SHSAT Math Quiz: One And Two Step Equations

19 questions · exam conditions

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One And Two Step EquationsQuestion 1 of 19

A cell phone plan costs $30 per month plus $8 per gigabyte (GB) for data used beyond a 2 GB allowance. A 15% tax is calculated on the entire monthly bill. If a customer's total bill after tax was $50.60, how many total gigabytes of data did they use that month?

1.75 GB

2.50 GB

3.75 GB

4.25 GB

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SHSAT Math Quiz

SHSAT Math Quiz: One And Two Step Equations

Practice One And Two Step Equations in SHSAT 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 One And Two Step Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for SHSAT 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

1.75 GB
2.50 GB
3.75 GB (correct answer)
4.25 GB

Explanation: Let

d

be the number of additional gigabytes used (beyond the 2 GB allowance). The cost before tax is the base fee plus the overage charge:

C = 30 + 8d

. The total bill is this cost plus a 15% tax, which means multiplying by 1.15. So,

50.60 = 1.15(30 + 8d)

. This is a two-step equation. First, divide by 1.15:

\frac{50.60}{1.15} = 44

. So,

44 = 30 + 8d

. Next, subtract 30:

14 = 8d

. Solve for

d

d = \frac{14}{8} = \frac{7}{4} = 1.75

. This is the additional data used. The question asks for the total data, which is the 2 GB allowance plus the additional data:

2 + 1.75 = 3.75

GB. Distractor A is the amount of additional data, not the total. Distractors B and D result from common errors in calculating with percentages or setting up the initial equation.

Question 2

If $\frac{3}{5}(2x - 7) = 9$ , what is the value of $4x - 14$ ?

11
15
22
30 (correct answer)

Explanation: We can solve this problem in two ways. Method 1: Solve for

x

first. Multiply both sides of

\frac{3}{5}(2x - 7) = 9

\frac{5}{3}

2x - 7 = 9 \cdot \frac{5}{3} = 3 \cdot 5 = 15

. This is a two-step equation. Add 7 to both sides:

2x = 22

. Divide by 2:

x=11

. Now substitute this into the expression

4x - 14

4(11) - 14 = 44 - 14 = 30

. Method 2: Notice the relationship between the expressions. The expression we want to find,

4x - 14

, is exactly twice the expression inside the parentheses,

2x-7

. From the first step of Method 1, we found that

2x - 7 = 15

. Therefore,

4x-14 = 2(2x-7) = 2(15) = 30

. Distractor A is the value of

x

. Distractor B is the value of

2x-7

. Distractor C is the value of

2x

Question 3

At a certain store, the price of a jacket is reduced by 20%. A sales tax of 5% is then applied to the discounted price. If the final price after the tax is $113.40, what was the original price of the jacket?

$135.00 (correct answer)
$136.08
$140.00
$141.75

Explanation: Let

P

be the original price of the jacket. A 20% reduction means the discounted price is

100\% - 20\% = 80\%

of the original price, or

0.80P

. A 5% sales tax is applied to this discounted price, which is equivalent to multiplying by 1.05. The final price is

1.05(0.80P)

. We are given that this final price is $113.40. So,

1.05(0.80P) = 113.40

. First, multiply the decimals on the left:

1.05 \times 0.80 = 0.84

. The equation is now

0.84P = 113.40

. This is a one-step equation. To find

P

, divide both sides by 0.84:

P = \frac{113.40}{0.84} = 135

. The original price was $135.00. Distractors often come from reversing the operations: dividing by 1.05 first and then by 0.80, or incorrectly adding/subtracting percentages.

Question 4

Let the operation $\$$ be defined for any number$ n $as$ $n = \frac{2n-5}{3} $. If$ $($(p)) = p-5 $, what is the value of$ p$?

4 (correct answer)
5
6
10

Explanation: This problem involves a nested application of a defined operation. First, find an expression for

\

($(p))

.

