ISEE Middle Level Quiz: Solving For An Unknown
20 questions · exam conditions
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Solving For An UnknownQuestion 1 of 20

In a triangle, the measure of the second angle is twice the measure of the first angle. The measure of the third angle is 20 degrees more than the first angle. What is the measure of the largest angle?

40 degrees
60 degrees
80 degrees
160 degrees
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ISEE Middle Level Quiz

ISEE Middle Level Quiz: Solving For An Unknown

Practice Solving For An Unknown 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 Solving For An Unknown, 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

In a triangle, the measure of the second angle is twice the measure of the first angle. The measure of the third angle is 20 degrees more than the first angle. What is the measure of the largest angle?

  1. 40 degrees
  2. 60 degrees
  3. 80 degrees (correct answer)
  4. 160 degrees
Explanation: Let the measure of the first angle be xx. Then the second angle is 2x2x and the third angle is x+20x + 20. The sum of the angles in a triangle is 180 degrees. So, x+2x+(x+20)=180x + 2x + (x + 20) = 180. Combine like terms: 4x+20=1804x + 20 = 180. Subtract 20 from both sides: 4x=1604x = 160. Divide by 4: x=40x = 40. The angles are x=40x=40 degrees, 2x=2(40)=802x = 2(40) = 80 degrees, and x+20=40+20=60x + 20 = 40 + 20 = 60 degrees. The largest of these is 80 degrees.

Question 2

A company has 45 employees. The number of employees in the sales department is one more than twice the number of employees in the production department. All other 8 employees are in administration. How many employees are in the sales department?

  1. 12
  2. 24
  3. 25 (correct answer)
  4. 37
Explanation: Let ss be the number of sales employees and pp be the number of production employees. The total number of employees is 45. There are 8 in administration, so s+p+8=45s + p + 8 = 45. This means s+p=37s + p = 37. We are also told that s=2p+1s = 2p + 1. Substitute this into the previous equation: (2p+1)+p=37(2p + 1) + p = 37. Combine like terms: 3p+1=373p + 1 = 37. Subtract 1: 3p=363p = 36. Divide by 3: p=12p = 12. There are 12 employees in production. To find the number of sales employees, use s=2p+1s = 2p + 1: s=2(12)+1=24+1=25s = 2(12) + 1 = 24 + 1 = 25. There are 25 employees in the sales department.

Question 3

At a carnival, ride tickets cost $2.00 each. The entrance fee is $8.00. If Marco spent a total of $34.00, how many ride tickets did he buy?

  1. 13 (correct answer)
  2. 17
  3. 21
  4. 26
Explanation: Let tt be the number of ride tickets Marco bought. The total cost is the sum of the entrance fee and the cost of the tickets. The equation is 2t+8=342t + 8 = 34. First, subtract the $8.00 entrance fee from the total amount spent: 2t=34−82t = 34 - 8, which is 2t=262t = 26. Then, divide the remaining amount by the cost per ticket to find the number of tickets: t=26/2=13t = 26 / 2 = 13. Marco bought 13 tickets.

Question 4

A streaming service charges $12 each month plus a $5 one-time fee. After a few months, the total cost is $53. Let $xbethenumberofmonthspaidfor.Thissituationisrepresentedbybe the number of months paid for. This situation is represented by12x + 5 = 53.If. If 12x + 5 = 53,whatis, what is x$?

  1. x=4x = 4 (correct answer)
  2. x=48x = 48
  3. x=5x = 5
  4. x=58x = 58
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 12x + 5 = 53, where the unknown variable is isolated by subtracting 5 from both sides to get 12x = 48 and then dividing by 12 to find x = 4. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students forget to subtract and divide 53 by something to get x = 48. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 5

A bus travels 150 miles in 3 hours at a constant speed. The driver keeps the same speed for the whole trip. Let xx be the speed in miles per hour. The situation can be written as 3x=1503x = 150. If 3x=1503x = 150, what is xx?

  1. x=450x = 450
  2. x=53x = 53
  3. x=50x = 50 (correct answer)
  4. x=47x = 47
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x = 150, where the unknown variable is isolated by dividing both sides by 3 to find x = 50. The correct answer, choice C, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 3 * 150 = 450. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 6

A recipe uses 2 cups of flour for 8 muffins. Sam wants to make 24 muffins and keeps the same recipe ratio. Let xx be the number of cups of flour needed. The situation can be written as 8x=488x = 48. Solve for xx in the equation 8x=488x = 48.

