Middle School Math Quiz: Rewrite Expressions In Different Forms

20 questions · exam conditions

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Rewrite Expressions In Different FormsQuestion 1 of 20

The cost to rent a car is $25 + 0.15m$ dollars, where $m$ is miles driven. A customer wants to spend exactly $40. Which form of the equation $ $25 + 0.15m = 40$ $ makes it easiest to see how many miles they can drive?

0.15m = 15

because it isolates the variable term from the constant

m = \frac{40 - 25}{0.15}

because it shows the calculation steps needed

25 = 40 - 0.15m

because it shows the base cost equals remaining budget

\frac{25 + 0.15m}{40} = 1

because it shows the ratio of cost to budget

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Middle School Math Quiz

Middle School Math Quiz: Rewrite Expressions In Different Forms

Practice Rewrite Expressions In Different Forms in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Rewrite Expressions In Different Forms, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$0.15m = 15$ because it isolates the variable term from the constant
$m = \frac{40 - 25}{0.15}$ because it shows the calculation steps needed (correct answer)
$25 = 40 - 0.15m$ because it shows the base cost equals remaining budget
$\frac{25 + 0.15m}{40} = 1$ because it shows the ratio of cost to budget

Explanation: The form

m = \frac{40 - 25}{0.15}

directly shows how to calculate the miles: take the budget (

40), subtract the base fee (

25), then divide by the per-mile rate ($0.15). This makes the solution process transparent. Choice A requires another step to solve for $

m

$. Choice C doesn't directly show the miles. Choice D creates unnecessary complexity with ratios.

Question 2

A rectangular garden has area $6x^2 + 9x$ square feet. Which factored form reveals the most useful information about possible dimensions?

$3x(2x + 3)$ because it shows one side is $3x$ and the other is $2x + 3$ (correct answer)
$x(6x + 9)$ because it shows one side is $x$ and the other is $6x + 9$
$3(2x^2 + 3x)$ because it shows the area is 3 times a simpler expression
$6x(x + 1.5)$ because it shows one side is $6x$ and the other is $x + 1.5$

Explanation: Factoring out the greatest common factor:

6x^2 + 9x = 3x(2x + 3)

. This form shows the dimensions as

3x

feet by

(2x + 3)

feet, both of which are reasonable expressions for side lengths. Choice B factors out only

x

, not the complete GCF. Choice C factors out 3 but doesn't reveal dimensional information. Choice D uses a decimal coefficient, which is less clean than the integer form.

Question 3

The temperature in degrees Fahrenheit can be converted to Celsius using $C = \frac{5}{9}(F - 32)$ . Which equivalent expression shows how much the Celsius temperature changes when Fahrenheit increases by 9 degrees?

$C = \frac{5F}{9} - \frac{160}{9}$ to show the linear relationship with slope $\frac{5}{9}$
$C = \frac{5F - 160}{9}$ to show the numerator contains both variable and constant terms
$9C = 5F - 160$ to show that 9-degree Fahrenheit changes equal 5-degree Celsius changes (correct answer)
$\frac{9C}{5} = F - 32$ to show the inverse relationship between the temperature scales

Explanation: The form

9C = 5F - 160

clearly shows that when

F

increases by 9, the term

5F

increases by

5 \times 9 = 45

, so

9C

increases by 45, meaning

C

increases by 5. This directly reveals the 9:5 ratio. Choice A shows the slope but not the specific 9-degree relationship. Choice B doesn't highlight the ratio. Choice D shows the inverse but doesn't emphasize the change relationship.

Question 4

A class is collecting cans. Last week they collected $p$ cans, but this week they collected 20% fewer. The amount this week is $p - 0.20p$ . Which equivalent expression makes it easiest to find the new amount?

$0.80p$ (correct answer)
$p - 20$
$1.20p$
$0.20p$

0.80p

shows equivalence by combining like terms p - 0.20p = 0.80p, makes it easiest to compute as 80% of p, and reveals the decrease relationship clearly. Common errors include

1.20p

confusing decrease with increase,

0.20p

which is only the reduction, or

p - 20

subtracting a flat 20 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 5

A snack costs $p$ dollars. The store adds 8% sales tax, so the total cost is $p+0.08p$ . Which equivalent expression shows the total as one multiplication?