$p = \frac{2p-5}{3}

. Now, apply the operation again to this result:

$($(p)) = $(\frac{2p-5}{3}) = \frac{2(\frac{2p-5}{3})-5}{3}

. To simplify the numerator:

\frac{2(2p-5)}{3} - \frac{15}{3} = \frac{4p-10-15}{3} = \frac{4p-25}{3}

. So,

$($(p)) = \frac{4p-25}{9}

. We are given that this equals

p-5

. So,

\frac{4p-25}{9} = p-5

. Multiply by 9:

4p-25 = 9(p-5) = 9p-45

. Add 45 to both sides:

4p+20 = 9p

. Subtract

from both sides:

20 = 5p

. Divide by 5:

p=4

. Distractor D comes from setting

$p = p-5$ instead of the nested operation. Distractor C results from an algebraic error in simplifying the nested function. Distractor B is close to the correct answer.

Question 5

A rental car company charges a flat fee of $45 plus $0.15 per mile driven. A second company charges a flat fee of $30 plus $0.20 per mile driven. For what number of miles driven is the cost from both companies exactly the same?

150 miles
250 miles
300 miles (correct answer)
450 miles

Explanation: Let

m

be the number of miles driven. The cost for the first company,

C_1

, can be modeled by the equation

C_1 = 45 + 0.15m

. The cost for the second company,

C_2

, can be modeled by

C_2 = 30 + 0.20m

. To find when the cost is the same, we set

C_1 = C_2

45 + 0.15m = 30 + 0.20m

. This is a two-step equation. First, subtract (0.15m) from both sides:

45 = 30 + 0.05m

. Next, subtract 30 from both sides:

15 = 0.05m

. To solve for

m

, divide by 0.05:

m = \frac{15}{0.05} = \frac{1500}{5} = 300

. The cost will be the same at 300 miles. The distractors represent common calculation errors, such as misplacing a decimal or subtracting costs instead of miles.

Question 6

At a school fundraiser, adult tickets cost $12 and student tickets cost $7. The total revenue from 150 tickets was $1400. To find the number of student tickets, $s$ , the equation $12(150-s) + 7s = 1400$ is used. Which of the following is the correct next step in solving this equation?

$1800 - 12s + 7s = 1400$ (correct answer)
$150 - 5s = 1400$
$12(150) - 5s = 1400$
$12(150) - 19s = 1400$

Explanation: The question asks for the correct next step in solving the given equation

12(150-s) + 7s = 1400

. The first step in simplifying this equation is to apply the distributive property to the term

12(150-s)

. This means multiplying 12 by both 150 and

-s

12 \times 150 = 1800

and

12 \times (-s) = -12s

. Substituting this back into the equation gives

1800 - 12s + 7s = 1400

. This matches choice A. Choice B incorrectly combines

-s

and

7s

before distributing. Choice C incorrectly combines

-12s

and

7s

into

-5s

without distributing the 12 to 150 first. Choice D incorrectly adds

-12s

and

-7s

instead of

7s

Question 7

A rectangle's length is $5x-4$ and its width is $2x+1$ . An equilateral triangle's side is $6x-5$ . The perimeters of the two shapes are equal. What is the area of a square whose side length is equal to the value of $x$ ?

$\frac{9}{4}$
$\frac{51}{2}$
$\frac{81}{16}$ (correct answer)
$\frac{17}{2}$

Explanation: First, set up expressions for the perimeters. The perimeter of the rectangle is

2(L+W) = 2((5x-4) + (2x+1)) = 2(7x-3) = 14x-6

. The perimeter of the equilateral triangle is

3s = 3(6x-5) = 18x-15

. Set the perimeters equal:

14x-6 = 18x-15

. This is a two-step equation. Add 15 to both sides:

14x+9 = 18x

. Subtract

14x

from both sides:

