ISEE Middle Level Mathematics Achievement Quiz: Solving For An Unknown

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 Mathematics Achievement.

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.

Question 1

1. 40 degrees
2. 60 degrees
3. 80 degrees (correct answer)
4. 160 degrees

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

Question 2

1. 12
2. 24
3. 25 (correct answer)
4. 37

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

Question 3

1. 13 (correct answer)
2. 17
3. 21
4. 26

Explanation: Let $t$ 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 $2t + 8 = 34$. First, subtract the $8.00 entrance fee from the total amount spent: $2t = 34 - 8$, which is $2t = 26$. Then, divide the remaining amount by the cost per ticket to find the number of tickets: $t = 26 / 2 = 13$. Marco bought 13 tickets.

Question 4

1. $x = 4$ (correct answer)
2. $x = 48$
3. $x = 5$
4. $x = 58$

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

1. $x = 450$
2. $x = 53$
3. $x = 50$ (correct answer)
4. $x = 47$

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

1. $x = 6$ (correct answer)
2. $x = 40$
3. $x = 8$
4. $x = 5$

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

1. $x = 28$
2. $x = 6$ (correct answer)
3. $x = 20$
4. $x = 8$

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

1. $x = 110$
2. $x = 9$
3. $x = 11$ (correct answer)
4. $x = 90$

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

1. $x = 11$
2. $x = 14$
3. $x = 7$ (correct answer)
4. $x = 9$

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

1. $x = 9$
2. $x = 8$ (correct answer)
3. $x = 32$
4. $x = 7$

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

1. 10
2. 16 (correct answer)
3. 20
4. 22

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

Question 12

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: $16 \times 5 = 80$. The sum of the four known numbers is $12 + 18 + 7 + 23 = 60$. To find the fifth number, subtract the sum of the four numbers from the total sum: $80 - 60 = 20$. The fifth number is 20.

Question 13

1. 4
2. 6 (correct answer)
3. 8
4. 10

Explanation: First, solve the equation for $x$. Subtract 5 from both sides: $\frac{3x}{4} = 11 - 5$, which is $\frac{3x}{4} = 6$. Multiply both sides by 4: $3x = 24$. Divide by 3: $x = 8$. The question asks for the value of $x - 2$. Substitute the value of $x$: $8 - 2 = 6$.

Question 14

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 $6(\frac{2}{3}m - 1) = 6(\frac{1}{2}m + 2)$, which simplifies to $4m - 6 = 3m + 12$. Subtract (3m) from both sides to get $m - 6 = 12$. Add 6 to both sides to find $m = 18$.

Question 15

1. $x = 8$ (correct answer)
2. $x = 200$
3. $x = 35$
4. $x = 10$

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

1. $x = 60$ (correct answer)
2. $x = 540$
3. $x = 63$
4. $x = 30$

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

1. $x = 8$
2. $x = 4$ (correct answer)
3. $x = 6$
4. $x = 12$

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

1. 2
2. 7
3. 8 (correct answer)
4. 16

Explanation: First, use the distributive property on the left side: $5y - 15 = 3y + 1$. Next, subtract $3y$ from both sides to get $2y - 15 = 1$. Then, add 15 to both sides: $2y = 16$. Finally, divide by 2 to find $y = 8$.

Question 19

1. 8 (correct answer)
2. 9
3. 10
4. 35

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

Question 20

1. -15
2. -3
3. -1 (correct answer)
4. 1

Explanation: First, distribute on both sides of the equation: $6x - 8 = 7x - 7$. Next, subtract $6x$ from both sides to get $-8 = x - 7$. Finally, add 7 to both sides to isolate $x$: $-8 + 7 = x$, which simplifies to $x = -1$.