 Classify Quadrilaterals

Classify the quadrilateral in as many ways as possible.

Question 7.

Answer: Square

Explanation: A square is a parallelogram that has four right angles and four sides of the same length.

Question 8.

Answer: Rectangle

Explanation: A rectangle is a parallelogram that has four right angle, Opposite sides are the same length

Question 9.

Answer: Square

Explanation: The diagonals of a square bisect each other and meet at 90°. The diagonals of a square bisect its angles. The opposite sides of a square are both parallel and equal in length. All four angles of a square are equal (each being 360°/4 = 90°, a right angle).

Question 10.

Answer: Isosceles trapezoid

Explanation: An isosceles trapezoid is a trapezoid whose non-parallel sides are congruent.

Question 11.

Answer: Rectangle

Explanation: A rectangle is a parallelogram that has four right angle, Opposite sides are the same length

Question 12.

Answer: Trapezoid

Explanation: A trapezoid is a quadrilateral that has exactly one pair of parallel sides.

Question 13.
Structure
Plot two more points in the coordinate plane to form a quadrilateral that has exactly two a rectangle. What two points can you plot to form a trapezoid? What two points can you plot to form a rhombus? Do not use the same pair of points twice.

Answer:

Question 14.
Reasoning
Can you draw a quadrilateral that has exactly two right angles? Explain.

Explanation: A quadrilateral with only 2 right angles and it is called a trapezoid .

Question 15.
Modeling Real Life
The dashed line shows how you break apart the graham cracker. Classify the new shape of each piece of the graham cracker.

Answer: Square

Explanation: The diagonals of a square bisect each other and meet at 90°.

14.3 Relate Quadrilaterals

Tell whether the statement is true or false.

Question 16.
All rectangles are quadrilaterals.

Answer: True

Explanation: A closed figure with four sides. For example, kites, parallelograms, rectangles, rhombuses, squares, and trapezoids are all quadrilaterals

Question 17.
Some parallelograms are squares.

Answer: True

Explanation: Squares fulfill all criteria of being a rectangle because all angles are right angle and opposite sides are equal. Similarly, they fulfill all criteria of a rhombus, as all sides are equal and their diagonals bisect each other.

Question 18.

All trapezoids are rectangles.

Answer: False

Explanation: Rectangles are defined as a four-sided polygon with two pairs of parallel sides. On the other hand, a trapezoid is defined as a quadrilateral with only one pair of parallel sides.

Question 19.
Some rectangles are rhombuses.

Answer: True

Explanation: A rectangle is a parallelogram with all its interior angles being 90 degrees. A rhombus is a parallelogram with all its sides equal. This means that for a rectangle to be a rhombus, its sides must be equal. A rectangle can be a rhombus only if has extra properties which would make it a square

Question 20.
Some squares are trapezoids.

Answer: True

Explanation: The definition of a trapezoid is that it is a quadrilateral that has at least one pair of parallel sides. A square, therefore, would be considered a trapezoid.

Question 21.
All quadrilaterals are squares.

Answer: False

Explanation : Quadrilateral: A closed figure with four sides. For example, kites, parallelograms, rectangles, Square: A rectangle with four sides of equal length. Trapezoid: A quadrilateral with at least one pair of parallel sides So, All quadrilaterals are not squares.

Classify Two-Dimensional Shapes Cumulative Practice 1-14

Question 1.
Which model shows 0.4 × 0.2?

Answer:

Question 2.
A triangle has angle measures of 82°, 53°, and 45°. Classify the triangle by its angles.

Answer: C

Explanation: An obtuse triangle has one angle measuring more than 90º but less than 180º (an obtuse angle). It is not possible to draw a triangle with more than one obtuse angle

Question 3.
Which expressions have an estimated difference of \(\frac{1}{2}\) ?

Answer:

Question 4.
A rectangular prism has a volume of 288 cubic centimeters. The height of the prism is 8 centimeters. The base is a square. What is a side length of the base?

Answer: A

Explanation: volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units. So,6 X 6 X 8 is equal to 288 cubic centimeters, Therefore the side length of the base is 6 cm.

Question 5.
A sandwich at a food stand costs $3.00. Each additional topping costs the same extra amount. The coordinate plan shows the costs, in dollars, of sandwiches with different numbers of additional toppings. What is the cost of a sandwich with 3 additional toppings?

Answer:

Question 6.
Which statements are true?

Answer:  The following statements are true
Option 2,option 3 and option 4 .

Explanation :
Option 2 :
All squares are rectangles are parallelograms is true, why because squares fulfill all criteria of being a rectangle because all angles are right angle and opposite sides are equal. Similarly, they fulfill all criteria of a rhombus, as all sides are equal and their diagonals bisect each other.
Option 3: All squares are rhombuses is true, why because All squares are rhombuses, but not all rhombuses are squares. The opposite interior angles of rhombuses are congruent.
Option 4:  Every trapezoid is a quadrilateral is true, why because Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides.