9 = 4x

, so

x = \frac{9}{4}

. The question asks for the area of a square with side length

x

. The area is

x^2 = (\frac{9}{4})^2 = \frac{81}{16}

. Distractor A is the value of

x

. Distractor B is the value of the perimeter (

14(\frac{9}{4})-6 = \frac{63}{2}-6 = \frac{51}{2}

). Distractor D is the side length of the triangle (

6(\frac{9}{4})-5 = \frac{27}{2}-5 = \frac{17}{2}

Question 8

Two cars, A and B, start at the same point and travel in the same direction. Car A travels at a constant speed of 50 miles per hour. Car B starts 30 minutes later and travels at a constant speed of 60 miles per hour. How many hours after Car A starts will Car B catch up to Car A?

1.5 hours
2.0 hours
2.5 hours
3.0 hours (correct answer)

Explanation: Let

t

be the number of hours Car A has been traveling. When Car B catches up, they will have traveled the same distance. The distance for Car A is

D_A = 50t

. Car B starts 30 minutes (0.5 hours) later, so its travel time is

t - 0.5

. The distance for Car B is

D_B = 60(t - 0.5)

. Set the distances equal:

50t = 60(t - 0.5)

. Distribute on the right side:

50t = 60t - 30

. This is a two-step equation. Subtract

60t

from both sides:

-10t = -30

. Divide by -10:

t = 3

. So, Car B will catch up to Car A 3 hours after Car A starts. Distractor C is the time that Car B has been traveling (

3 - 0.5 = 2.5

hours). Distractors A and B result from errors in setting up the time relationship, for example using

t+0.5

or making a calculation mistake.

Question 9

A number is tripled, then 8 is subtracted from the result. If this final result equals the original number increased by 4, what was the original number?

2
4
6 (correct answer)
8

Explanation: Let x = the original number. The equation is: 3x - 8 = x + 4. Solving: 3x - x = 4 + 8 → 2x = 12 → x = 6. Verify: Triple 6 gives 18, subtract 8 gives 10. Original number plus 4: 6 + 4 = 10 ✓. Choice A results from solving 2x = 4 instead of 2x = 12. Choice B results from confusing the final result with the original number. Choice D results from the value 4 + 8 = 12, but forgetting to divide by 2.

Question 10

The sum of three consecutive integers is 54. What is the largest of these three integers?

17
18
19 (correct answer)
20

Explanation: Let the three consecutive integers be n, n+1, and n+2. Their sum is: n + (n+1) + (n+2) = 54 → 3n + 3 = 54 → 3n = 51 → n = 17. Therefore, the three integers are 17, 18, and 19, making the largest one 19. Choice A gives the smallest integer instead of the largest. Choice B gives the middle integer. Choice D results from setting up the equation as 3n = 54, giving n = 18, then incorrectly adding 2 to get the largest.

Question 11

Solve for $y$ in $3y = -12$ .

-4 (correct answer)
4
-9
36

Explanation: When you encounter a linear equation like

3y = -12

, you're dealing with one of the most fundamental algebra skills: isolating a variable. The goal is to get

y

by itself on one side of the equation. To solve

3y = -12

, you need to undo the multiplication of

y

by 3. Since multiplication and division are inverse operations, divide both sides by 3:

\frac{3y}{3} = \frac{-12}{3}

The 3's cancel on the left side, leaving

y = -4

. Let's examine why the other answers are incorrect: A) -4 is correct, as shown above. B) 4 represents a sign error. You might get this if you incorrectly think that dividing a negative number by a positive number gives a positive result. Remember:

\frac{-12}{3} = -4

, not

+4

. C) -9 suggests you subtracted 3 from -12 instead of dividing by 3. This shows confusion between the operations needed to solve

3y = -12

versus

y + 3 = -12

. D) 36 indicates you multiplied both sides by 3 instead of dividing:

3y \times 3 = -12 \times 3

, giving

9y = -36

. This moves you further from the solution rather than closer. Strategy tip: Always verify your answer by substituting back into the original equation. Here:

3(-4) = -12

✓. This quick check catches sign errors and operation mistakes, which are the most common traps in basic equation solving.

