Math worksheet with fraction problems and solutions.

Let’s solve each problem step by step. We’ll go one at a time, carefully checking our work.

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Problem 1:
Simplify: $\frac{3}{x} + \frac{2}{x}$

Since the denominators are the same, we add the numerators:

$\frac{3 + 2}{x} = \frac{5}{x}$

✔ Final Answer for #1: $\boxed{\frac{5}{x}}$

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Problem 2:
Simplify: $\frac{4}{x+1} - \frac{1}{x+1}$

Same denominator → subtract numerators:

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

✔ Final Answer for #2: $\boxed{\frac{3}{x+1}}$

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Problem 3:
Simplify: $\frac{x}{x-2} + \frac{2}{x-2}$

Same denominator → add numerators:

$\frac{x + 2}{x - 2}$

Cannot simplify further (no common factors).

✔ Final Answer for #3: $\boxed{\frac{x+2}{x-2}}$

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Problem 4:
Simplify: $\frac{5}{y} - \frac{3}{y}$

Same denominator → subtract:

$\frac{5 - 3}{y} = \frac{2}{y}$

✔ Final Answer for #4: $\boxed{\frac{2}{y}}$

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Problem 5:
Simplify: $\frac{a}{a+3} + \frac{6}{a+3}$

Add numerators:

$\frac{a + 6}{a + 3}$

No simplification possible.

✔ Final Answer for #5: $\boxed{\frac{a+6}{a+3}}$

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Problem 6:
Simplify: $\frac{7}{m} - \frac{2}{m}$

Subtract numerators:

$\frac{7 - 2}{m} = \frac{5}{m}$

✔ Final Answer for #6: $\boxed{\frac{5}{m}}$

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Problem 7:
Simplify: $\frac{n^2}{n-1} - \frac{1}{n-1}$

Subtract numerators:

$\frac{n^2 - 1}{n - 1}$

Factor numerator: $n^2 - 1 = (n - 1)(n + 1)$

So:

$\frac{(n - 1)(n + 1)}{n - 1} = n + 1$ (as long as $n \ne 1$)

✔ Final Answer for #7: $\boxed{n + 1}$

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Problem 8:
Simplify: $\frac{p^2 - 9}{p - 3}$

Factor numerator: $p^2 - 9 = (p - 3)(p + 3)$

So:

$\frac{(p - 3)(p + 3)}{p - 3} = p + 3$ (as long as $p \ne 3$)

✔ Final Answer for #8: $\boxed{p + 3}$

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Problem 9:
Simplify: $\frac{q^2 + 4q + 4}{q + 2}$

Factor numerator: $q^2 + 4q + 4 = (q + 2)^2$

So:

$\frac{(q + 2)^2}{q + 2} = q + 2$ (as long as $q \ne -2$)

✔ Final Answer for #9: $\boxed{q + 2}$

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Problem 10:
Simplify: $\frac{r^2 - 5r + 6}{r - 2}$

Factor numerator: Find two numbers that multiply to 6 and add to -5 → -2 and -3

So: $r^2 - 5r + 6 = (r - 2)(r - 3)$

Then:

$\frac{(r - 2)(r - 3)}{r - 2} = r - 3$ (as long as $r \ne 2$)

✔ Final Answer for #10: $\boxed{r - 3}$

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Problem 11:
Simplify: $\frac{s^2 - 16}{s + 4}$

Factor numerator: $s^2 - 16 = (s - 4)(s + 4)$

So:

$\frac{(s - 4)(s + 4)}{s + 4} = s - 4$ (as long as $s \ne -4$)

✔ Final Answer for #11: $\boxed{s - 4}$

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Problem 12:
Simplify: $\frac{t^2 + 6t + 9}{t + 3}$

Factor numerator: $t^2 + 6t + 9 = (t + 3)^2$

So:

$\frac{(t + 3)^2}{t + 3} = t + 3$ (as long as $t \ne -3$)

✔ Final Answer for #12: $\boxed{t + 3}$

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Problem 13:
Simplify: $\frac{u^2 - 4u - 12}{u - 6}$

Factor numerator: Find two numbers that multiply to -12 and add to -4 → -6 and +2

Wait: (-6) * (+2) = -12, but (-6) + (+2) = -4 → yes!

