Middle School Math Quiz: Solve Linear Equations With Rationals

18 questions · exam conditions

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Solve Linear Equations With RationalsQuestion 1 of 18

Solve for $x$ .

$1.5(x-2)-0.75x=6$

x=14

x=12

x=10

x=8

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Middle School Math Quiz

Middle School Math Quiz: Solve Linear Equations With Rationals

Practice Solve Linear Equations With Rationals 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 Solve Linear Equations With Rationals, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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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

Solve for $x$ .

$1.5(x-2)-0.75x=6$

$x=14$
$x=12$ (correct answer)
$x=10$
$x=8$

Explanation: This problem tests solving linear equations with decimal coefficients using the distributive property. Process: distribute

1.5(x-2) = 1.5x - 3

, combine like terms (

1.5x - 0.75x = 0.75x

), isolate variable, solve. For

1.5(x-2) - 0.75x = 6

: distribute to get

1.5x - 3 - 0.75x = 6

, combine x terms to get

0.75x - 3 = 6

, add 3 to both sides to get

0.75x = 9

, divide by 0.75 to get

x = 12

. The correct answer is

x=12

. Common errors include arithmetic mistakes with decimals or forgetting to distribute the negative sign.

Question 2

Solve for $x$ . (You will need to distribute and then collect like terms.)

$\frac{1}{4}(2x-12)+\frac{3}{8}x=3$

$x=\frac{48}{7}$ (correct answer)
$x=\frac{12}{7}$
$x=\frac{24}{7}$
$x=7$

Explanation: This problem tests solving linear equations with multiple fractions requiring distribution and combining like terms. Process: distribute 1/4(2x-12) = x/2 - 3, combine all x terms and constants, then solve. Starting with 1/4(2x-12) + 3x/8 = 3, distribute to get x/2 - 3 + 3x/8 = 3. To combine x terms, find LCD of 2 and 8, which is 8: 4x/8 + 3x/8 = 7x/8. The equation becomes 7x/8 - 3 = 3. Add 3 to both sides: 7x/8 = 6. Multiply by 8/7: x = 48/7. Common errors include incorrect distribution of fractions or adding fractions without common denominators. Steps: distribute carefully, find LCD for combining terms, isolate variable, and simplify final answer.

Question 3

When solving $0.3(x-2) - 0.5(2x+1) = 1.2$ , Alex converts to fractions. Which equation is equivalent to the original?

$\frac{3(x-2) + 5(2x+1)}{10} = \frac{12}{10}$
$\frac{3(x-2) - 5(2x+1)}{100} = \frac{120}{100}$
$\frac{30(x-2) - 50(2x+1)}{10} = \frac{12}{10}$
$\frac{3(x-2) - 5(2x+1)}{10} = \frac{12}{10}$ (correct answer)

Explanation: Since

0.3=\frac{3}{10}

0.5=\frac{5}{10}

, and

1.2=\frac{12}{10}

, substituting gives

\frac{3}{10}(x-2)-\frac{5}{10}(2x+1)=\frac{12}{10}

, and combining the left side over the common denominator 10 gives

\frac{3(x-2)-5(2x+1)}{10}=\frac{12}{10}

. Choice A is wrong because it adds the two terms instead of subtracting them. Choice B is wrong because it treats the decimals as hundredths instead of tenths, giving the wrong denominator and numerator. Choice C is wrong because it multiplies the terms by 30 and 50 instead of correctly converting

0.3

and

0.5

to tenths.