Question 12

What value of $x$ satisfies the equation $x + 7 = 15$ ?

8 (correct answer)
22
-8
108

Explanation: When you encounter a linear equation like

x + 7 = 15

, you're working with one of the most fundamental concepts in algebra: isolating the variable. Your goal is to get

x

by itself on one side of the equation. To solve this, you need to "undo" what's being done to

x

. Since 7 is being added to

x

, you subtract 7 from both sides to maintain the equation's balance:

x + 7 = 15

x + 7 - 7 = 15 - 7

x = 8

You can verify this by substituting back:

8 + 7 = 15

✓ Looking at the wrong answers reveals common mistakes: Choice B (22) comes from adding instead of subtracting:

15 + 7 = 22

. This happens when students forget they need to do the opposite operation to isolate the variable. Choice C (-8) results from subtracting 7 from the left side but forgetting to subtract it from the right side, or from incorrectly thinking

x = 7 - 15

. Choice D (108) seems to come from an entirely different calculation, possibly multiplying the numbers instead of using proper algebraic steps. Study tip: Always remember the golden rule of equations: whatever you do to one side, you must do to the other. When solving

x + a = b

, subtract

a

from both sides to get

x = b - a

. Always check your answer by substituting it back into the original equation.

Question 13

The sum of one-third of a number $n$ and seven is equal to the difference when 1 is subtracted from one-half of the same number. What is the value of $n$ ?

42
48 (correct answer)
54
60

Explanation: Translate the words into a mathematical equation. 'One-third of a number

n

and seven' is

\frac{1}{3}n + 7

. 'The difference when 1 is subtracted from one-half of the same number' is

\frac{1}{2}n - 1

. Set these two expressions equal:

\frac{1}{3}n + 7 = \frac{1}{2}n - 1

. To solve this equation, first eliminate the fractions by multiplying the entire equation by the least common multiple of 3 and 2, which is 6.

6(\frac{1}{3}n) + 6(7) = 6(\frac{1}{2}n) - 6(1)

, which simplifies to

2n + 42 = 3n - 6

. This is a two-step equation. Subtract

2n

from both sides:

42 = n - 6

. Add 6 to both sides:

48 = n

. The value of

n

is 48. The distractors come from common errors, such as reversing the subtraction (

1 - \frac{1}{2}n

) or making sign errors when clearing fractions.

Question 14

The average (arithmetic mean) of three numbers, $a$ , $b$ , and $c$ , is $M$ . The average of $a$ and $b$ is $N$ . If $c = \frac{1}{2}M$ , what is $N$ in terms of $M$ ?

$M(5/4)$ (correct answer)
$M(3/2)$
$M - 1/4$
$M/2$

Explanation: From the given information, we can write two equations. First,

\frac{a+b+c}{3} = M \implies a+b+c = 3M

. Second,

\frac{a+b}{2} = N \implies a+b = 2N

. We can substitute the second equation into the first:

2N + c = 3M

. We are also given that

c = \frac{1}{2}M

. Substitute this into the equation:

2N + \frac{1}{2}M = 3M

. This is a two-step equation to solve for

N

. First, subtract

\frac{1}{2}M

from both sides:

2N = 3M - \frac{1}{2}M = \frac{6}{2}M - \frac{1}{2}M = \frac{5}{2}M

. Second, divide both sides by 2:

N = \frac{5}{4}M

. The distractors represent common algebraic errors, such as incorrectly combining terms or isolating the wrong variable.

Question 15

What is the value of $n$ that satisfies the equation $\frac{3n - 5(n-2)}{4} = \frac{1}{2} - n$ ?

