Printable math worksheet for solving multi-step equations with fractions.

Let's solve each of the multi-step equations involving fractions step by step. I'll go through each one carefully, showing all steps and simplifications.

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1) $ 3\left(a - \frac{2}{3}\right) = \frac{3}{4}a + 2\frac{1}{4} $

Step 1: Distribute the 3 on the left side:
$$
3a - 3 \cdot \frac{2}{3} = \frac{3}{4}a + 2\frac{1}{4}
$$
$$
3a - 2 = \frac{3}{4}a + 2\frac{1}{4}
$$

Step 2: Convert mixed number to improper fraction:
$$
2\frac{1}{4} = \frac{9}{4}
$$
So:
$$
3a - 2 = \frac{3}{4}a + \frac{9}{4}
$$

Step 3: Eliminate fractions by multiplying every term by the least common denominator (LCD), which is 4:
$$
4(3a - 2) = 4\left(\frac{3}{4}a + \frac{9}{4}\right)
$$
$$
12a - 8 = 3a + 9
$$

Step 4: Subtract $3a$ from both sides:
$$
9a - 8 = 9
$$

Step 5: Add 8 to both sides:
$$
9a = 17
$$

Step 6: Divide by 9:
$$
a = \frac{17}{9}
$$

✔ Answer: $ a = \frac{17}{9} $

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

Wait — this equation has both x and z, but no indication that they are related. This seems like a typo or error. Likely, it should be the same variable.

Assuming it’s supposed to be:
$$
\frac{x}{2} - \frac{3}{5} = -\frac{2}{3}x + \frac{1}{6}
$$

Let’s solve this corrected version:

Step 1: Get all terms with $x$ on one side:
$$
\frac{x}{2} + \frac{2}{3}x = \frac{1}{6} + \frac{3}{5}
$$

Step 2: Find LCD for $x$-terms: LCD of 2 and 3 is 6.
$$
\frac{3}{6}x + \frac{4}{6}x = \frac{7}{6}x
$$

Step 3: Right side: LCD of 6 and 5 is 30:
$$
\frac{1}{6} = \frac{5}{30}, \quad \frac{3}{5} = \frac{18}{30} \Rightarrow \frac{23}{30}
$$

So:
$$
\frac{7}{6}x = \frac{23}{30}
$$

Step 4: Multiply both sides by reciprocal of $\frac{7}{6}$:
$$
x = \frac{23}{30} \cdot \frac{6}{7} = \frac{138}{210} = \frac{23}{35}
$$

✔ Answer: $ x = \frac{23}{35} $

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3) $ \frac{7}{4}x - 3 = 2 + \frac{9}{4}x $

Step 1: Move all $x$-terms to one side:
$$
\frac{7}{4}x - \frac{9}{4}x = 2 + 3
$$
$$
-\frac{2}{4}x = 5 \Rightarrow -\frac{1}{2}x = 5
$$

Step 2: Multiply both sides by $-2$:
$$
x = -10
$$

✔ Answer: $ x = -10 $

---

4) $ \frac{3c + 8}{3} = \frac{1}{2}c + \frac{c}{4} $

Step 1: Simplify right side:
$$
\frac{1}{2}c + \frac{1}{4}c = \frac{3}{4}c
$$

So:
$$
\frac{3c + 8}{3} = \frac{3}{4}c
$$

Step 2: Multiply both sides by 12 (LCD of 3 and 4):
$$
12 \cdot \frac{3c + 8}{3} = 12 \cdot \frac{3}{4}c
$$
$$
4(3c + 8) = 9c
$$
$$
12c + 32 = 9c
$$

Step 3: Subtract $9c$:
$$
3c + 32 = 0
\Rightarrow 3c = -32
\Rightarrow c = -\frac{32}{3}
$$

✔ Answer: $ c = -\frac{32}{3} $

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5) $ \frac{1}{3} - \frac{2}{9}m = 15 + m $

Step 1: Move all $m$-terms to one side:
$$
-\frac{2}{9}m - m = 15 - \frac{1}{3}
$$

Convert $m$ to ninths:
$$
-\frac{2}{9}m - \frac{9}{9}m = -\frac{11}{9}m
$$

Right side:
$$
15 - \frac{1}{3} = \frac{45}{3} - \frac{1}{3} = \frac{44}{3}
$$

So:
$$
-\frac{11}{9}m = \frac{44}{3}
$$

Step 2: Multiply both sides by $-\frac{9}{11}$:
$$
m = \frac{44}{3} \cdot \left(-\frac{9}{11}\right) = -\frac{396}{33} = -12
$$

✔ Answer: $ m = -12 $

---

6) $ \frac{1}{2}(q + 1) = \frac{4}{3} - q $

Step 1: Distribute on left:
$$
\frac{1}{2}q + \frac{1}{2} = \frac{4}{3} - q
$$

Step 2: Move all $q$-terms to left:
$$
\frac{1}{2}q + q = \frac{4}{3} - \frac{1}{2}
$$
$$
\frac{3}{2}q = \frac{8}{6} - \frac{3}{6} = \frac{5}{6}
$$

Step 3: Multiply both sides by $\frac{2}{3}$:
$$
q = \frac{5}{6} \cdot \frac{2}{3} = \frac{10}{18} = \frac{5}{9}
$$

✔ Answer: $ q = \frac{5}{9} $

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7) $ \frac{1}{6}r + 2 = 4\frac{1}{3}r + \frac{8}{3} $

First, convert $4\frac{1}{3}$ to improper fraction:
$$
4\frac{1}{3} = \frac{13}{3}
$$

So:
$$
\frac{1}{6}r + 2 = \frac{13}{3}r + \frac{8}{3}
$$

Step 1: Move $r$-terms to one side:
$$
\frac{1}{6}r - \frac{13}{3}r = \frac{8}{3} - 2
$$

Convert $ \frac{13}{3} $ to sixths: $ \frac{26}{6} $
So:
$$
\frac{1}{6}r - \frac{26}{6}r = -\frac{25}{6}r
$$

Right side:
$$
\frac{8}{3} - 2 = \frac{8}{3} - \frac{6}{3} = \frac{2}{3}
$$

So:
$$
-\frac{25}{6}r = \frac{2}{3}
$$

Step 2: Multiply both sides by $-\frac{6}{25}$:
$$
r = \frac{2}{3} \cdot \left(-\frac{6}{25}\right) = -\frac{12}{75} = -\frac{4}{25}
$$

✔ Answer: $ r = -\frac{4}{25} $

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8) $ \frac{2}{3} - \frac{2}{3}y + \frac{1}{3}y + 4 = 0 $

Step 1: Combine like terms:
$$
\left(-\frac{2}{3}y + \frac{1}{3}y\right) + \left(\frac{2}{3} + 4\right) = 0
$$
$$
-\frac{1}{3}y + \left(\frac{2}{3} + \frac{12}{3}\right) = 0
\Rightarrow -\frac{1}{3}y + \frac{14}{3} = 0
$$

Step 2: Subtract $\frac{14}{3}$:
$$
-\frac{1}{3}y = -\frac{14}{3}
$$

Step 3: Multiply both sides by $-3$:
$$
y = 14
$$

✔ Answer: $ y = 14 $

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✔ Final Answers:

1) $ a = \frac{17}{9} $
2) $ x = \frac{23}{35} $
3) $ x = -10 $
4) $ c = -\frac{32}{3} $
5) $ m = -12 $
6) $ q = \frac{5}{9} $
7) $ r = -\frac{4}{25} $
8) $ y = 14 $

Let me know if you'd like these formatted in a printable answer sheet!