Question 4

Solve for $x$ (distribute and then combine like terms): $2.5(x-1.2)-0.5x=6.4$

$x=\frac{57}{10}$
$x=\frac{47}{10}$ (correct answer)
$x=\frac{37}{10}$
$x=\frac{27}{10}$

Explanation: This problem tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. Process: distribute a(bx+c)=abx+ac (2/3(x-6)=2x/3-4), collect like terms (2x+3x=5x, 1/2x+1/4x=3/4x using common denominator), isolate variable (move x terms one side, constants other), solve (divide both sides). Clearing fractions by multiplying by LCD simplifies arithmetic (equation with 1/2 and 3/4: multiply by 4 converts to integers). For this specific equation, distribute 2.5 to (x-1.2) to get 2.5x - 3, then subtract 0.5x yielding 2.5x - 0.5x - 3 = 6.4; combine to 2x - 3 = 6.4, add 3 to get 2x = 9.4, then divide by 2 to find x = 4.7 or 47/10. The correct process yields the answer x = 47/10. A common error is miscalculating the distribution like 2.5*1.2 as 2.5 instead of 3, leading to wrong values like 37/10. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator (1/2+1/3≠2/5), sign errors moving terms, dividing wrong (x/4=2 → x=8 not 0.5).

Question 5

Solve for $x$ : $\frac{5}{6}x-\frac{1}{3}(x-9)=12$

$x=18$ (correct answer)
$x=27$
$x=9$
$x=15$

Explanation: Distribute

-\frac{1}{3}

across

(x-9)

-\frac{1}{3}x + 3

. The equation becomes

\frac{5}{6}x - \frac{1}{3}x + 3 = 12

. Combining

\frac{5}{6}x - \frac{1}{3}x

using a common denominator of 6 gives

\frac{1}{2}x

, so

\frac{1}{2}x + 3 = 12

. Subtracting 3 gives

\frac{1}{2}x = 9

, and multiplying both sides by 2 gives

x = 18

. Choice C (

x=9

) comes from forgetting to multiply by 2 in the last step.

Question 6

Sarah needs to solve $2.5(x + 4) - 1.8x = 3.2$ . After distributing and combining like terms, she gets $0.7x + k = 3.2$ . What is the value of $k$ ?

$k = 8$
$k = 10$ (correct answer)
$k = 6$
$k = 12$

Explanation: Distributing:

2.5(x + 4) = 2.5x + 10

. The equation becomes

2.5x + 10 - 1.8x = 3.2

. Combining like terms:

(2.5 - 1.8)x + 10 = 0.7x + 10

. Therefore

k = 10

. Choice A results from calculation error in distribution. Choice C comes from forgetting to distribute to the constant. Choice D comes from doubling the distributed constant.

Question 7

Solve for $x$ . Multiply by an LCD to eliminate fractions: $\frac{5}{6}x - \frac{1}{3} = \frac{1}{2}x + 3$

$x=12$
$x=10$ (correct answer)
$x=\frac{10}{3}$
$x=8$

Explanation: This problem tests solving linear equations with fraction coefficients by clearing fractions using LCD. Process: multiply entire equation by LCD of 6 to eliminate fractions, then solve the resulting integer equation. To solve:

\frac{5}{6}x - \frac{1}{3} = \frac{1}{2}x + 3

→ multiply by 6:

5x - 2 = 3x + 18

→

5x - 3x = 18 + 2

→

2x = 20

→

x = 10

. Let me verify:

\frac{5}{6}(10) - \frac{1}{3} = \frac{50}{6} - \frac{2}{6} = \frac{48}{6} = 8

, and

\frac{1}{2}(10) + 3 = 5 + 3 = 8

✓. The correct answer is x = 10. Steps: (1) identify LCD as 6, (2) multiply every term by 6, (3) simplify to get 5x - 2 = 3x + 18, (4) collect x terms on left, (5) collect constants on right, (6) divide by 2. Common errors: forgetting to multiply every term by LCD, arithmetic mistakes when clearing fractions.

Question 8

The equation $\frac{1}{3}(6x-9) + \frac{1}{2}(4x-8) = 4x + k$ is an identity (true for all values of $x$ ). What is the value of $k$ ?