$8p$
$1.08p$ (correct answer)
$p(0.08)$
$p+0.8$

Explanation: This question tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, a $50 item with 8% tax is 50 + 0.08(50) = 50 + 4 = 54, or rewritten as 50(1 + 0.08) = 50(1.08) = 54, showing the total as 108% of original. The correct rewriting is p + 0.08p = 1.08p, which shows equivalence by combining like terms and reveals the utility of one multiplication for total cost including tax. A common error is p(0.08) only the tax amount, or p + 0.8 adding flat 0.8, or 8p multiplying by 8 incorrectly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 6

A video game has a 25% off sale. If the original price is $p$ , the sale price is $p-0.25p$ . Which rewrite shows the sale price as "multiply by the part that remains," so it's quicker to compute?

$0.75p$ (correct answer)
$1.25p$
$p-25$
$0.25p$

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a video game price p discounted by 25%, the sale price p - 0.25p can be factored as p(1 - 0.25) = 0.75p, showing it's equivalent to multiplying by the remaining 75% for quicker computation. The correct rewriting is 0.75p, which shows equivalence through factoring and highlights the utility of multiplying by the portion that remains after the discount. A common error is selecting 1.25p, which would be for a 25% increase instead of decrease (sign error), or 0.25p which is just the discount amount, forgetting the original minus that (should be 0.75p), or p-25 which subtracts a flat 25 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if p=100, does 0.75(100)=100-0.25(100)? 75=75✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Question 7

A notebook costs $6. A store is having a 25% off sale. The sale price can be written as $6 - 0.25(6)$ . Which equivalent expression shows the discount as "multiply by the part you pay" and makes it easier to calculate?

$6(1.25)$
$6(0.25)$
$6(0.75)$ (correct answer)
$6 - 25$

6 - 25

which subtracts a flat 25 incorrectly ignoring percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 8

A school club buys 12 identical water bottles for a trip and 8 more for new members. Each bottle costs $3.50$ . The total cost is $3.5\times 12 + 3.5\times 8$ . Which rewrite makes the calculation easiest?

$3.5(12-8)$
$3.5\times (12\times 8)$
$3.5(12+8)$ (correct answer)
$(3.5\times 12)\times 8$

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: cost of 12 items at $3 each plus 8 at $3: 3×12 + 3×8=3(12+8)=3(20)=60 (factoring 3 simplifies); or 3.5×12 + 3.5×8=3.5(20)=70 (rewrite makes addition easy first). The correct rewriting 3.5(12+8) shows equivalence by factoring out 3.5 using distributive property, makes calculation easiest by adding 12+8=20 then 3.5×20=70, and is useful for mental math. Common errors include 3.5(12-8) changing to subtraction, (3.5×12)×8 which associates differently and complicates, or 3.5×(12×8) which multiplies counts first incorrectly for total cost. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 9

A video game has a score bonus that adds 5% more points to your current score $s$ . The new score is written as $s + 0.05s$ . Which rewrite best shows the relationship as one multiplication?

$5s$
$1.05s$ (correct answer)
$0.05s$
$s + 5$

1.05s

shows equivalence by factoring out s to combine terms as s(1+0.05), reveals the relationship as a single multiplication for the total score after bonus, and simplifies computation. Common errors include

0.05s

forgetting the original score,

5s

misinterpreting percent as multiplying by 5, or

s + 5

adding a flat 5 instead of 5%. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 10

A science club's supply amount increases by 20%. If the original amount is $a$ , the new amount is $a+0.2a$ . Which equivalent expression correctly shows this as a single multiplication and explains the relationship between adding 20% and multiplying?

$a(0.2)$
$2a$
$1.2a$ (correct answer)
$0.8a$

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a supply amount a increased by 20%, the new amount a + 0.2a can be factored as a(1 + 0.2) = 1.2a, explaining that adding 20% is equivalent to multiplying by 1.2. The correct rewriting is 1.2a, which shows equivalence by combining terms and reveals the multiplication relationship for percentage increase. A common error is choosing a(0.2) which is only the increase (forgetting original), or 0.8a which is for decrease, or 2a which doubles instead of adding 20%. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in a=10: 1.2(10)=12, 10+0.2(10)=10+2=12✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Question 11

A rectangle has length $l$ and width $w$ . Its perimeter is written as $2l+2w$ . Which equivalent expression best shows the relationship "perimeter is twice the sum of length and width"?

