Math worksheet with ten fraction equations to solve.

Let's solve each of these equations with fractions step by step. We'll simplify, eliminate denominators (when possible), and solve for $ x $. Answers will be rounded to 2 decimal places if needed.

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

Step 1: Find the LCD of 5 and 4 → 20

Multiply both sides by 20:
$$
20\left(\frac{3 - 2x}{5} - \frac{4x - 2}{4}\right) = 20(-8)
$$
$$
4(3 - 2x) - 5(4x - 2) = -160
$$
$$
12 - 8x - 20x + 10 = -160
$$
$$
-28x + 22 = -160
$$
$$
-28x = -182
$$
$$
x = \frac{-182}{-28} = \frac{91}{14} = 6.5
$$

✔ Answer: $ x = 6.50 $

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② $\frac{4}{3x} - \frac{1}{x - 1} = \frac{1}{4x}$

Step 1: LCD is $ 12x(x - 1) $

Multiply both sides by $ 12x(x - 1) $:
$$
12x(x - 1)\left(\frac{4}{3x} - \frac{1}{x - 1}\right) = 12x(x - 1)\left(\frac{1}{4x}\right)
$$

Break it down:
- $ 12x(x - 1) \cdot \frac{4}{3x} = 4 \cdot 4(x - 1) = 16(x - 1) $
- $ 12x(x - 1) \cdot \left(-\frac{1}{x - 1}\right) = -12x $
- RHS: $ 12x(x - 1) \cdot \frac{1}{4x} = 3(x - 1) $

So:
$$
16(x - 1) - 12x = 3(x - 1)
$$
$$
16x - 16 - 12x = 3x - 3
$$
$$
4x - 16 = 3x - 3
$$
$$
x = 13
$$

⚠️ Check: $ x = 13 $ → no division by zero (since $ x \neq 0, 1 $) → valid

✔ Answer: $ x = 13.00 $

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③ $6\left(\frac{1}{3}x - \frac{3}{5}\right) = \frac{5}{6}$

Distribute 6:
$$
6 \cdot \frac{1}{3}x - 6 \cdot \frac{3}{5} = \frac{5}{6}
$$
$$
2x - \frac{18}{5} = \frac{5}{6}
$$

Convert to common denominator or decimals:

$ \frac{18}{5} = 3.6 $, $ \frac{5}{6} \approx 0.8333 $

$$
2x = 3.6 + 0.8333 = 4.4333
$$
$$
x = \frac{4.4333}{2} = 2.2166...
$$

Round to 2 decimal places:

✔ Answer: $ x = 2.22 $

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④ $2x + \frac{9}{5} = -\frac{11}{5}(x + 51)$

Distribute RHS:
$$
2x + \frac{9}{5} = -\frac{11}{5}x - \frac{561}{5}
$$

Multiply all terms by 5 to eliminate denominators:
$$
10x + 9 = -11x - 561
$$

Bring like terms together:
$$
10x + 11x = -561 - 9
$$
$$
21x = -570
$$
$$
x = \frac{-570}{21} = -27.142857...
$$

Round to 2 decimal places:

✔ Answer: $ x = -27.14 $

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⑤ $\frac{2(x - 1)}{3} + \frac{2}{3} = \frac{9}{2}(x - 7)$

Left side: factor $ \frac{2}{3} $:
$$
\frac{2(x - 1) + 2}{3} = \frac{2x - 2 + 2}{3} = \frac{2x}{3}
$$

So equation becomes:
$$
\frac{2x}{3} = \frac{9}{2}(x - 7)
$$

Multiply both sides by 6 (LCD of 3 and 2):
$$
6 \cdot \frac{2x}{3} = 6 \cdot \frac{9}{2}(x - 7)
$$
$$
4x = 27(x - 7)
$$
$$
4x = 27x - 189
$$
$$
-23x = -189
$$
$$
x = \frac{189}{23} \approx 8.2174
$$

✔ Answer: $ x = 8.22 $

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⑥ $\frac{3}{2} - \frac{7}{4}x = -\frac{3}{8}(x + 2)$