$k = -3$
$k = -5$
$k = -7$ (correct answer)
$k = -1$

Explanation: When you see an equation described as an "identity," it means the equation is true for all possible values of the variable. To find the unknown constant, you need to simplify both sides and make them equivalent. Start by distributing the fractions on the left side. For

\frac{1}{3}(6x-9)

, multiply each term:

\frac{1}{3} \cdot 6x - \frac{1}{3} \cdot 9 = 2x - 3

. For

\frac{1}{2}(4x-8)

, you get

\frac{1}{2} \cdot 4x - \frac{1}{2} \cdot 8 = 2x - 4

. Now the equation becomes:

2x - 3 + 2x - 4 = 4x + k

Combine like terms on the left:

4x - 7 = 4x + k

Since this must be true for all values of

x

, the coefficients of

x

must match (they already do:

4x = 4x

), and the constant terms must be equal. Therefore,

k = -7

. Looking at the wrong answers: Choice A (

k = -3

) likely comes from only using the first fraction's constant term. Choice B (

k = -5

) might result from incorrectly combining

-3

and

-4

-3 + (-2) = -5

instead of

-3 + (-4) = -7

. Choice D (

k = -1

) could come from mistakenly calculating

-4 - (-3) = -1

. The correct answer is C:

k = -7

. Study tip: For identity problems, always simplify completely and remember that corresponding terms on both sides must be exactly equal. Double-check your arithmetic when combining constants.

Question 9

When solving $\frac{3x-2}{4} = \frac{x+6}{3} - \frac{x-1}{2}$ , Marcus clears the fractions by multiplying both sides by 12. After expanding but before combining like terms, his equation is:

$9x - 6 = 4(x+6) - 6(x-1)$
$3(3x-2) = 4(x+6) - 6(x-1)$
$9x - 6 = 4x + 24 - 6x + 6$ (correct answer)
$3(3x-2) = 4(x+6) + 6(x-1)$

Explanation: Multiplying every term by 12 clears the fractions:

\frac{3x-2}{4}\cdot12=3(3x-2)

\frac{x+6}{3}\cdot12=4(x+6)

, and

\frac{x-1}{2}\cdot12=6(x-1)

. "After expanding" means the parentheses have been distributed, but like terms have not yet been combined:

3(3x-2)=9x-6

and

4(x+6)-6(x-1)=4x+24-6x+6

, giving

9x-6=4x+24-6x+6

. Choice A expands only the left side and leaves the right side in factored form. Choice B has cleared the fractions but stopped before expanding, so its parentheses are still intact. Choice D changes the subtraction in front of

6(x-1)

to addition, flipping the sign of the last group.

Question 10

Solve for $x$ . Clear fractions if it helps: $\frac{5}{6}x - \frac{1}{3}(x-9) = 12$

$x=18$ (correct answer)
$x=9$
$x=15$
$x=27$

Explanation: This problem tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. Process: distribute

a(bx+c) = abx + ac

(

\frac{2}{3}(x-6) = \frac{2x}{3} - 4

), collect like terms (

2x + 3x = 5x

\frac{1}{2}x + \frac{1}{4}x = \frac{3}{4}x

using common denominator), isolate variable (move x terms one side, constants other), solve (divide both sides). Clearing fractions by multiplying by LCD simplifies arithmetic (equation with

\frac{1}{2}

and

\frac{3}{4}

: multiply by 4 converts to integers). For this specific equation, distribute

-\frac{1}{3}

to (x-9) to get

- \frac{1}{3}x + 3

, then add

\frac{5}{6}x

yielding

\frac{5}{6}x - \frac{1}{3}x + 3 = 12

; combine to

\frac{1}{2}x + 3 = 12

, subtract 3 to get

\frac{1}{2}x = 9

, then multiply by 2 to find x = 18. The correct process yields the answer

x = 18

. A common error is distributing with the wrong sign, like subtracting instead of adding 3, leading to wrong values like

x = 9

. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator (

\frac{1}{2} + \frac{1}{3} \neq \frac{2}{5}

), sign errors moving terms, dividing wrong (

\frac{x}{4} = 2 \rightarrow x=8

not 0.5).

