Solving Equations with Fractions Worksheets - Math Monks - Free Printable

Problem: Solve each equation. Correct your answer to 2 decimal places, if needed.

We will solve each equation step by step.

---

#### Equation 1:
\[
\frac{3 - 2x}{5} - \frac{4x - 2}{4} = -8
\]

1. Find a common denominator for the fractions, which is \(20\):
\[
\frac{4(3 - 2x)}{20} - \frac{5(4x - 2)}{20} = -8
\]
2. Simplify the numerators:
\[
\frac{12 - 8x}{20} - \frac{20x - 10}{20} = -8
\]
3. Combine the fractions:
\[
\frac{(12 - 8x) - (20x - 10)}{20} = -8
\]
\[
\frac{12 - 8x - 20x + 10}{20} = -8
\]
\[
\frac{22 - 28x}{20} = -8
\]
4. Eliminate the denominator by multiplying both sides by 20:
\[
22 - 28x = -160
\]
5. Solve for \(x\):
\[
-28x = -160 - 22
\]
\[
-28x = -182
\]
\[
x = \frac{-182}{-28}
\]
\[
x = 6.5
\]

Answer:
\[
\boxed{6.50}
\]

---

#### Equation 2:
\[
\frac{4}{3x} - \frac{1}{x - 1} = \frac{1}{4x}
\]

1. Find a common denominator for the fractions, which is \(12x(x - 1)\):
\[
\frac{4 \cdot 4(x - 1)}{12x(x - 1)} - \frac{1 \cdot 12x}{12x(x - 1)} = \frac{1 \cdot 3(x - 1)}{12x(x - 1)}
\]
2. Simplify the numerators:
\[
\frac{16(x - 1)}{12x(x - 1)} - \frac{12x}{12x(x - 1)} = \frac{3(x - 1)}{12x(x - 1)}
\]
3. Combine the fractions:
\[
\frac{16(x - 1) - 12x}{12x(x - 1)} = \frac{3(x - 1)}{12x(x - 1)}
\]
\[
\frac{16x - 16 - 12x}{12x(x - 1)} = \frac{3(x - 1)}{12x(x - 1)}
\]
\[
\frac{4x - 16}{12x(x - 1)} = \frac{3(x - 1)}{12x(x - 1)}
\]
4. Since the denominators are the same, equate the numerators:
\[
4x - 16 = 3(x - 1)
\]
\[
4x - 16 = 3x - 3
\]
5. Solve for \(x\):
\[
4x - 3x = -3 + 16
\]
\[
x = 13
\]

Answer:
\[
\boxed{13.00}
\]

---

#### Equation 3:
\[
6\left(\frac{1}{3}x - \frac{3}{5}\right) = \frac{5}{6}
\]

1. Distribute the 6 inside the parentheses:
\[
6 \cdot \frac{1}{3}x - 6 \cdot \frac{3}{5} = \frac{5}{6}
\]
\[
2x - \frac{18}{5} = \frac{5}{6}
\]
2. Eliminate the fractions by finding a common denominator (30):
\[
2x - \frac{18}{5} = \frac{5}{6}
\]
\[
2x - \frac{18 \cdot 6}{30} = \frac{5 \cdot 5}{30}
\]
\[
2x - \frac{108}{30} = \frac{25}{30}
\]
3. Combine the terms:
\[
2x = \frac{25}{30} + \frac{108}{30}
\]
\[
2x = \frac{133}{30}
\]
4. Solve for \(x\):
\[
x = \frac{133}{30 \cdot 2}
\]
\[
x = \frac{133}{60}
\]
\[
x \approx 2.22
\]

Answer:
\[
\boxed{2.22}
\]

---

#### Equation 4:
\[
2x + \frac{9}{5} = -\frac{11}{5}(x + 51)
\]

1. Distribute \(-\frac{11}{5}\) on the right-hand side:
\[
2x + \frac{9}{5} = -\frac{11}{5}x - \frac{11}{5} \cdot 51
\]
\[
2x + \frac{9}{5} = -\frac{11}{5}x - \frac{561}{5}
\]
2. Eliminate the fractions by multiplying through by 5:
\[
5 \cdot 2x + 5 \cdot \frac{9}{5} = 5 \cdot \left(-\frac{11}{5}x\right) + 5 \cdot \left(-\frac{561}{5}\right)
\]
\[
10x + 9 = -11x - 561
\]
3. Combine like terms:
\[
10x + 11x = -561 - 9
\]
\[
21x = -570
\]
4. Solve for \(x\):
\[
x = \frac{-570}{21}
\]
\[
x \approx -27.14
\]

Answer:
\[
\boxed{-27.14}
\]

---

#### Equation 5:
\[
\frac{2(x - 1)}{3} + \frac{2}{3} = \frac{9}{2}(x - 7)
\]

1. Simplify the left-hand side:
\[
\frac{2(x - 1) + 2}{3} = \frac{9}{2}(x - 7)
\]
\[
\frac{2x - 2 + 2}{3} = \frac{9}{2}(x - 7)
\]
\[
\frac{2x}{3} = \frac{9}{2}(x - 7)
\]
2. Eliminate the fractions by multiplying through by 6:
\[
6 \cdot \frac{2x}{3} = 6 \cdot \frac{9}{2}(x - 7)
\]
\[
4x = 27(x - 7)
\]
3. Distribute on the right-hand side:
\[
4x = 27x - 189
\]
4. Combine like terms:
\[
4x - 27x = -189
\]
\[
-23x = -189
\]
5. Solve for \(x\):
\[
x = \frac{-189}{-23}
\]
\[
x \approx 8.22
\]

