This handy Spectrum Math Grade 7 Answer Key Chapter 3 Pretest provides detailed answers for the workbook questions

Expressions, Equations, and Inequalities

Rewrite each expression using the property indicated.

Question 1.
a. associative: (5 + 6) + 7
_____________
Answer: 5 + (6 + 7)
According to the associative principle of addition, when adding three integers, the outcome will always be the same regardless of how the numbers are grouped. If there are three numbers, x, y and z, the associative property of addition implies that x + (y + z) = (x + y) + z. The grouping of addends does not change the sum.
(5 + 6) + 7 = 5 + (6 + 7)
18 = 18

b. identity: 56 × 1
_____________
According to the identity property of multiplication, if a number is multiplied by 1 (one), the result will be the original number. This property is applied when numbers are multiplied by 1. If there is a number, x then the identity property implies that x × 1 = x.
56 × 1 = 56

Question 2.
a. zero: 0 ÷ 4
According to the zero property of division, if 0(zero) is divided by any other number, the result will be zero. If there is a number, x then the zero property of division implies that 0 ÷ x = 0.
0 ÷ 4 = 0

b. commutative: 8 × 9
According to the commutative property of multiplication, changing the order of the numbers we are multiplying does not change the product. If there are two numbers, x and y, the commutative property of multiplication implies that x × y = y × x.
8 × 9 = 9 × 8
72 = 72

Question 3.
a. distributive: 3 × (5 – 2)
_____________
Answer: (3 × 5) – (3 × 2)
The distributive property states that multiplying the sum of two or more addends by a number yields the same outcome as multiplying each addend separately by the number and combining the resulting products. a × (b – c) = (a × b) – (a × c) is the rule for distributive property.
3 × (5 – 2) = (3 × 5) – (3 × 2)
9 = 9

b. associative: (7 × 2) × 3
_____________
Answer: 7 × (2 × 3)
According to the associative principle of addition, when adding three integers, the outcome will always be the same regardless of how the numbers are grouped. If there are three numbers, x, y and z, the associative property of addition implies that x × (y × z) = (x × y) × z. The grouping of addends does not change the sum.
(7 × 2) × 3 = 7 × (2 × 3)
42 = 42

Write each phrase as an expression or equation.

Question 4.
a. five less than a number
_____________
Let x be the number.
The expression for ‘five less than a number’ can be given as x -5

b. eight more than a number
___________________
Let x be the number.
The expression for ‘eight more than a number’ can be given as x + 8

Question 5.
a. number divided by six
_____________
Let x be the number.
The expression for ‘number divided by six’ can be given as x ÷ 6

b. the product of two and a number
_____________
Let x be the number.
The expression for ‘the product of two and a number’ can be given as 2 × x

Question 6.
a. the sum of 3 and a number is 12
_____________
Answer: 3 + x = 12
Let x be the number.
The equation for ‘the sum of 3 and a number is 12’ can be given as 3 + x = 12

b. six less than a number is nineteen
_____________
Answer: x – 6 = 19
Let x be the number.
The equation for ‘six less than a number is nineteen’ can be given as x – 6 = 19

Question 7.
a. thirty divided by a number is three
_____________
Answer: 30 ÷ x = 3
Let x be the number.
The equation for ‘thirty divided by a number is three’ can be given as 30 ÷ x = 3

b. the product of 5 and a number is fifteen
_____________
Answer: 5 ×  x = 15
Let x be the number.
The equation for ‘the product of 5 and a number is fifteen’ can be given as 5 ×  x = 15

Question 8.
a. the product of 5 and a number
_____________
Let x be the number.
The expression for ‘the product of 5 and a number’ can be given as 5 × x

b. the sum of 6 and a number is 16
_____________
Let x be the number.
The equation for ‘the sum of 6 and a number is 16′ can be given as 6 + x= 16

Question 9.
a. 19 less than a number
_____________
Let x be the number.
The expression for ’19 less than a number’ can be given as 19 < x

b. 27 divided by a number is 9
_____________
Answer: 27 ÷ x = 9
Let x be the number.
The equation for ’27 divided by a number is 9′ can be given as 27 ÷ x = 9

Question 10.
a. 12 less than a number is 5
_____________
Answer: 12 < x = 5
Let x be the number.
The equation for ’12 less than a number is 5′ can be given as 12 < x = 5

b. the product of 6 and a number is 72
_____________
Answer: 6 ×  x = 72
Let x be the number.
The equation for ‘the product of 6 and a number is 72’ can be given as 6 ×  x = 72

Solve each problem.

