ISEE Middle Level Quiz: One And Two Step Equations
19 questions · exam conditions
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One And Two Step EquationsQuestion 1 of 19

A store owner buys a tablet for a certain cost, cc. To set the selling price, she marks up the cost by 25%. If the final selling price is $300, what was the original cost, cc?

$225
$240
$275
$375
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ISEE Middle Level Quiz

ISEE Middle Level Quiz: One And Two Step Equations

Practice One And Two Step Equations in ISEE Middle Level 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 ISEE Middle Level.

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

A store owner buys a tablet for a certain cost, cc. To set the selling price, she marks up the cost by 25%. If the final selling price is $300, what was the original cost, cc?

  1. $225
  2. $240 (correct answer)
  3. $275
  4. $375
Explanation: The selling price is the original cost plus a 25% markup. This can be written as c+0.25c=300c + 0.25c = 300, or 1.25c=3001.25c = 300. To find the original cost cc, divide the selling price by 1.25: c=300÷1.25=240c = 300 \div 1.25 = 240. The original cost was $240. Distractor A is a common error where one finds 25% of the selling price (0.25×300=750.25 \times 300 = 75) and subtracts it from the selling price (300−75=225300 - 75 = 225). Distractor D is another common error where one finds 25% of the cost and adds it to the selling price.

Question 2

The sum of three consecutive odd integers is 87. What is the largest of these three integers?

  1. 27
  2. 29
  3. 30
  4. 31 (correct answer)
Explanation: Let the three consecutive odd integers be nn, n+2n+2, and n+4n+4. Their sum is n+(n+2)+(n+4)=87n + (n+2) + (n+4) = 87. Combining like terms gives 3n+6=873n + 6 = 87. Subtracting 6 from both sides gives 3n=813n = 81. Dividing by 3 gives n=27n = 27. This is the smallest of the three integers. The integers are 27, 29, and 31. The largest is 31. Distractor A is the smallest integer. Distractor B is the middle integer (which is also the average, 87÷3=2987 \div 3 = 29). Distractor C is incorrect because the integer must be odd.

Question 3

The result of dividing a number pp by -6 is 8. What is the result of dividing pp by 4?

  1. 12
  2. -12 (correct answer)
  3. -48
  4. 5.33
Explanation: First, find the value of pp. The problem states p−6=8\frac{p}{-6} = 8. To solve for pp, multiply both sides by -6: p=8×(−6)=−48p = 8 \times (-6) = -48. Next, divide pp by 4: −484=−12\frac{-48}{4} = -12. The final result is -12. Distractor C is the value of pp. Distractor A is the result of a sign error when finding pp (if p=48p=48) or when performing the final division. Distractor D does not follow from a likely calculation error.

Question 4

Jamal owes his friend $60. He pays her $15 at the end of the first week, and then $5 each week after that until the debt is fully paid. How many weeks in total will it take him to pay off the entire debt?

  1. 9
  2. 10 (correct answer)
  3. 11
  4. 12
Explanation: After the first week's payment of 15, Jamal has \(60 - $15 = $45) left to pay. He pays this remaining amount at a rate of $5 per week. The number of 5 payments needed is \(45 \div $5 = 9) weeks. The total time is the first week (for the $15 payment) plus the 9 subsequent weeks (for the $5 payments). So, 1+9=101 + 9 = 10 weeks in total. Distractor A is the number of weeks after the first payment. Distractor D is from dividing the total debt by the smaller payment amount (60÷5=1260 \div 5 = 12).

Question 5

A pizza is cut into 8 equal slices. After Maya and her friends eat some slices, 38\frac{3}{8} of the pizza remains. Let nn be the number of slices that were eaten.

Column A nn

Column B 5

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater.
  3. The two quantities are equal. (correct answer)
  4. The relationship cannot be determined from the information given.
Explanation: If 38\frac{3}{8} of the pizza remains, then the fraction of the pizza that was eaten is 1−38=581 - \frac{3}{8} = \frac{5}{8}. Since the pizza had 8 slices, the number of slices eaten, nn, is 58×8=5\frac{5}{8} \times 8 = 5. So, the value in Column A is 5. The value in Column B is also 5. Therefore, the two quantities are equal. A common mistake is to assume n=3n=3, which would make Column B greater.

Question 6

A submarine is at a depth of 240 meters below sea level. It begins to ascend at a constant rate of 12 meters per minute. How many minutes will it take for the submarine to reach a depth of 60 meters below sea level?