6
-4 (correct answer)
$\frac{4}{3}$
4

Explanation: First, simplify the numerator on the left side of the equation:

3n - 5(n-2) = 3n - 5n + 10 = -2n + 10

. The equation becomes

\frac{-2n+10}{4} = \frac{1}{2} - n

. The fraction on the left can be simplified by dividing the numerator and denominator by 2:

\frac{-n+5}{2}

. So,

\frac{-n+5}{2} = \frac{1}{2} - n

. To eliminate the denominators, multiply the entire equation by 2:

-n+5 = 1 - 2n

. Now, this is a two-step equation. Add

2n

to both sides:

n+5 = 1

. Subtract 5 from both sides:

n = -4

. Distractor A results from a sign error in the initial distribution:

3n-5n-10

instead of

3n-5n+10

, leading to

n=6

. Distractor C comes from incorrectly moving terms, e.g.,

-n+2n = 1+5

. Distractor D is a sign error in the final step.

Question 16

If $x$ and $y$ are positive integers such that $3x-7=14$ and $2xy=84$ , what is the value of $y$ ?

6 (correct answer)
7
12
14

Explanation: This is a two-part problem. First, we must solve the one-step equation for

x

. Given

3x-7=14

, add 7 to both sides to get

3x = 21

. Then divide by 3 to find

x=7

. Now, substitute this value of

x

into the second equation:

2(7)y = 84

. This simplifies to

14y = 84

. To find

y

, divide both sides by 14:

y = \frac{84}{14} = 6

. The value of

y

is 6. Distractor B is the value of

x

. Distractor C is the value of

xy

. Distractor D is the coefficient of

y

in the second step.

Question 17

The sum of the ages of three siblings, Ann, Bob, and Chloe, is 52. Ann is 5 years younger than Bob. Chloe's age is 3 years more than twice Ann's age. How old is Bob?

11
16 (correct answer)
25
29

Explanation: Let

A

B

, and

C

be the ages of Ann, Bob, and Chloe, respectively. We are given three relationships: 1)

A+B+C = 52

. 2)

A = B - 5

. 3)

C = 2A + 3

. Our goal is to find

B

. We can express all ages in terms of a single variable. Since A is in both (2) and (3), let's use A. From (2), we can write

B = A+5

. Now substitute the expressions for

B

and

C

into the first equation:

A + (A+5) + (2A+3) = 52

. Combine like terms:

4A + 8 = 52

. This is a two-step equation. Subtract 8 from both sides:

4A = 44

. Divide by 4:

A = 11

. This is Ann's age. The question asks for Bob's age. Using

B = A+5

, we find Bob's age is

B = 11+5 = 16

. Distractor A is Ann's age. Distractor C is Chloe's age (

2(11)+3=25

). Distractor D is a calculation error.

Question 18

If $3(x - 4) + 2x = 18$ , what is the value of $2x + 1$ ?

5
7
13 (correct answer)
15

Explanation: First solve for x: 3(x - 4) + 2x = 18 → 3x - 12 + 2x = 18 → 5x - 12 = 18 → 5x = 30 → x = 6. Then 2x + 1 = 2(6) + 1 = 13. Choice A results from finding x - 1 instead of 2x + 1. Choice B results from solving incorrectly to get x = 3. Choice D results from finding 2x + 3 instead of 2x + 1.

Question 19

If $5x - 2(x + 4) = 3x - 16$ , what is the value of $x$ ?

All real numbers satisfy this equation
No solution exists for this equation (correct answer)
$x = 4$
$x = -4$

Explanation: Expand the left side: 5x - 2(x + 4) = 5x - 2x - 8 = 3x - 8. The equation becomes 3x - 8 = 3x - 16. Subtracting 3x from both sides: -8 = -16, which is false. Therefore, no solution exists. Choice A would occur if both sides simplified to identical expressions. Choice C comes from incorrectly solving as if -8 = -16 + 2x, giving x = 4. Choice D comes from a sign error in the expansion or solving process.