This handy Spectrum Math Grade 7 Answer Key Chapter 1 Posttest provides detailed answers for the workbook questions

Check What You Learned

Evaluate each expression.

Question 1.
a. opposite of -54 _____
opposite of -54 is 54
-54 and 54 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

b. opposite of 19 ____
opposite of 19 is -19
-19 and 19 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

c. opposite of 31 ____
opposite of 31 is -31
-31 and 31 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

Question 2.
a. opposite of -6 ____
opposite of -6 is 6
-6 and 6 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

b. opposite of 21 ____
opposite of 21 is -21
-21 and 21 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

c. opposite of -10 ____
opposite of -10 is 10
-10 and 10 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

Question 3.
a. opposite of 54 ____
opposite of 54 is -54
-54 and 54 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

b. opposite of -34 ___
opposite of -34 is 34
-34 and 34 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

c. opposite of 86 ____
opposite of 86 is -86
-86 and 86 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

Question 4.
a. |-35| = ____
|-35| = 35
-35 and 35 are absolute value because they are the same distance from zero on opposite sides of the number line.
Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

b. -|-43| = ____
-|-43| = -43
-43 and 43 are absolute value because they are the same distance from zero on opposite sides of the number line.
The absolute value of a number is a number that is the same distance from zero on a number line. The distance is always a positive quantity (or zero). Absolute value is shown by vertical bars on each side of the number.

c. |35| = ____
|35| = 35
-35 and 35 are absolute value because they are the same distance from zero on opposite sides of the number line.
The absolute value of a number is a number that is the same distance from zero on a number line. The distance is always a positive quantity (or zero). Absolute value is shown by vertical bars on each side of the number.

Question 5.
a. -|75| = ___
-|75| = -75
-75 and 75 are absolute value because they are the same distance from zero on opposite sides of the number line.
The absolute value of a number is a number that is the same distance from zero on a number line. The distance is always a positive quantity (or zero). Absolute value is shown by vertical bars on each side of the number.

b. -|83| = ___
-|83| = -83
-83 and 83 are absolute value because they are the same distance from zero on opposite sides of the number line.
The absolute value of a number is a number that is the same distance from zero on a number line. The distance is always a positive quantity (or zero). Absolute value is shown by vertical bars on each side of the number.

c. -|99| = ____
-|99| = -99
-99 and 99 are absolute value because they are the same distance from zero on opposite sides of the number line.
The absolute value of a number is a number that is the same distance from zero on a number line. The distance is always a positive quantity (or zero). Absolute value is shown by vertical bars on each side of the number.

Identify the property of addition described as commutative, associative, or identity.

Question 6.
When two numbers are added, the sum is the same regardless of the order of addends.
_________
When two numbers are added, the sum is the same regardless of the order of addends =  commutative property
According to the commutative property of addition, the sum is unaffected by changes in the order of the numbers being added. The commutative property of addition can be defined as the fact that adding two integers in any sequence results in the same result. Therefore, the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

Question 7.
When three or more are grouped numbers are added, the sum is the same regardless of how the addends are grouped.
______________
When three or more are grouped numbers are added, the sum is the same regardless of how the addends are grouped = associative property
The associative property of addition is a mathematical statement that states that the arrangement of three or more integers does not change their sum. This means that no matter how the numbers are grouped, the sum of three or more integers remains the same.

Question 8.
The sum of any number and zero is the original number.
___________
The sum of any number and zero is the original number. = identity property
An identity in mathematics is a number, n, that results in the same number, n, when other numbers are added to it. The identity of the additive is always zero. This brings us to the identity property of addition, which simply states that when you add zero to any number, it equals the number itself.