  1. 5
  2. 15 (correct answer)
  3. 20
  4. 25
Explanation: Let the initial depth be -240 meters and the final depth be -60 meters. The rate of change is +12 meters per minute. The equation is −240+12m=−60-240 + 12m = -60, where mm is the number of minutes. To solve for mm, add 240 to both sides: 12m=−60+24012m = -60 + 240, which is 12m=18012m = 180. Then, divide by 12: m=180÷12=15m = 180 \div 12 = 15. It will take 15 minutes. Distractor C comes from dividing the initial depth by the rate (240÷12=20240 \div 12 = 20). Distractor A comes from dividing the final depth by the rate (60÷12=560 \div 12 = 5). Distractor D comes from adding the absolute values of the depths and dividing by the rate ((240+60)÷12=25(240+60) \div 12 = 25).

Question 7

Ana is twice as old as her brother, Sam. In 4 years, the sum of their ages will be 32. How old is Ana now?

  1. 8
  2. 12
  3. 16 (correct answer)
  4. 20
Explanation: Let Sam's current age be SS. Ana's current age is 2S2S. In 4 years, Sam will be S+4S+4 and Ana will be 2S+42S+4. The sum of their ages in 4 years will be (S+4)+(2S+4)=32(S+4) + (2S+4) = 32. Combine like terms: 3S+8=323S + 8 = 32. Subtract 8 from both sides: 3S=243S = 24. Divide by 3: S=8S = 8. This is Sam's current age. The question asks for Ana's current age, which is 2S=2(8)=162S = 2(8) = 16. Distractor A is Sam's current age. Distractor B is Sam's age in 4 years. Distractor D is Ana's age in 4 years.

Question 8

A sequence of numbers is generated by the rule 3n−53n - 5, where nn is the position of the number in the sequence. Which term in the sequence has a value of 49?

  1. 16th
  2. 18th (correct answer)
  3. 54th
  4. 142nd
Explanation: We are looking for the value of nn when the term's value is 49. Set up the equation: 3n−5=493n - 5 = 49. Add 5 to both sides: 3n=543n = 54. Divide by 3: n=18n = 18. So, the 18th term in the sequence has a value of 49. Distractor C is the intermediate value of 3n3n. Distractor D is the value of the 49th term (3(49)−5=147−5=1423(49) - 5 = 147 - 5 = 142). Distractor A is a result of an estimation or calculation error.

Question 9

The perimeter of a rectangle is 54 cm. The length of the rectangle is 3 cm more than twice its width. What is the width of the rectangle in centimeters?

  1. 8 (correct answer)
  2. 17
  3. 19
  4. 25.5
Explanation: Let ww be the width. The length ll is 2w+32w + 3. The formula for the perimeter of a rectangle is P=2(l+w)P = 2(l + w). Substitute the given values and expressions: 54=2((2w+3)+w)54 = 2((2w + 3) + w). Simplify the expression inside the parentheses: 54=2(3w+3)54 = 2(3w + 3). Distribute the 2: 54=6w+654 = 6w + 6. Subtract 6 from both sides: 48=6w48 = 6w. Divide by 6: w=8w = 8. The width is 8 cm. Distractor B (17) comes from using l+w=54l + w = 54 instead of 2(l+w)=542(l + w) = 54. Distractor C (19) is the length. Distractor D (25.5) comes from solving 2w+3=542w + 3 = 54 directly.

Question 10

When a number kk is divided by 4 and then 9 is added to the result, the answer is 15. What is the value of k−2k - 2?

  1. 22 (correct answer)
  2. 24
  3. 49
  4. 94
Explanation: First, set up the equation to find kk: k4+9=15\frac{k}{4} + 9 = 15. Subtract 9 from both sides: k4=6\frac{k}{4} = 6. Multiply both sides by 4: k=24k = 24. The question asks for the value of k−2k - 2, not kk. So, substitute the value of kk: 24−2=2224 - 2 = 22. Distractor B is the value of kk. Distractor C results from misinterpreting the order of operations as k+94=15\frac{k+9}{4}=15, which gives k+9=60k+9=60 and k=51k=51, so k−2=49k-2=49. Distractor D results from adding 9 instead of subtracting: k4=15+9=24\frac{k}{4}=15+9=24, so k=96k=96 and k−2=94k-2=94.

Question 11

The formula to convert temperature from Celsius (C) to Fahrenheit (F) is F=95C+32F = \frac{9}{5}C + 32. If the temperature is 86°F, what is the temperature in degrees Celsius?