Question 11

Solve for $x$ . Be sure to use the distributive property and combine like terms:

$\frac{3}{4}(x-8)+\frac{1}{2}x=10$

$x=8$
$x=\frac{16}{5}$
$x=\frac{64}{5}$ (correct answer)
$x=\frac{32}{5}$

Explanation: This problem tests solving linear equations with fraction coefficients using distributive property and combining like terms. Process: distribute a(bx+c)=abx+ac, collect like terms using common denominators, isolate variable by moving x terms to one side and constants to the other, then solve by dividing both sides. For this equation: distribute 3/4(x-8) = 3x/4 - 6, so the equation becomes 3x/4 - 6 + x/2 = 10. To combine x terms, find common denominator: 3x/4 + x/2 = 3x/4 + 2x/4 = 5x/4. The equation is now 5x/4 - 6 = 10, so 5x/4 = 16, giving x = 64/5. Common errors include distributing incorrectly (forgetting to multiply both terms) or adding fractions without finding common denominators.

Question 12

A science club is mixing solutions. The equation for the amount of concentrate is

$0.25(4x-8)+1.5=5.5$

Solve for $x$ .

$x=6$ (correct answer)
$x=4$
$x=5$
$x=8$

Explanation: This problem tests solving linear equations with decimals in a real-world context about mixing solutions. Process: distribute 0.25(4x-8) = x - 2, combine constants, isolate variable, and solve. Starting with 0.25(4x-8) + 1.5 = 5.5, distribute to get x - 2 + 1.5 = 5.5. Simplify left side: x - 0.5 = 5.5. Add 0.5 to both sides: x = 6. Common errors include incorrect distribution (0.25 × 4x = x, not 0.25x) or decimal arithmetic mistakes. Steps: (1) distribute carefully with decimals, (2) combine constants, (3) isolate x, (4) check by substituting back: 0.25(4·6-8) + 1.5 = 0.25(16) + 1.5 = 4 + 1.5 = 5.5 ✓.

Question 13

Solve for $x$ (distribute and combine like terms): $0.6(x+5)-0.2x=4$ .

$x=3.5$
$x=2.5$ (correct answer)
$x=1.5$
$x=4.5$

Explanation: This question tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. The process involves distributing a(bx+c)=abx+ac (e.g., 2/3(x-6)=2x/3-4), collecting like terms (2x+3x=5x, 1/2x+1/4x=3/4x using common denominator), isolating the variable (move x terms one side, constants other), and solving (divide both sides); clearing fractions by multiplying by LCD simplifies arithmetic (equation with 1/2 and 3/4: multiply by 4 converts to integers). For the equation 0.6(x+5)-0.2x=4, distribute to get 0.6x + 3 - 0.2x = 4, combine x terms to 0.4x + 3 = 4, subtract 3 to get 0.4x = 1, and divide by 0.4 to find x=2.5. The correct process yields x=2.5, which matches choice B. A common error is mishandling the subtraction of 0.2x, such as treating it as addition, or decimal division errors. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator (1/2+1/3≠2/5), sign errors moving terms, dividing wrong (x/4=2 → x=8 not 0.5).

Question 14

Solve for $x$ (clear fractions if you want): $\frac{1}{2}x + \frac{3}{4} = \frac{5}{8}x - \frac{1}{4}.$

$x=6$
$x=2$
$x=8$ (correct answer)
$x=4$

Explanation: This question tests solving linear equations with fraction coefficients using combining like terms. Process: collect like terms (

\frac{1}{2} x - \frac{5}{8} x = -\frac{1}{8} x

), isolate variable (move x terms one side, constants other), solve (divide both sides). Clearing fractions by multiplying by LCD 8 simplifies arithmetic (equation with 1/2, 3/4, 5/8, 1/4: multiply by 8 converts to integers). For the equation (