Question 9.
a. 4 + 1o = 10 + 4 ________________
4 + 10 = 10 + 4 = commutative property
According to the commutative property of addition, the sum is unaffected by changes in the order of the numbers being added. The commutative property of addition can be defined as the fact that adding two integers in any sequence results in the same result. Therefore, the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

b. 1 + (-1) = 0 ____
1 + (-1) = 0
According to the commutative property of addition, the sum is unaffected by changes in the order of the numbers being added. The commutative property of addition can be defined as the fact that adding two integers in any sequence results in the same result. Therefore, the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

Question 10.
a. (1 + 8) + 2 = 1 + (8 + 2) _________
(1 + 8) + 2 = 1 + (8 + 2) = associative property
The associative property of addition is a mathematical statement that states that the arrangement of three or more integers does not change their sum. This means that no matter how the numbers are grouped, the sum of three or more integers remains the same.

b. 3 + 5 = 5 + 3 _____
3 + 5 = 5 + 3 = commutative property
According to the commutative property of addition, the sum is unaffected by changes in the order of the numbers being added. The commutative property of addition can be defined as the fact that adding two integers in any sequence results in the same result. Therefore, the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

Question 11.
a. 8 + 0 = 8 _____
8 + 0 = 8 = identity property
An identity in mathematics is a number, n, that results in the same number, n, when other numbers are added to it. The identity of the additive is always zero. This brings us to the identity property of addition, which simply states that when you add zero to any number, it equals the number itself.

b. 2 + (6 + 4) = (2 + 6) + 4 _____
2 + (6 + 4) = (2 + 6) + 4 = associative property
The associative property of addition is a mathematical statement that states that the arrangement of three or more integers does not change their sum. This means that no matter how the numbers are grouped, the sum of three or more integers remains the same.

Question 12.
a. 12 + 9 = 9 + 12 _____
12 + 9 = 9 + 12 = commutative property
According to the commutative property of addition, the sum is unaffected by changes in the order of the numbers being added. The commutative property of addition can be defined as the fact that adding two integers in any sequence results in the same result. Therefore, the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

b. (8 + 5) + 3 = 8 + (5 + 3) ________
(8 + 5) + 3 = 8 + (5 + 3) = associative property
The associative property of addition is a mathematical statement that states that the arrangement of three or more integers does not change their sum. This means that no matter how the numbers are grouped, the sum of three or more integers remains the same.

Add or subtract. Write fractions in simplest form.

Question 13.
a.

Answer: 1$$\frac{61}{56}$$
$$\frac{3}{8}$$ + 1$$\frac{5}{7}$$
Partition the fractions and whole numbers to add them separately.
= (0 + 1) + $$\frac{3}{8}$$ + $$\frac{5}{7}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 1 + [$$\frac{3}{8}$$ x $$\frac{7}{7}$$] + [$$\frac{5}{7}$$  x $$\frac{8}{8}$$]
= 1 + $$\frac{21}{56}$$ + $$\frac{40}{56}$$
= 1 + $$\frac{21 + 40}{56}$$
After simplification,
= 1 + $$\frac{61}{56}$$
Therefore, the result is given by,
= 1$$\frac{61}{56}$$

b.

Answer: 5$$\frac{7}{12}$$
2$$\frac{1}{4}$$ + 3$$\frac{1}{3}$$
Partition the fractions and whole numbers to add them separately.
= (2 + 3) + $$\frac{1}{4}$$ + $$\frac{1}{3}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 5 + [$$\frac{1}{4}$$ x $$\frac{3}{3}$$] + [$$\frac{1}{3}$$  x $$\frac{4}{4}$$]
= 5 + $$\frac{3}{12}$$ + $$\frac{4}{12}$$
= 5 + $$\frac{3 + 4}{12}$$
After simplification,
= 5 + $$\frac{7}{12}$$
Therefore, the result is given by,
= 5$$\frac{7}{12}$$

c.

Answer: 3$$\frac{41}{24}$$
1$$\frac{5}{6}$$ + 2$$\frac{7}{8}$$
Partition the fractions and whole numbers to add them separately.
= (1 + 2) + $$\frac{5}{6}$$ + $$\frac{7}{8}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 3 + [$$\frac{5}{6}$$ x $$\frac{8}{8}$$] + [$$\frac{7}{8}$$  x $$\frac{6}{6}$$]
= 3 + $$\frac{40}{48}$$ + $$\frac{42}{48}$$
= 3 + $$\frac{40 + 42}{48}$$
After simplification,
= 3 + $$\frac{82}{48}$$
Therefore, the result is given by,
= 3$$\frac{41}{24}$$

d.