  1. 30 (correct answer)
  2. 48
  3. 54
  4. 97.2
Explanation: Substitute 86 for F in the formula: 86=95C+3286 = \frac{9}{5}C + 32. First, subtract 32 from both sides: 86−32=95C86 - 32 = \frac{9}{5}C, which simplifies to 54=95C54 = \frac{9}{5}C. To solve for C, multiply both sides by the reciprocal of 95\frac{9}{5}, which is 59\frac{5}{9}: C=54×59C = 54 \times \frac{5}{9}. This can be calculated as (54÷9)×5=6×5=30(54 \div 9) \times 5 = 6 \times 5 = 30. The temperature is 30°C. Distractor C is the intermediate result before multiplying by the reciprocal. Distractor D is the result of multiplying 54 by 95\frac{9}{5} instead of 59\frac{5}{9}.

Question 12

An isosceles triangle has a perimeter of 39 inches. The base is 6 inches longer than each of the two equal sides. What is the length of the base?

  1. 11 inches
  2. 15 inches
  3. 17 inches (correct answer)
  4. 21 inches
Explanation: Let ss be the length of each of the two equal sides. The base bb is s+6s + 6. The perimeter is the sum of the three sides: s+s+b=39s + s + b = 39. Substitute the expression for bb: s+s+(s+6)=39s + s + (s + 6) = 39. Combine like terms: 3s+6=393s + 6 = 39. Subtract 6 from both sides: 3s=333s = 33. Divide by 3: s=11s = 11. This is the length of an equal side. The question asks for the length of the base, which is s+6=11+6=17s + 6 = 11 + 6 = 17 inches. Distractor A is the length of one of the equal sides. Distractors B and D are calculation errors.

Question 13

A number nn is decreased by 6, and the result is multiplied by -3. The final result is 24.

Column A nn

Column B -2

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater.
  3. The two quantities are equal. (correct answer)
  4. The relationship cannot be determined from the information given.
Explanation: First, set up the equation based on the information given: −3(n−6)=24-3(n - 6) = 24. To solve for nn, first divide both sides by -3: n−6=24÷(−3)n - 6 = 24 \div (-3), which simplifies to n−6=−8n - 6 = -8. Then, add 6 to both sides: n=−8+6n = -8 + 6, which gives n=−2n = -2. Therefore, the value in Column A is -2. The value in Column B is also -2. The two quantities are equal. A common mistake is a sign error in distribution (−3n−18=24  ⟹  −3n=42  ⟹  n=−14-3n - 18 = 24 \implies -3n = 42 \implies n = -14, leading to B) or in solving (n−6=−8  ⟹  n=−14n - 6 = -8 \implies n = -14, leading to B).

Question 14

A phone company charges $0.20 for the first minute of a call and $0.08 for each additional minute. If a call costs $1.24, what was the total duration of the call in minutes?

  1. 12
  2. 13
  3. 14 (correct answer)
  4. 15.5
Explanation: Let mm be the total number of minutes. The cost is for the first minute plus the additional minutes. The cost of the additional minutes is (1.24−1.24 - 0.20 = 1.04\). The number of additional minutes is the cost of additional minutes divided by the rate: \(1.04 \div 0.08 = 13\). These are the additional minutes. The total duration is the first minute plus the additional minutes: 1 + 13 = 14 minutes. Distractor B is the number of additional minutes, not the total duration. Distractor A is an estimation error. Distractor D is the result of dividing the total cost by the additional minute rate \(1.24 / 0.08 = 15.5), which ignores the special rate for the first minute.

Question 15

Which of the following situations is best modeled by the equation 25−2x=925 - 2x = 9?

  1. A plant that was 9 inches tall grows 2 inches per week for xx weeks and is now 25 inches tall.
  2. A shirt originally costing xx dollars is discounted by $2 to a sale price of $9. Its original price was $25.
  3. Mark has $25. He buys 2 books that each cost xx dollars and has $9 in change remaining. (correct answer)
  4. Jan started with 25 cookies. She gave away xx cookies to two friends and had 9 left.
Explanation: The equation 25−2x=925 - 2x = 9 represents a starting amount of 25, from which an amount of 2x2x is subtracted, leaving a result of 9. Choice C matches this perfectly: Mark starts with $25, the cost of the two books is 2x2x, and his remaining change is $9. Choice A is modeled by 9+2x=259 + 2x = 25. Choice B is not modeled correctly by any simple equation provided. Choice D is modeled by 25−x=925 - x = 9 if she gave away x cookies total, or 25−2x=925-2x=9 if she gave x cookies to each of two friends. The phrasing in D is ambiguous, while C is precise and correct.