Question 11.
Alicia had $22 to spend on pencils. If each pencil costs$ 1.50, how many pencils can she buy?
Let p represent the number of pencils.
Equation or Inequality: _____________
Answer: Inequality: p × $1.50 =$22
Alicia had $22 to spend on pencils. each pencil costs$ 1.50
Let p represent the number of pencils.
Inequality: p × $1.50 =$22
p = $$\frac{22}{1.5}$$
p = 14.666
Therefore, Alicia can buy 14 pencils.

Question 12.
The sum of three consecutive numbers is 51. What is the smallest of these numbers?
Let n represent the smallest number of the set.
Equation or Inequality: ___________
The smallest of these numbers is _________.
Answer: Equation: x + x+1+ x+2 = 51
The smallest of these numbers is 16
The sum of three consecutive numbers is 51.
Let the numbers be x, x+1 and x+2.
Then the sum is : Equation: x + x+1+ x+2 = 51
3x + 3 = 51
3x = 51 – 3 = 48
x = 48 ÷ 3
x = 16
Therefore, The smallest of these numbers is 16

Question 13.
Mark bought 8 boxes. A week later, half of all his boxes were destroyed in a fire. There are now only 20 boxes left. With how many did he start?
Let b represent how many boxes he started with.
Equation or inequality: ___________
Mark began with ____________________ boxes.
Answer: Equation: $$\frac{b + 8}{2}$$ = 20
Mark began with 32 boxes.
Let b represent how many boxes he started with.
Mark bought 8 boxes.
So he have b + 8 books at present.
A week later, half of all his boxes were destroyed in a fire.
There are now only 20 boxes left.
The number boxed Mark began with = $$\frac{b + 8}{2}$$ = 20
b + 8 = 20 × 2
b + 8 = 40
b = 40 – 8
b = 32
Therefore, Mark began with 32 boxes.

Question 14.
Jillian sold half of her comic books and then bought 15 more. She now has 30. With how many did she begin?
Let c represent the number of comic books with which she began.
Equation or inequality: ___________
Jillian began with ____________________ CDs.
Answer: Equation: $$\frac{c}{2}$$ + 15 = 30
Jillian began with 30 CDs.
Jillian sold half of her comic books and then bought 15 more.
She now has 30.
Let c represent the number of comic books with which she began.
The number of books she begin with = $$\frac{c}{2}$$ + 15 = 30
$$\frac{c}{2}$$  = 30 -15
$$\frac{c}{2}$$ = 15
c = 15 × 2
c =30
Therefore, Jillian began with 30 CDs.

Question 15.
On Tuesday, Shan ice bought 5 new pens. On Wednesday, half of all the pens that she had were accidentally thrown away. On Thursday, there were only 16 left. How many did she have on Monday?
Let p represent the number of pens she had on Monday.
Equation or inequality: ___________
Shanice had ___________________ pens on Monday.
Answer: Equation: $$\frac{p+5}{2}$$ = 16
Shanice had 27 pens on Monday.
Let p represent the number of pens she had on Monday.
On Tuesday, Shan ice bought 5 new pens. So she had p + 5 pens on Tuesday
On Wednesday, half of all the pens that she had were accidentally thrown away. So, she had $$\frac{p+5}{2}$$
On Thursday, there were only 16 left.
$$\frac{p+5}{2}$$ = 16
p + 5 = 16 × 2
p + 5 = 32
p = 32- 5
p = 27
Therefore, Shanice had 27 pens on Monday.