Answer:
\[
\boxed{8.22}
\]

---

#### Equation 6:
\[
\frac{3}{2} - \frac{7}{4}x = -\frac{3}{8}(x + 2)
\]

1. Distribute \(-\frac{3}{8}\) on the right-hand side:
\[
\frac{3}{2} - \frac{7}{4}x = -\frac{3}{8}x - \frac{3}{8} \cdot 2
\]
\[
\frac{3}{2} - \frac{7}{4}x = -\frac{3}{8}x - \frac{3}{4}
\]
2. Eliminate the fractions by multiplying through by 8:
\[
8 \cdot \frac{3}{2} - 8 \cdot \frac{7}{4}x = 8 \cdot \left(-\frac{3}{8}x\right) + 8 \cdot \left(-\frac{3}{4}\right)
\]
\[
12 - 14x = -3x - 6
\]
3. Combine like terms:
\[
12 + 6 = -3x + 14x
\]
\[
18 = 11x
\]
4. Solve for \(x\):
\[
x = \frac{18}{11}
\]
\[
x \approx 1.64
\]

Answer:
\[
\boxed{1.64}
\]

---

#### Equation 7:
\[
\frac{9}{x + 3} = \frac{16}{2x - 4}
\]

1. Cross-multiply to eliminate the fractions:
\[
9(2x - 4) = 16(x + 3)
\]
2. Distribute on both sides:
\[
18x - 36 = 16x + 48
\]
3. Combine like terms:
\[
18x - 16x = 48 + 36
\]
\[
2x = 84
\]
4. Solve for \(x\):
\[
x = \frac{84}{2}
\]
\[
x = 42
\]

Answer:
\[
\boxed{42.00}
\]

---

#### Equation 8:
\[
\frac{3(x + 1)}{5} = \frac{7 - 4x}{6}
\]

1. Cross-multiply to eliminate the fractions:
\[
6 \cdot 3(x + 1) = 5 \cdot (7 - 4x)
\]
\[
18(x + 1) = 5(7 - 4x)
\]
2. Distribute on both sides:
\[
18x + 18 = 35 - 20x
\]
3. Combine like terms:
\[
18x + 20x = 35 - 18
\]
\[
38x = 17
\]
4. Solve for \(x\):
\[
x = \frac{17}{38}
\]
\[
x \approx 0.45
\]

Answer:
\[
\boxed{0.45}
\]

---

#### Equation 9:
\[
\frac{4x - 4}{8} - \frac{4x - 2}{6} = \frac{6x + 7}{25}
\]

1. Simplify the fractions:
\[
\frac{4(x - 1)}{8} - \frac{4x - 2}{6} = \frac{6x + 7}{25}
\]
\[
\frac{x - 1}{2} - \frac{4x - 2}{6} = \frac{6x + 7}{25}
\]
2. Find a common denominator for the left-hand side, which is 6:
\[
\frac{3(x - 1)}{6} - \frac{4x - 2}{6} = \frac{6x + 7}{25}
\]
\[
\frac{3(x - 1) - (4x - 2)}{6} = \frac{6x + 7}{25}
\]
\[
\frac{3x - 3 - 4x + 2}{6} = \frac{6x + 7}{25}
\]
\[
\frac{-x - 1}{6} = \frac{6x + 7}{25}
\]
3. Cross-multiply to eliminate the fractions:
\[
25(-x - 1) = 6(6x + 7)
\]
\[
-25x - 25 = 36x + 42
\]
4. Combine like terms:
\[
-25x - 36x = 42 + 25
\]
\[
-61x = 67
\]
5. Solve for \(x\):
\[
x = \frac{-67}{61}
\]
\[
x \approx -1.10
\]

Answer:
\[
\boxed{-1.10}
\]

---

#### Equation 10:
\[
\frac{21(x + 4)}{54} = \frac{2(x + 9) + 11}{6}
\]

1. Simplify the fractions:
\[
\frac{7(x + 4)}{18} = \frac{2(x + 9) + 11}{6}
\]
\[
\frac{7(x + 4)}{18} = \frac{2x + 18 + 11}{6}
\]
\[
\frac{7(x + 4)}{18} = \frac{2x + 29}{6}
\]
2. Cross-multiply to eliminate the fractions:
\[
6 \cdot 7(x + 4) = 18 \cdot (2x + 29)
\]
\[
42(x + 4) = 18(2x + 29)
\]
3. Distribute on both sides:
\[
42x + 168 = 36x + 522
\]
4. Combine like terms:
\[
42x - 36x = 522 - 168
\]
\[
6x = 354
\]
5. Solve for \(x\):
\[
x = \frac{354}{6}
\]
\[
x = 59
\]

Answer:
\[
\boxed{59.00}
\]

---

Final Answers:

\[
\boxed{6.50, 13.00, 2.22, -27.14, 8.22, 1.64, 42.00, 0.45, -1.10, 59.00}
\]