Answer: 6$$\frac{9}{8}$$
4$$\frac{3}{4}$$ + 2$$\frac{3}{8}$$
Partition the fractions and whole numbers to add them separately.
= (4 + 2) + $$\frac{3}{4}$$ + $$\frac{3}{8}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 6 + [$$\frac{3}{4}$$ x $$\frac{8}{8}$$] + [$$\frac{3}{8}$$  x $$\frac{4}{4}$$]
= 6 + $$\frac{24}{32}$$ + $$\frac{12}{32}$$
= 6 + $$\frac{24 + 12}{32}$$
After simplification,
= 6 + $$\frac{36}{32}$$
Therefore, the result is given by,
= 6$$\frac{9}{8}$$

Question 14.
a.

Answer: 3$$\frac{5}{12}$$
4$$\frac{2}{3}$$ – 1$$\frac{1}{4}$$
Partition the fractions and whole numbers to subtract them separately.
= (4 – 1) + [$$\frac{2}{3}$$ – $$\frac{1}{4}$$]
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 3 + [$$\frac{2}{3}$$ x $$\frac{4}{4}$$] – [$$\frac{1}{4}$$  x $$\frac{3}{3}$$]
= 3 + $$\frac{8}{12}$$ – $$\frac{3}{12}$$
= 3 + $$\frac{8 – 3}{12}$$
After simplification,
= 3 + $$\frac{5}{12}$$
Therefore, the result is given by,
= 3$$\frac{5}{12}$$

b.

Answer: $$\frac{7}{16}$$
$$\frac{7}{8}$$ – $$\frac{1}{2}$$
Partition the fractions and whole numbers to subtract them separately.
= $$\frac{7}{8}$$ – $$\frac{1}{2}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= [$$\frac{7}{8}$$ x $$\frac{2}{2}$$] – [$$\frac{1}{2}$$  x $$\frac{8}{8}$$]
= $$\frac{14}{16}$$ – $$\frac{7}{16}$$
= $$\frac{14 – 7}{16}$$
After simplification,
= $$\frac{7}{16}$$

c.

Answer: 2$$\frac{31}{70}$$
4$$\frac{3}{10}$$ – 1$$\frac{6}{7}$$
Partition the fractions and whole numbers to subtract them separately.
= (4 – 1) + [$$\frac{3}{10}$$ – $$\frac{6}{7}$$]
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 3 + [$$\frac{3}{10}$$ x $$\frac{7}{7}$$] – [$$\frac{6}{7}$$  x $$\frac{10}{10}$$]
= 3 + $$\frac{21}{70}$$ – $$\frac{60}{70}$$
= 2 + $$\frac{91}{70}$$ – $$\frac{60}{70}$$
= 2 + $$\frac{91 – 60}{70}$$
After simplification,
= 2 + $$\frac{31}{70}$$
Therefore, the result is given by,
= 2$$\frac{31}{70}$$

d.

Answer: 3$$\frac{13}{12}$$
5$$\frac{1}{4}$$ – 2$$\frac{5}{6}$$
Partition the fractions and whole numbers to subtract them separately.
= (5 – 2 ) + $$\frac{1}{4}$$ – $$\frac{5}{6}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 3 + [$$\frac{1}{4}$$ x $$\frac{6}{6}$$] – [$$\frac{5}{6}$$  x $$\frac{4}{4}$$]
= 3 + $$\frac{6}{24}$$ – $$\frac{20}{24}$$
= 3 + $$\frac{6 + 20}{24}$$
After simplification,
= 3 + $$\frac{26}{24}$$
Therefore, the result is given by,
= 3$$\frac{13}{12}$$

Question 15.
a. -6 + 4 = ____
-6 + 4 = -6 – (-4) = -2
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

b. 7 + (-3) = ____
7 + (-3) = 7 – 3 = 4
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

c. -5 + (-2) = ____
-5 + (-2) = – 5 -2 = -7
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

Question 16.
a. -9 + 12 = ___
-9 + 12 = -9 – (-12) = 3
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

b. 8 + (-11) = ____
8 + (-11) = 8 – 11 = -3
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

c. -4 + (-8) = ____
-4 + (-8) = -4  – 8 = -12
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

Question 17.
a. 13 – 16 = ____
13 – 16 = 13 + (-16) = -3
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

b. 9 – (-8) = ___
9 – (-8) = 9 + 8 = 17
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

c. -3 – 7 = ____
-3 – 7 = -3 + (-7) = -10
Subtraction is the same as the process of adding the additive inverse, or opposite, of a number to another number.