  1. x=6x = 6 (correct answer)
  2. x=40x = 40
  3. x=8x = 8
  4. x=5x = 5
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 8x = 48, where the unknown variable is isolated by dividing both sides by 8 to find x = 6. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 8 * 5 = 40. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 7

A runner completes 24 miles over 4 days, running the same distance each day. No rest days are included in the total. Let xx be the miles run each day. The situation is modeled by 4x=244x = 24. If 4x=244x = 24, what is xx?

  1. x=28x = 28
  2. x=6x = 6 (correct answer)
  3. x=20x = 20
  4. x=8x = 8
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 4x = 24, where the unknown variable is isolated by dividing both sides by 4 to find x = 6. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 4 * 7 = 28. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 8

A class orders 9 identical pizzas for a total of $99. Each pizza costs the same amount, and delivery is free. Let $xbethecostofonepizza.Thesituationisrepresentedbybe the cost of one pizza. The situation is represented by9x = 99.Solvefor. Solve for xintheequationin the equation9x = 99$.

  1. x=110x = 110
  2. x=9x = 9
  3. x=11x = 11 (correct answer)
  4. x=90x = 90
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 9x = 99, where the unknown variable is isolated by dividing both sides by 9 to find x = 11. The correct answer, choice C, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students divide incorrectly, such as 99 / 11 = 9. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 9

A taxi charges a $4 start fee plus $2 per mile. Jordan's ride costs $18 total, including the start fee. Let $xbethenumberofmilestraveled.Thissituationisrepresentedbybe the number of miles traveled. This situation is represented by2x + 4 = 18.If. If 2x + 4 = 18,whatis, what is x$?

  1. x=11x = 11
  2. x=14x = 14
  3. x=7x = 7 (correct answer)
  4. x=9x = 9
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 2x + 4 = 18, where the unknown variable is isolated by subtracting 4 from both sides to get 2x = 14 and then dividing by 2 to find x = 7. The correct answer, choice C, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students subtract incorrectly, such as 18 - 4 = 14 but then divide by something else leading to x = 11. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 10

A bookstore sells a novel for xdollars,andabookmarkcosts$2.Avabuys4novelsand1bookmarkfor$34total.Allnovelshavethesameprice.Thesituationismodeledbyx dollars, and a bookmark costs $2. Ava buys 4 novels and 1 bookmark for $34 total. All novels have the same price. The situation is modeled by 4x + 2 = 34.Find. Find xifif4x + 2 = 34$.

  1. x=9x = 9
  2. x=8x = 8 (correct answer)
  3. x=32x = 32
  4. x=7x = 7
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 4x + 2 = 34, where the unknown variable is isolated by subtracting 2 from both sides to get 4x = 32 and then dividing by 4 to find x = 8. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice C is incorrect due to a common arithmetic error where students forget to subtract and divide 34 by something to get x = 32. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 11

Sam is 4 years younger than twice his sister's age. The sum of their ages is 26. How old is Sam?

  1. 10
  2. 16 (correct answer)
  3. 20
  4. 22
Explanation: Let Sam's age be SS and his sister's age be RR. From the problem, we can write two equations: S=2R−4S = 2R - 4 and S+R=26S + R = 26. Substitute the first equation into the second: (2R−4)+R=26(2R - 4) + R = 26. Combine like terms: 3R−4=263R - 4 = 26. Add 4 to both sides: 3R=303R = 30. Divide by 3 to find the sister's age, R=10R = 10. Now find Sam's age using S=2R−4S = 2R - 4: S=2(10)−4=20−4=16S = 2(10) - 4 = 20 - 4 = 16. Sam is 16 years old.

Question 12

The mean of five numbers is 16. Four of the numbers are 12, 18, 7, and 23. What is the fifth number?

  1. 15
  2. 16
  3. 20 (correct answer)
  4. 30
Explanation: The mean of a set of numbers is the sum of the numbers divided by the count of the numbers. The sum of the five numbers is the mean multiplied by the count: 16×5=8016 \times 5 = 80. The sum of the four known numbers is 12+18+7+23=6012 + 18 + 7 + 23 = 60. To find the fifth number, subtract the sum of the four numbers from the total sum: 80−60=2080 - 60 = 20. The fifth number is 20.

Question 13

If 3x4+5=11\frac{3x}{4} + 5 = 11, what is the value of x−2x - 2?

