Fractions Worksheets with Answer Key - Free Printable

To solve the given problems involving algebraic fractions, we need to simplify each expression step by step. Let's go through each problem systematically.

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Problem 1: Simplify \( \frac{3x}{7} + \frac{x+3}{7} \)

Since the denominators are the same, we can combine the numerators directly:
\[
\frac{3x}{7} + \frac{x+3}{7} = \frac{3x + (x + 3)}{7}
\]
Simplify the numerator:
\[
3x + x + 3 = 4x + 3
\]
So the expression becomes:
\[
\frac{4x + 3}{7}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{4x + 3}{7}}
\]

---

Problem 2: Simplify \( \frac{8x}{3} + \frac{x-3}{12} \)

The denominators are different, so we need a common denominator. The least common multiple (LCM) of 3 and 12 is 12. Rewrite each fraction with the common denominator:
\[
\frac{8x}{3} = \frac{8x \cdot 4}{3 \cdot 4} = \frac{32x}{12}
\]
\[
\frac{x-3}{12} \text{ remains as it is.}
\]
Now add the fractions:
\[
\frac{32x}{12} + \frac{x-3}{12} = \frac{32x + (x - 3)}{12}
\]
Simplify the numerator:
\[
32x + x - 3 = 33x - 3
\]
So the expression becomes:
\[
\frac{33x - 3}{12}
\]
Factor out the greatest common divisor (GCD) of the numerator and denominator:
\[
\frac{33x - 3}{12} = \frac{3(11x - 1)}{12} = \frac{11x - 1}{4}
\]

Answer:
\[
\boxed{\frac{11x - 1}{4}}
\]

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Problem 3: Simplify \( \frac{9x}{8} - \frac{x-5}{16} \)

The denominators are different, so we need a common denominator. The LCM of 8 and 16 is 16. Rewrite each fraction with the common denominator:
\[
\frac{9x}{8} = \frac{9x \cdot 2}{8 \cdot 2} = \frac{18x}{16}
\]
\[
\frac{x-5}{16} \text{ remains as it is.}
\]
Now subtract the fractions:
\[
\frac{18x}{16} - \frac{x-5}{16} = \frac{18x - (x - 5)}{16}
\]
Simplify the numerator:
\[
18x - x + 5 = 17x + 5
\]
So the expression becomes:
\[
\frac{17x + 5}{16}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{17x + 5}{16}}
\]

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Problem 4: Simplify \( \frac{x+7}{3} + \frac{3x-2}{18} \)

The denominators are different, so we need a common denominator. The LCM of 3 and 18 is 18. Rewrite each fraction with the common denominator:
\[
\frac{x+7}{3} = \frac{(x+7) \cdot 6}{3 \cdot 6} = \frac{6(x+7)}{18} = \frac{6x + 42}{18}
\]
\[
\frac{3x-2}{18} \text{ remains as it is.}
\]
Now add the fractions:
\[
\frac{6x + 42}{18} + \frac{3x - 2}{18} = \frac{(6x + 42) + (3x - 2)}{18}
\]
Simplify the numerator:
\[
6x + 3x + 42 - 2 = 9x + 40
\]
So the expression becomes:
\[
\frac{9x + 40}{18}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{9x + 40}{18}}
\]

---

Problem 5: Simplify \( \frac{7x}{8} + \frac{3x}{10} - \frac{x}{5} \)

The denominators are different, so we need a common denominator. The LCM of 8, 10, and 5 is 40. Rewrite each fraction with the common denominator:
\[
\frac{7x}{8} = \frac{7x \cdot 5}{8 \cdot 5} = \frac{35x}{40}
\]
\[
\frac{3x}{10} = \frac{3x \cdot 4}{10 \cdot 4} = \frac{12x}{40}
\]
\[
\frac{x}{5} = \frac{x \cdot 8}{5 \cdot 8} = \frac{8x}{40}
\]
Now combine the fractions:
\[
\frac{35x}{40} + \frac{12x}{40} - \frac{8x}{40} = \frac{35x + 12x - 8x}{40}
\]
Simplify the numerator:
\[
35x + 12x - 8x = 39x
\]
So the expression becomes:
\[
\frac{39x}{40}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{39x}{40}}
\]

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Problem 6: Simplify \( \frac{2x}{11} - \frac{5(x-2)}{11} \)

The denominators are the same, so we can combine the numerators directly:
\[
\frac{2x}{11} - \frac{5(x-2)}{11} = \frac{2x - 5(x-2)}{11}
\]
Distribute the 5 in the second term:
\[
2x - 5(x-2) = 2x - 5x + 10 = -3x + 10
\]
So the expression becomes:
\[
\frac{-3x + 10}{11}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{-3x + 10}{11}}
\]

---

Problem 7: Simplify \( \frac{2}{x} - \frac{3}{x} + \frac{7}{2x} \)

The denominators are different, so we need a common denominator. The LCM of \(x\) and \(2x\) is \(2x\). Rewrite each fraction with the common denominator:
\[
\frac{2}{x} = \frac{2 \cdot 2}{x \cdot 2} = \frac{4}{2x}
\]
\[
\frac{3}{x} = \frac{3 \cdot 2}{x \cdot 2} = \frac{6}{2x}
\]
\[
\frac{7}{2x} \text{ remains as it is.}
\]
Now combine the fractions:
\[
\frac{4}{2x} - \frac{6}{2x} + \frac{7}{2x} = \frac{4 - 6 + 7}{2x}
\]
Simplify the numerator:
\[
4 - 6 + 7 = 5
\]
So the expression becomes:
\[
\frac{5}{2x}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{5}{2x}}
\]

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Problem 8: Simplify \( \frac{4x}{7} - \frac{x-3}{8} \)

The denominators are different, so we need a common denominator. The LCM of 7 and 8 is 56. Rewrite each fraction with the common denominator:
\[
\frac{4x}{7} = \frac{4x \cdot 8}{7 \cdot 8} = \frac{32x}{56}
\]
\[
\frac{x-3}{8} = \frac{(x-3) \cdot 7}{8 \cdot 7} = \frac{7(x-3)}{56} = \frac{7x - 21}{56}
\]
Now subtract the fractions:
\[
\frac{32x}{56} - \frac{7x - 21}{56} = \frac{32x - (7x - 21)}{56}
\]
Simplify the numerator:
\[
32x - 7x + 21 = 25x + 21
\]
So the expression becomes:
\[
\frac{25x + 21}{56}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{25x + 21}{56}}
\]

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Final Answers:

1. \(\boxed{\frac{4x + 3}{7}}\)
2. \(\boxed{\frac{11x - 1}{4}}\)
3. \(\boxed{\frac{17x + 5}{16}}\)
4. \(\boxed{\frac{9x + 40}{18}}\)
5. \(\boxed{\frac{39x}{40}}\)
6. \(\boxed{\frac{-3x + 10}{11}}\)
7. \(\boxed{\frac{5}{2x}}\)
8. \(\boxed{\frac{25x + 21}{56}}\)