Solve each problem.

Question 18.
A large patio brick weighs 4$$\frac{3}{8}$$ pounds. A small patio brick weighs 2$$\frac{1}{3}$$ pounds. How much more does the large brick weigh?
The large brick weighs ____ pounds more.
Answer: 2$$\frac{17}{24}$$
A large patio brick weighs 4$$\frac{3}{8}$$ pounds
A small patio brick weighs 2$$\frac{1}{3}$$ pounds
The more does the large brick weigh = large patio brick – small patio brick
= 4$$\frac{3}{8}$$ – 2$$\frac{1}{3}$$
4$$\frac{3}{8}$$ – 2$$\frac{1}{3}$$
Partition the fractions and whole numbers to subtract them separately.
= (4 – 2) + $$\frac{3}{8}$$ – $$\frac{1}{3}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 2 + [$$\frac{3}{8}$$ x $$\frac{3}{3}$$] – [$$\frac{1}{3}$$  x $$\frac{8}{8}$$]
= 2 + $$\frac{9}{24}$$ – $$\frac{8}{24}$$
= 2 + $$\frac{9 + 8 }{24}$$
After simplification,
=2 + $$\frac{17}{24}$$
Therefore, the result is given by,
= 2$$\frac{17}{24}$$
Therefore, the large brick weighs 2$$\frac{17}{24}$$pounds more.

Question 19.
A small bottle holds $$\frac{1}{3}$$ of a liter. A large bottle holds 4$$\frac{1}{2}$$ liters. How much more does the large bottle hold?
The large bottle holds ____ liters more.
Answer: 4$$\frac{1}{6}$$
A small bottle holds $$\frac{1}{3}$$ of a liter
A large bottle holds 4$$\frac{1}{2}$$ liters
The number of more does the large bottle hold = 4$$\frac{1}{2}$$ – $$\frac{1}{3}$$
4$$\frac{1}{2}$$ – $$\frac{1}{3}$$
Partition the fractions and whole numbers to subtract them separately.
= (4 – 0) + $$\frac{1}{2}$$ – $$\frac{1}{3}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 4 + [$$\frac{1}{2}$$ x $$\frac{3}{3}$$] – [$$\frac{1}{3}$$  x $$\frac{2}{2}$$]
= 4 + $$\frac{3}{6}$$ – $$\frac{2}{6}$$
= 4 + $$\frac{3 – 2 }{6}$$
After simplification,
=4 + $$\frac{1}{6}$$
Therefore, the result is given by,
= 4$$\frac{1}{6}$$
The large bottle holds  4$$\frac{1}{6}$$ liters more.

Question 20.
The basketball team practiced 3$$\frac{1}{4}$$ hours on Monday and 2$$\frac{1}{3}$$ hours on Tuesday. How many hours has the team practiced so far this week?
The team has practiced ____ hours this week.
Answer: 5$$\frac{7}{24}$$
The basketball team practiced 3$$\frac{1}{4}$$ hours on Monday and 2$$\frac{1}{3}$$ hours on Tuesday. Therefore, total hours practiced by team so far this week = number of hours practiced on monday + number of hours practiced on tuesday
=  3$$\frac{1}{4}$$ + 2$$\frac{1}{3}$$
Partition the fractions and whole numbers to add them separately.
= (3 + 2) + $$\frac{1}{4}$$ + $$\frac{1}{3}$$
To add fractions or mixed numbers when the denominators are different, rename the fractions in such a way by multiplying denominator of one fraction with another, to make the denominators same.
= 5 + [$$\frac{1}{4}$$ x $$\frac{3}{3}$$] + [$$\frac{1}{3}$$  x $$\frac{4}{4}$$]
= 5 + $$\frac{3}{24}$$ + $$\frac{4}{24}$$
= 5 + $$\frac{3 + 4 }{24}$$
After simplification,
=5 + $$\frac{7}{24}$$
Therefore, the result is given by,
= 5$$\frac{7}{24}$$