\frac{1}{2} x + \frac{3}{4} = \frac{5}{8} x - \frac{1}{4}

), subtract (

\frac{1}{2} x

) to get (

\frac{3}{4} = \frac{1}{8} x - \frac{1}{4}

), add (

\frac{1}{4}

) to get (

1 = \frac{1}{8} x

), multiply by 8 to find (

x = 8

). The correct process and answer is

x = 8

, verified by substitution: left (

\frac{1}{2}(8) + 0.75 = 4 + 0.75 = 4.75

), right (

\frac{5}{8}(8) - 0.25 = 5 - 0.25 = 4.75

). A common error is incorrect fraction addition, like

\frac{3}{4} + \frac{1}{4}

\frac{3}{8}

. Steps: (1) clear fractions (multiply by 8:

4x + 6 = 5x - 2

), (2) collect like terms, (3) isolate x (subtract 4x:

6 = x - 2

, add 2:

x=8

), (4) check. Common errors: adding fractions without common denominator, sign errors moving terms, dividing wrong.

Question 15

Solve for $x$ . Be sure to use the distributive property and combine like terms:

$\frac{3}{4}(x-8)+\frac{1}{2}x=10$

$x=8$
$x=\frac{64}{5}$ (correct answer)
$x=\frac{16}{5}$
$x=\frac{32}{5}$

Question 16

A science club is mixing solutions. The equation for the amount of concentrate is $0.25(4x-8)+1.5=5.5$ Solve for $x$ .

$x=6$ (correct answer)
$x=5$
$x=4$
$x=8$

Explanation: This problem tests solving linear equations with decimals in a real-world context about mixing solutions. Process: distribute

0.25(4x-8) = x - 2

, combine constants, isolate variable, and solve. Starting with

0.25(4x-8) + 1.5 = 5.5

, distribute to get

x - 2 + 1.5 = 5.5

. Simplify left side:

x - 0.5 = 5.5

. Add 0.5 to both sides:

x = 6

. Common errors include incorrect distribution (

0.25 \times 4x = x

, not

0.25x

) or decimal arithmetic mistakes. Steps: (1) distribute carefully with decimals, (2) combine constants, (3) isolate x, (4) check by substituting back:

0.25(4\cdot6-8) + 1.5 = 0.25(16) + 1.5 = 4 + 1.5 = 5.5

✓.

Question 17

Solve for $x$ . (You will need to distribute and then collect like terms.)

$\frac{1}{4}(2x-12)+\frac{3}{8}x=3$

$x=\frac{48}{7}$ (correct answer)
$x=7$
$x=\frac{24}{7}$
$x=\frac{12}{7}$

Question 18

A school club sells snack packs. The cost (in dollars) is modeled by the equation $0.5(x+6)+1.25=0.8x-0.25$ . Solve for $x$ .

$x=20$
$x=10$
$x=5$
$x=15$ (correct answer)

Explanation: This question tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. The process involves distributing a(bx+c)=abx+ac (e.g., 2/3(x-6)=2x/3-4), collecting like terms (2x+3x=5x, 1/2x+1/4x=3/4x using common denominator), isolating the variable (move x terms one side, constants other), and solving (divide both sides); clearing fractions by multiplying by LCD simplifies arithmetic (equation with 1/2 and 3/4: multiply by 4 converts to integers). For the equation 0.5(x+6) + 1.25 = 0.8x - 0.25, distribute to get 0.5x + 3 + 1.25 = 0.8x - 0.25, combine constants to 0.5x + 4.25 = 0.8x - 0.25, subtract 0.5x from both sides to get 4.25 = 0.3x - 0.25, add 0.25 to both sides to get 4.5 = 0.3x, and divide by 0.3 to find x=15. The correct process yields x=15, which matches choice C. A common error is mishandling decimals, such as subtracting 0.5 from 0.8 incorrectly as 0.2 instead of 0.3, or forgetting to add the constants properly. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator (1/2+1/3≠2/5), sign errors moving terms, dividing wrong (x/4=2 → x=8 not 0.5).