So: $u^2 - 4u - 12 = (u - 6)(u + 2)$

Then:

$\frac{(u - 6)(u + 2)}{u - 6} = u + 2$ (as long as $u \ne 6$)

✔ Final Answer for #13: $\boxed{u + 2}$

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Problem 14:
Simplify: $\frac{v^2 + 2v - 8}{v + 4}$

Factor numerator: Find two numbers that multiply to -8 and add to +2 → +4 and -2

Check: 4 * (-2) = -8, 4 + (-2) = 2 → yes!

So: $v^2 + 2v - 8 = (v + 4)(v - 2)$

Then:

$\frac{(v + 4)(v - 2)}{v + 4} = v - 2$ (as long as $v \ne -4$)

✔ Final Answer for #14: $\boxed{v - 2}$

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Problem 15:
Simplify: $\frac{w^2 - w - 6}{w - 3}$

Factor numerator: Find two numbers that multiply to -6 and add to -1 → -3 and +2

Check: (-3)*2 = -6, (-3)+2 = -1 → yes!

So: $w^2 - w - 6 = (w - 3)(w + 2)$

Then:

$\frac{(w - 3)(w + 2)}{w - 3} = w + 2$ (as long as $w \ne 3$)

✔ Final Answer for #15: $\boxed{w + 2}$

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Problem 16:
Simplify: $\frac{x^2 + 5x + 6}{x + 2}$

Factor numerator: Two numbers that multiply to 6 and add to 5 → 2 and 3

So: $x^2 + 5x + 6 = (x + 2)(x + 3)$

Then:

$\frac{(x + 2)(x + 3)}{x + 2} = x + 3$ (as long as $x \ne -2$)

✔ Final Answer for #16: $\boxed{x + 3}$

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Problem 17:
Simplify: $\frac{y^2 - 7y + 10}{y - 5}$

Factor numerator: Two numbers that multiply to 10 and add to -7 → -5 and -2

Check: (-5)*(-2)=10, (-5)+(-2)=-7 → yes!

So: $y^2 - 7y + 10 = (y - 5)(y - 2)$

Then:

$\frac{(y - 5)(y - 2)}{y - 5} = y - 2$ (as long as $y \ne 5$)

✔ Final Answer for #17: $\boxed{y - 2}$

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Problem 18:
Simplify: $\frac{z^2 + z - 12}{z + 4}$

Factor numerator: Two numbers that multiply to -12 and add to +1 → +4 and -3

Check: 4*(-3) = -12, 4 + (-3) = 1 → yes!

So: $z^2 + z - 12 = (z + 4)(z - 3)$

Then:

$\frac{(z + 4)(z - 3)}{z + 4} = z - 3$ (as long as $z \ne -4$)

✔ Final Answer for #18: $\boxed{z - 3}$

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Problem 19:
Simplify: $\frac{a^2 - 9a + 18}{a - 6}$

Factor numerator: Two numbers that multiply to 18 and add to -9 → -6 and -3

Check: (-6)*(-3)=18, (-6)+(-3)=-9 → yes!

So: $a^2 - 9a + 18 = (a - 6)(a - 3)$

Then:

$\frac{(a - 6)(a - 3)}{a - 6} = a - 3$ (as long as $a \ne 6$)

✔ Final Answer for #19: $\boxed{a - 3}$

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Problem 20:
Simplify: $\frac{b^2 + 8b + 15}{b + 5}$

Factor numerator: Two numbers that multiply to 15 and add to 8 → 3 and 5

So: $b^2 + 8b + 15 = (b + 3)(b + 5)$

Then:

$\frac{(b + 3)(b + 5)}{b + 5} = b + 3$ (as long as $b \ne -5$)

✔ Final Answer for #20: $\boxed{b + 3}$

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Final Answer:
1. $\boxed{\frac{5}{x}}$
2. $\boxed{\frac{3}{x+1}}$
3. $\boxed{\frac{x+2}{x-2}}$
4. $\boxed{\frac{2}{y}}$
5. $\boxed{\frac{a+6}{a+3}}$
6. $\boxed{\frac{5}{m}}$
7. $\boxed{n + 1}$
8. $\boxed{p + 3}$
9. $\boxed{q + 2}$
10. $\boxed{r - 3}$
11. $\boxed{s - 4}$
12. $\boxed{t + 3}$
13. $\boxed{u + 2}$
14. $\boxed{v - 2}$
15. $\boxed{w + 2}$
16. $\boxed{x + 3}$
17. $\boxed{y - 2}$
18. $\boxed{z - 3}$
19. $\boxed{a - 3}$
20. $\boxed{b + 3}$