This handy Spectrum Math Grade 7 Answer Key Chapter 3 Lesson 3.2 Solving Problems with Equivalent Expressions provides detailed answers for the workbook questions.
Spectrum Math Grade 7 Chapter 3 Lesson 3.2 Solving Problems with Equivalent Expressions Answers Key
Sometimes, it is easier to solve equations by writing them in different ways.
A number increased by 10% can be written as:
- n + (0.10 × n)
- 1.10 × n
A number divided by 7 equals 3 can be written as:
- n ÷ 7 = 3
- 3 × 7 = n
Write two equivalent expressions for each statement.
Question 1.
a. a number decreased by 7%
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Answer: n – (0.07 × n) = 0.93 × n
a number decreased by 7% can be written as n – (0.07 × n)
By taking n common in the above expression, it will become n[1 – 0.07 × 1] = n [0.93]
Therefore, n – (0.07 × n) = 0.93 × n
b. 9 times the sum of 7 and a number
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Answer: 9 × (7 + n)= (9 × 7) + (9 × n) = 63 + (9 × n)
9 times the sum of 7 and a number can be written as 9 × (7 + n)
The above expression can be given as (9 × 7) + (9 × n) = 63 + (9 × n) according to distributive property
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.
Therefore, 9 × (7 + n) = (9 × 7) + (9 × n) = 63 + (9 × n)
Question 2.
a. $25 plus a 5% tip
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Answer: 25 + (0.05 × 25) = 26.25
$25 plus a 5% tip can be written as 25 + (0.05 × 25)
as 5% is 0.05,
Now solving the above expression, 25 + (0.05 × 25) = 25 + (1.25) = 26.25
Therefore, 25 + (0.05 × 25) = 26.25
b. the sum of a number and 4 times the number
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Answer: n + (4 × n) = 5n
the sum of a number and 4 times the number can be written as n + (4 × n)
By taking n common in the above expression, it will become n[1 + 4 × 1] = n [5]
Therefore, n + (4 × n) = 5n
Question 3.
a. number divided by 5 equals 9
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Answer: n ÷ 5 = 9 implies n = 5 × 9 = 45
number divided by 5 equals 9 can be written as n ÷ 5 = 9
If we take 5 to the other side, the division becomes multiplication.
which is n = 5 × 9 = 45
Therefore, n ÷ 5 = 9 implies n = 5 × 9 = 45
b. a number increased by \(\frac{1}{5}\)
Answer: n + \(\frac{1}{5}\) = n + 0.2
a number increased by \(\frac{1}{5}\) can be written as n + \(\frac{1}{5}\)
\(\frac{1}{5}\) can be represented as 0.2 in decimals
So, n + \(\frac{1}{5}\) = n + 0.2
Question 4.
a. 12 times the difference of 15 and a number
Answer: 12 × (15 – n) = (12 × 15) – (12× n) = 180 – (12× n)
12 times the difference of 15 and a number can be written as 12 × (15 – n)
The above expression can be given as (12 × 15) – (12× n) = 180 – (12× n) according to distributive property
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.
Therefore, 12 × (15 – n) = (12 × 15) – (12× n) = 180 – (12× n)
b. $44 plus a 20% tip
Answer: 44 + (0.2 × 44) = 52.8
$44 plus a 20% tip can be written as 44 + (0.2 × 44)
Now solving the above expression, 44 + (0.2 × 44) = 44 + (8.8) = 52.8
Therefore, 44 + (0.2 × 44) = 52.8
Question 5.
a. the sum of 7 and a number times 10
Answer: (7 + n ) × 10 = (7 × 10) + (n × 10) = 70 + (10n)
the sum of 7 and a number times 10 can be written as (7 + n ) × 10
The above expression can be given as (7 × 10) + (n × 10) = 70 + (10n) according to distributive property
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.
Therefore, (7 + n ) × 10 = (7 × 10) + (n × 10) = 70 + (10n)
b. a number decreased by 3\(\frac{1}{4}\)
Answer: n – 3\(\frac{1}{4}\) = n – 3.25
a number decreased by 3\(\frac{1}{4}\) can be written as n – 3\(\frac{1}{4}\)
3\(\frac{1}{4}\) = \(\frac{13}{4}\) in improper fraction and 3.25 in decimals
So, n – 3\(\frac{1}{4}\) = n – 3.25