Multiply both sides by 8 (LCD of 2, 4, 8):

$$
8\left(\frac{3}{2} - \frac{7}{4}x\right) = 8\left(-\frac{3}{8}(x + 2)\right)
$$
$$
12 - 14x = -3(x + 2)
$$
$$
12 - 14x = -3x - 6
$$
$$
12 + 6 = -3x + 14x
$$
$$
18 = 11x
$$
$$
x = \frac{18}{11} \approx 1.6364
$$

✔ Answer: $ x = 1.64 $

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⑦ $\frac{9}{x + 3} = \frac{16}{2x - 4}$

Note: $ 2x - 4 = 2(x - 2) $

Cross-multiply:
$$
9(2x - 4) = 16(x + 3)
$$
$$
18x - 36 = 16x + 48
$$
$$
18x - 16x = 48 + 36
$$
$$
2x = 84
$$
$$
x = 42
$$

Check: $ x = 42 $ → $ x + 3 = 45 $, $ 2x - 4 = 80 $ → no division by zero → valid

✔ Answer: $ x = 42.00 $

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⑧ $\frac{3(x + 1)}{5} = \frac{7 - 4x}{6}$

Cross-multiply:
$$
6 \cdot 3(x + 1) = 5(7 - 4x)
$$
$$
18(x + 1) = 35 - 20x
$$
$$
18x + 18 = 35 - 20x
$$
$$
18x + 20x = 35 - 18
$$
$$
38x = 17
$$
$$
x = \frac{17}{38} \approx 0.4474
$$

✔ Answer: $ x = 0.45 $

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⑨ $\frac{4x - 4}{8} - \frac{4x - 2}{6} = \frac{6x + 7}{25}$

Simplify fractions:
- $ \frac{4x - 4}{8} = \frac{4(x - 1)}{8} = \frac{x - 1}{2} $
- $ \frac{4x - 2}{6} = \frac{2(2x - 1)}{6} = \frac{2x - 1}{3} $

So:
$$
\frac{x - 1}{2} - \frac{2x - 1}{3} = \frac{6x + 7}{25}
$$

LCD of 2, 3, 25 → 150

Multiply all terms by 150:
$$
150 \cdot \left(\frac{x - 1}{2} - \frac{2x - 1}{3}\right) = 150 \cdot \frac{6x + 7}{25}
$$
$$
75(x - 1) - 50(2x - 1) = 6(6x + 7)
$$
$$
75x - 75 - 100x + 50 = 36x + 42
$$
$$
-25x - 25 = 36x + 42
$$
$$
-25x - 36x = 42 + 25
$$
$$
-61x = 67
$$
$$
x = -\frac{67}{61} \approx -1.0984
$$

✔ Answer: $ x = -1.10 $

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⑩ $\frac{21(x + 4)}{54} = \frac{2(x + 9) + 11}{6}$

Simplify both sides:

LHS: $ \frac{21(x + 4)}{54} = \frac{7(x + 4)}{18} $ (divide numerator and denominator by 3)

RHS: $ \frac{2x + 18 + 11}{6} = \frac{2x + 29}{6} $

Now:
$$
\frac{7(x + 4)}{18} = \frac{2x + 29}{6}
$$

Multiply both sides by 18:
$$
7(x + 4) = 3(2x + 29)
$$
$$
7x + 28 = 6x + 87
$$
$$
7x - 6x = 87 - 28
$$
$$
x = 59
$$

✔ Answer: $ x = 59.00 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| ① | $ x = 6.50 $ |
| ② | $ x = 13.00 $ |
| ③ | $ x = 2.22 $ |
| ④ | $ x = -27.14 $ |
| ⑤ | $ x = 8.22 $ |
| ⑥ | $ x = 1.64 $ |
| ⑦ | $ x = 42.00 $ |
| ⑧ | $ x = 0.45 $ |
| ⑨ | $ x = -1.10 $ |
| ⑩ | $ x = 59.00 $ |

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