$(2l)(2w)$
$2(l+w)$ (correct answer)
$l+w+2$
$2(l-w)$

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, a rectangle's perimeter 2l + 2w can be factored as 2(l + w), clearly showing it's twice the sum of length and width. The correct rewriting is 2(l + w), which demonstrates equivalence by factoring out 2 and reveals the geometric relationship effectively. A common error is selecting (2l)(2w) which multiplies to area instead of perimeter, or l + w + 2 which adds an extra 2 incorrectly, or 2(l - w) which subtracts leading to difference. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in l=5, w=3: 2(5+3)=16, 2(5)+2(3)=10+6=16✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Question 12

A school store increases the price of a notebook by 5%. The new price can be written as $p+0.05p$ . Which equivalent expression shows this as "multiply the original price by one number," making it easier to calculate?

$p(0.05)$
$0.05p$
$5p$
$1.05p$ (correct answer)

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a notebook price p increased by 5%, the new price p + 0.05p can be factored as p(1 + 0.05) = 1.05p, showing it's equivalent to multiplying the original price by 1.05 for quick calculation. The correct rewriting is 1.05p, which shows equivalence by combining like terms and reveals the utility of a single multiplication for efficiency. A common error is choosing 0.05p or p(0.05), which only represents the increase amount, forgetting to include the original p (should be 1.05p), or mistaking it for 5p which multiplies by 5 instead of 1.05. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if p=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Question 13

A science kit costs $40. The store marks it up by 30%, so the new price is $40 + 0.3(40)$ . Which rewrite shows the price as "multiply by a single factor" and helps you compute quickly?

$40(1.3)$ (correct answer)
$0.7(40)$
$40(0.3)$
$40 + 0.3$

7 \times 23 + 7 \times 77

7(23+77)=7(100)=700

easier mentally), (2) reveal relationship (

a + 0.05a = a(1+0.05) = 1.05a

shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (

p + 0.08p = 1.08p

shows total with 8% tax). Apply distributive

a(b+c)=ab+ac

both directions: expanding (multiply out) or factoring (pull out common factor). For example, a $50 item with 30% markup is

50 + 0.3(50) = 50 + 15 = 65

, or rewritten as

50(1 + 0.3) = 50(1.3) = 65

, showing quick multiplication by 1.3. The correct rewriting is

40 + 0.3(40) = 40(1.3)

, which shows equivalence by factoring out 40 and helps compute quickly as a single multiplication factor. A common error is

40(0.3)

only the markup, or

0.7(40)

confusing with discount, or

40 + 0.3

adding flat 0.3. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does

1.05(10)=10+0.05(10)

10.5=10.5\checkmark

), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by

(1+r)

as decimal (increase by 15% →

\times 1.15

), decrease by r% means multiply by

(1-r)

(decrease by 20% →

\times 0.80

). Mistakes: forgetting original amount (

a+0.1a \neq 0.1a

=1.1a

), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 14

A video game item costs $p$ coins. During a sale, the price is reduced by 25%, so the sale price is $p-0.25p$ . Which rewrite shows the sale price as a single multiplication (and makes the meaning clearer)?

$0.75p$ (correct answer)
$p-25$
$1.25p$
$0.25p$

Explanation: This question tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, if an item costs 100 coins with 25% off, it's 100 - 0.25(100) = 100 - 25 = 75, or rewritten as 100(1 - 0.25) = 100(0.75) = 75, showing multiplication by 0.75 for the discount. The correct rewriting is p - 0.25p = 0.75p, which shows equivalence by factoring out p and clarifies the sale price as 75% of original, useful for quick mental math. A common error is choosing 1.25p, confusing decrease with increase, or 0.25p only the discount amount, or p - 25 subtracting a flat 25 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 15

A student is doing mental math to find 15% of 80. Which rewrite makes the calculation easiest without a calculator?

$15\times 80$
$80\div 15$
$0.15\times 80$
$(10\%+5\%)\times 80=8+4$ (correct answer)

Explanation: This question tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, 15% of 100 is 0.15(100)=15, or rewritten as (10% + 5%)(100)=10 + 5=15, breaking it into easier percentages for mental math. The correct rewriting is (10% + 5%) × 80 = 8 + 4, which shows equivalence by decomposing 15% and makes calculation easiest by adding simple amounts without a calculator. A common error is 0.15 × 80, which is correct but not the rewritten form for ease, or 15 × 80=1200 way off, or 80 ÷ 15 confusing with division. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 16

A school club has $a$ members. If the club gains 5% more members, the new number of members can be written as $a+0.05a$ . Which equivalent expression shows that a 5% increase is the same as multiplying by one number (and makes it easier to calculate)?