Question 16

Leo is thinking of a number. If he multiplies his number by 4 and then adds 9, he gets the same result as when he multiplies his number by 2 and adds 15. What is Leo's number?

  1. 3 (correct answer)
  2. 4
  3. 6
  4. 12
Explanation: Let the number be xx. The problem can be translated into the equation 4x+9=2x+154x + 9 = 2x + 15. To solve for xx, first gather the variable terms on one side by subtracting 2x2x from both sides: 2x+9=152x + 9 = 15. Next, isolate the variable term by subtracting 9 from both sides: 2x=62x = 6. Finally, divide by 2: x=3x = 3. Distractor C comes from an error where the constants are added instead of subtracted: 2x=15+9=242x = 15+9=24, so x=12x=12. Distractor D is that value. Distractor B is a guess. The error for D is 4x−2x=15+9  ⟹  2x=24  ⟹  x=124x-2x = 15+9 \implies 2x=24 \implies x=12. The error for C (6) is 4x+2x=15+9  ⟹  6x=24  ⟹  x=44x+2x = 15+9 \implies 6x=24 \implies x=4. No, wait. Error for D: 4x−2x=15+9  ⟹  2x=24  ⟹  x=124x-2x = 15+9 \implies 2x=24 \implies x=12. Error for B: 4x+2x=15+9  ⟹  6x=24  ⟹  x=44x+2x=15+9 \implies 6x=24 \implies x=4. Error for C: 4x+2x=15−9  ⟹  6x=6  ⟹  x=14x+2x=15-9 \implies 6x=6 \implies x=1. The provided distractors seem to have a mixup. Let's re-calculate: 4x+9=2x+15  ⟹  2x=6  ⟹  x=34x+9=2x+15 \implies 2x=6 \implies x=3. Correct is A. Distractor D (12) from 2x=15+92x = 15+9. Let's get C=6. Maybe 2x=15−9=62x=15-9=6 but they forget to divide by 2, so x=6. That's plausible. So A is correct. B is a guess. C is from stopping at an intermediate step. D is from a sign error.

Question 17

A baker starts with a full 50-pound bag of flour. He uses 2122\frac{1}{2} pounds of flour for each cake he bakes. At the end of the day, he has 20 pounds of flour left. How many cakes did he bake?

  1. 8
  2. 12 (correct answer)
  3. 20
  4. 28
Explanation: First, determine how much flour was used. The baker started with 50 pounds and had 20 pounds left, so he used 50−20=3050 - 20 = 30 pounds of flour. Each cake requires 2122\frac{1}{2} or 2.5 pounds of flour. To find the number of cakes, divide the total flour used by the amount per cake: 30÷2.5=1230 \div 2.5 = 12. The baker made 12 cakes. Distractor A is from dividing 20 by 2.5. Distractor C is from dividing 50 by 2.5, ignoring the remaining flour. Distractor D is from adding the remaining flour instead of subtracting: (50+20)÷2.5=28(50+20) \div 2.5 = 28.

Question 18

Given the equation y=2x+5y = 2x + 5.

Column A The value of xx when y=1y = 1

Column B The value of yy when x=−1x = -1

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater. (correct answer)
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Explanation: For Column A, substitute y=1y=1 into the equation: 1=2x+51 = 2x + 5. Subtract 5 from both sides: −4=2x-4 = 2x. Divide by 2: x=−2x = -2. So, the quantity in Column A is -2. For Column B, substitute x=−1x=-1 into the equation: y=2(−1)+5y = 2(-1) + 5. Simplify: y=−2+5y = -2 + 5, so y=3y = 3. The quantity in Column B is 3. Since 3>−23 > -2, the quantity in Column B is greater.

Question 19

If 12 is subtracted from 4 times a number, the result is 20. What is the number?

  1. -2
  2. 2
  3. 8 (correct answer)
  4. 17
Explanation: Let the number be nn. The phrase '4 times a number' is 4n4n. '12 is subtracted from' this means 4n−124n - 12. The result is 20, so the equation is 4n−12=204n - 12 = 20. Add 12 to both sides: 4n=324n = 32. Divide by 4: n=8n = 8. Distractor B results from subtracting 12 from 20 instead of adding: 4n=20−12=8  ⟹  n=24n = 20 - 12 = 8 \implies n=2. Distractor A results from a sign error: 4n=12−20=−8  ⟹  n=−24n = 12-20 = -8 \implies n=-2. Distractor D results from misreading the problem as 'a number minus 12, times 4': 4(n−12)=20  ⟹  n−12=5  ⟹  n=174(n-12)=20 \implies n-12=5 \implies n=17.