  1. 4
  2. 6 (correct answer)
  3. 8
  4. 10
Explanation: First, solve the equation for xx. Subtract 5 from both sides: 3x4=11−5\frac{3x}{4} = 11 - 5, which is 3x4=6\frac{3x}{4} = 6. Multiply both sides by 4: 3x=243x = 24. Divide by 3: x=8x = 8. The question asks for the value of x−2x - 2. Substitute the value of xx: 8−2=68 - 2 = 6.

Question 14

What is the value of mm in the equation 23m−1=12m+2\frac{2}{3}m - 1 = \frac{1}{2}m + 2?

  1. -6
  2. 3
  3. 6
  4. 18 (correct answer)
Explanation: To solve this equation, first eliminate the fractions by multiplying both sides by the least common denominator of 3 and 2, which is 6. This gives 6(23m−1)=6(12m+2)6(\frac{2}{3}m - 1) = 6(\frac{1}{2}m + 2), which simplifies to 4m−6=3m+124m - 6 = 3m + 12. Subtract (3m) from both sides to get m−6=12m - 6 = 12. Add 6 to both sides to find m=18m = 18.

Question 15

A baker makes 5 identical loaves and sells them for $40 total. Each loaf has the same price, and there are no coupons. Let $xbethepriceofoneloaf.Thesituationisrepresentedbybe the price of one loaf. The situation is represented by5x = 40.Determinethevalueof. Determine the value of xwhenwhen5x = 40$.

  1. x=8x = 8 (correct answer)
  2. x=200x = 200
  3. x=35x = 35
  4. x=10x = 10
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 5x = 40, where the unknown variable is isolated by dividing both sides by 5 to find x = 8. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 5 * 40 = 200. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 16

A car travels at a constant speed for 3 hours and goes 180 miles. The driver does not stop, and the distance stays the same the whole time. Let xx be the car's speed in miles per hour. The situation can be written as 3x=1803x = 180. Solve for xx in the equation 3x=1803x = 180.

  1. x=60x = 60 (correct answer)
  2. x=540x = 540
  3. x=63x = 63
  4. x=30x = 30
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x = 180, where the unknown variable is isolated by dividing both sides by 3 to find x = 60. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 3 * 180 = 540. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 17

Maya buys 3 notebooks and 1 binder, and the total is $18. The binder costs $6. The notebooks all cost the same amount, and sales tax is already included. The situation can be modeled by 3x+6=183x + 6 = 18, where xx is the price of one notebook. If 3x+6=183x + 6 = 18, what is xx?

  1. x=8x = 8
  2. x=4x = 4 (correct answer)
  3. x=6x = 6
  4. x=12x = 12
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x + 6 = 18, where the unknown variable is isolated by subtracting 6 from both sides to get 3x = 12 and then dividing by 3 to find x = 4. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students add 6 instead of subtracting, leading to 3x = 24 and x = 8. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

Question 18

Solve for yy: 5(y−3)=3y+15(y - 3) = 3y + 1

  1. 2
  2. 7
  3. 8 (correct answer)
  4. 16
Explanation: First, use the distributive property on the left side: 5y−15=3y+15y - 15 = 3y + 1. Next, subtract 3y3y from both sides to get 2y−15=12y - 15 = 1. Then, add 15 to both sides: 2y=162y = 16. Finally, divide by 2 to find y=8y = 8.

Question 19

What is the value of zz in the equation 36z+1=4\frac{36}{z+1} = 4?

  1. 8 (correct answer)
  2. 9
  3. 10
  4. 35
Explanation: To solve for zz, you can first multiply both sides of the equation by z+1z+1 to get 36=4(z+1)36 = 4(z+1). Then, divide both sides by 4 to get 9=z+19 = z+1. Finally, subtract 1 from both sides to find z=8z = 8. Alternatively, recognize that if 3636 divided by a number is 4, that number must be 9. So, z+1=9z+1=9, which means z=8z=8.

Question 20

What is the solution to the equation 2(3x−4)=7(x−1)2(3x - 4) = 7(x - 1)?

  1. -15
  2. -3
  3. -1 (correct answer)
  4. 1
Explanation: First, distribute on both sides of the equation: 6x−8=7x−76x - 8 = 7x - 7. Next, subtract 6x6x from both sides to get −8=x−7-8 = x - 7. Finally, add 7 to both sides to isolate xx: −8+7=x-8 + 7 = x, which simplifies to x=−1x = -1.