$0.05a$
$a+5a$
$1.5a$
$1.05a$ (correct answer)

Explanation: This question tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, if a club has 100 members, a 5% gain is 100 + 0.05(100) = 100 + 5 = 105, or rewritten as 100(1 + 0.05) = 100(1.05) = 105, showing the multiplication by 1.05 directly. The correct rewriting is a + 0.05a = 1.05a, which shows equivalence by factoring out a and reveals the utility of multiplying by 1.05 for quick calculation of percentage increases. A common error is choosing 0.05a, forgetting the original a and only accounting for the increase, or 1.5a confusing 5% with 50%, or a + 5a which is 6a, misinterpreting 5% as 5 times. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Question 17

A gym charges a monthly fee that can be expressed as $45 + 0.1(m-100)$ dollars, where $m$ is the number of minutes used. A second expression for the same plan is $45+0.1m-10$ . For which values of $m$ are these two expressions equal?

Only when $m=100$ , since that is when the expression's parentheses become zero
Only when $m=0$ , since that is when both expressions have no minutes-based charge
For all values of $m$ , since the two expressions simplify to the same expression (correct answer)
Only when $m=200$ , since that produces the same total fee under both formulas

Explanation: Expand the first expression: 45+0.1(m-100)=45+0.1m-10=35+0.1m. The second expression, 45+0.1m-10, also simplifies to 35+0.1m. Since both expressions simplify to the exact same expression, they are equal for every value of m, not just one specific value, matching choice C. Choice A picks out one specific value where the parentheses happen to equal zero, but the expressions are equal for every value of m, not just that one. Choice B and choice D each pick another single value where the expressions happen to match numerically, which is true for every value of m, not just those specific ones.

Question 18

A library charges $c$ dollars per bookmark, and a student buys 4 bookmarks. The total cost is written as $c+c+c+c$ . Which expression is an equivalent rewrite that shows the total as "4 times $c$ "?

$4c$ (correct answer)
$c^4$
$c/4$
$4+c$

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, buying 4 bookmarks at c dollars each, the total c + c + c + c can be combined as 4c, showing it's 4 times the cost per bookmark. The correct rewriting is 4c, which demonstrates equivalence by combining like terms and clearly shows the multiplication relationship. A common error is selecting c^4 which exponents instead of multiplies, or 4 + c which adds instead, or c/4 which divides wrongly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in c=2: 4(2)=8, 2+2+2+2=8✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Question 19

A snack pack costs $40 and the store adds a 30% markup. The selling price is $40+0.3(40)$ . Which equivalent expression is the most efficient rewrite for calculating the selling price?

$0.3(80)$
$40(0.3)$
$40(1.3)$ (correct answer)
$40+0.3$

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a snack pack at $40 with 30% markup, the selling price 40 + 0.3(40) can be rewritten as 40(1 + 0.3) = 40(1.3), making it efficient to calculate as a single multiplication. The correct rewriting is 40(1.3), which demonstrates equivalence by factoring out 40 and shows the utility for quick mental math on the marked-up price. A common error is choosing 40(0.3) which is only the markup amount (forgetting the original), or 0.3(80) which equals the markup but not the total, or 40+0.3 which adds a flat 0.3 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: 40(1.3)=52, 40+0.3(40)=40+12=52✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Question 20

A rectangle has length $l$ and width $w$ . Its perimeter can be written as $2l + 2w$ . Which rewrite shows the meaning more clearly as "twice the sum of length and width"?

$l + w$
$(2l)(2w)$
$2(l-w)$
$2(l+w)$ (correct answer)

Explanation: Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: perimeter of rectangle with sides 5 and 3 is 2×5 + 2×3=2(5+3)=2(8)=16 (factoring 2 shows twice the sum); or length l width w: 2l + 2w=2(l+w) (rewrite clarifies the structure). The correct rewriting 2(l+w) shows equivalence by factoring out 2 from both terms, clearly reveals the meaning as twice the sum of length and width, and simplifies understanding of the perimeter formula. Common errors include l + w which omits the doubling, (2l)(2w) which multiplies instead of adding, or 2(l-w) which subtracts incorrectly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).