Question 7.
Your friend makes a volcano for a science project. She uses 10 cups of vinegar. How many pints of vinegar does he use?

Answer: Option B

Explanation: 1 cup is equal to  0.5 pints, therefore 10 cups are equal to 5 pints.

Question 8.
The volume of the rectangular prism is 432 cubic centimeters. What is the length of the prism?

Answer: Option A

Explanation: volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units.
So, 6 cm X 9 cm X 8 cm is equal to 432 cubic centimeters.
Therefore the length of prim is 9 cm.

Question 9.
Descartes draws a pentagon by plotting another point in the coordinate plane and connecting the points. Which coordinates could he use?

Answer:

Question 10.
Newton rides to the dog park in a taxi. He owes the driver $12. He calculates the driver’s tip by multiplying $12 by 0.15. How much does he pay the driver, including the tip?

Answer: Option  C

Explanation: Driver cost $12 + ($12 X 0.15 )= 12+1.8 =13.8
Therefore answer is $13.8.

Question 11.
A quadrilateral has four sides with the same length, two pairs of parallel sides, and four 90° angles. Classify the quadrilateral in as many ways as possible.

Answer: Square, Parallelogram

Explanation: A quadrilateral has four sides with the same length, two pairs of parallel sides and four 90° angles is called as square. All squares are parallelograms.

Question 12.
Which ordered pair represents the location of a point shown in the coordinate plane?

Answer:

Question 13.
What is the product of 5,602 and 17?

Answer: 95234

Explanation: 5602 X 17 is equal to 95234.

Question 14.
Which pair of points do not lie on a line that is perpendicular to the x-axis?

Answer:

Question 15.
Newton has a gift in the shape of a rectangular prism that has a volume of 10,500 cubic inches. The box he uses to ship the gift is shown.

Part A What is the volume of the box?
Part B What is the volume, in cubic inches, of the space inside the box that is not taken up by gift? Explain.?

Answer:

Question 16.
Which expressions have a product greater than \(\frac{5}{6}=\)?

Answer:

Question 17.
Newton is thinking of a shape that has 4 sides, only one pair of parallel sides, and angle measures of 90°, 40°, 140°, and 90°. Which is Newton’s shape?

Answer: D

Explanation: Trapezoid Only one pair of opposite sides is parallel.

Question 18.
Which rectangular prisms have a volume of 150 cubic feet?

Answer: Option 1

Explanation: volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic feet.
So,2 ft X 25 ft X 3 ft is equal to 150 cubic ft, Therefore the right answer is option 1.

Classify Two-Dimensional Shapes Steam Performance Task 1-14

Each student in your grade makes a constellation display by making holes for the stars of a constellation on each side of the display. Each display is a rectangular prism with a square base.

Question 1.
Your science teacher orders a display for each student. The diagram shows the number of packages that can fit in a shipping box.
a. How many displays come in one box?

b. There are 108 students in your grade. How many boxes of displays does your teacher order? Explain.
c. The volume of the shipping box is 48,000 cubic inches. What is the volume of each display?
d. The height of each display is 15 inches. What are the dimensions of the square base?
e. Estimate the dimensions of the shipping box.
f. You paint every side of the display except the bottom. What is the total area you will paint?
g. You need a lantern to light up your display. Does the lantern fit inside of your display? Explain.

Question 2.
On one side of your display, you create an image of the constellation Libra. Each square on the grid is 1 square inch.
a. Classify the triangle formed by the points of the constellation.

Answer: Equiangular triangle
Explanation: In an equilateral triangle, all the lengths of the sides are equal. In such a case, each of the interior angles

b. What are the coordinates of the points of the constellation?
c. What is the height of the constellation on your display?

Question 3.
You use the coordinate plane to create the image of the Big Dipper.
a. Plot the points A(6, 2), B(8, 2), C(7, 6), D(5, 5), E(7, 9), F(6, 12), and G(4, 14).
b. Draw lines connecting the points of quadrilateral ABCD. Draw \(\overline{C E}\), \(\overline{E F}\) and \(\overline{F G}\).

Answer:

c. Is quadrilateral ABCD a trapezoid? How do you know?

Answer: Yes ABCD is a trapezoid because all sides are not equal and only one pair has parallel sides.

Question 4.
Use the Internet or some other resource to learn more about constellations. Write one interesting thing you learn.

Answer: A constellation is a group of stars that appears to form a pattern or picture like Orion the Great Hunter, Leo the Lion, or Taurus the Bull. Constellations are easily recognizable patterns that help people orient themselves using the night sky. There are 88 “official” constellations.

Conclusion:

Sharpen your math skills by practicing the problems from Big Ideas Math Book 5th Grade Answer Key Chapter 14 Classify Two-Dimensional Shapes. All the solutions of Grade 5 Chapter 14 Classify Two-Dimensional Shapes are prepared by the math professionals. Thus you can prepare effectively and score good marks in the exams.