Fractions Worksheets with Answer Key - Free Printable

Let's solve each of these algebraic fraction problems step by step. We'll simplify each expression and write the answers in lowest terms.

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1. $\frac{3x}{7} + \frac{x + 3}{7}$

Since both fractions have the same denominator, we can combine the numerators:

$$
\frac{3x + (x + 3)}{7} = \frac{3x + x + 3}{7} = \frac{4x + 3}{7}
$$

✔ Answer: $\boxed{\frac{4x + 3}{7}}$

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2. $\frac{8x}{3} + \frac{x - 3}{12}$

Find the LCM of denominators 3 and 12 → LCM is 12.

Convert first fraction:
$$
\frac{8x}{3} = \frac{8x \cdot 4}{3 \cdot 4} = \frac{32x}{12}
$$

Now add:
$$
\frac{32x}{12} + \frac{x - 3}{12} = \frac{32x + x - 3}{12} = \frac{33x - 3}{12}
$$

Simplify numerator and denominator by factoring:
$$
\frac{3(11x - 1)}{12} = \frac{11x - 1}{4}
$$

✔ Answer: $\boxed{\frac{11x - 1}{4}}$

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3. $\frac{9x}{8} - \frac{x - 5}{16}$

LCM of 8 and 16 is 16.

Convert first fraction:
$$
\frac{9x}{8} = \frac{9x \cdot 2}{16} = \frac{18x}{16}
$$

Now subtract:
$$
\frac{18x}{16} - \frac{x - 5}{16} = \frac{18x - (x - 5)}{16} = \frac{18x - x + 5}{16} = \frac{17x + 5}{16}
$$

✔ Answer: $\boxed{\frac{17x + 5}{16}}$

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4. $\frac{x + 7}{3} + \frac{3x - 2}{18}$

LCM of 3 and 18 is 18.

Convert first fraction:
$$
\frac{x + 7}{3} = \frac{(x + 7) \cdot 6}{18} = \frac{6x + 42}{18}
$$

Add:
$$
\frac{6x + 42}{18} + \frac{3x - 2}{18} = \frac{6x + 42 + 3x - 2}{18} = \frac{9x + 40}{18}
$$

Check if reducible: 9 and 18 share a factor, but $9x + 40$ doesn't factor with 18 evenly. So leave as is.

✔ Answer: $\boxed{\frac{9x + 40}{18}}$

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5. $\frac{7x}{8} + \frac{3x}{10} - \frac{x}{5}$

Find LCM of 8, 10, and 5 → LCM is 40

Convert each term:

- $\frac{7x}{8} = \frac{7x \cdot 5}{40} = \frac{35x}{40}$
- $\frac{3x}{10} = \frac{3x \cdot 4}{40} = \frac{12x}{40}$
- $\frac{x}{5} = \frac{x \cdot 8}{40} = \frac{8x}{40}$

Now compute:
$$
\frac{35x}{40} + \frac{12x}{40} - \frac{8x}{40} = \frac{35x + 12x - 8x}{40} = \frac{39x}{40}
$$

✔ Answer: $\boxed{\frac{39x}{40}}$

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6. $\frac{2x}{11} - \frac{5(x - 2)}{11}$

Same denominator, so combine numerators:

$$
\frac{2x - 5(x - 2)}{11} = \frac{2x - 5x + 10}{11} = \frac{-3x + 10}{11}
$$

Or write as: $\frac{10 - 3x}{11}$

✔ Answer: $\boxed{\frac{10 - 3x}{11}}$

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7. $\frac{2}{x} - \frac{3}{x} + \frac{7}{2x}$

First combine like terms:
$$
\left(\frac{2 - 3}{x}\right) + \frac{7}{2x} = \frac{-1}{x} + \frac{7}{2x}
$$

Now find common denominator: $2x$

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

So:
$$
\frac{-2}{2x} + \frac{7}{2x} = \frac{5}{2x}
$$

✔ Answer: $\boxed{\frac{5}{2x}}$

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8. $\frac{4x}{7} - \frac{x - 3}{8}$

LCM of 7 and 8 is 56

Convert both:

- $\frac{4x}{7} = \frac{4x \cdot 8}{56} = \frac{32x}{56}$
- $\frac{x - 3}{8} = \frac{(x - 3) \cdot 7}{56} = \frac{7x - 21}{56}$

Now subtract:
$$
\frac{32x}{56} - \frac{7x - 21}{56} = \frac{32x - (7x - 21)}{56} = \frac{32x - 7x + 21}{56} = \frac{25x + 21}{56}
$$

No common factors between numerator and denominator.

✔ 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. $\boxed{\frac{10 - 3x}{11}}$
7. $\boxed{\frac{5}{2x}}$
8. $\boxed{\frac{25x + 21}{56}}$

Let me know if you'd like explanations for any specific steps!