Simplifying Complex Fractions Worksheet featuring eight problems for practice.

Let’s solve each problem one by one, step by step. We’ll simplify complex fractions by finding common denominators, combining terms, and reducing where possible.

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Problem ①:
\[
\frac{5}{x - 1} - \frac{3}{x + 1}
\]

We need a common denominator: \((x - 1)(x + 1)\)

Rewrite both fractions:

\[
= \frac{5(x + 1)}{(x - 1)(x + 1)} - \frac{3(x - 1)}{(x - 1)(x + 1)}
\]

Expand numerators:

\[
= \frac{5x + 5 - (3x - 3)}{(x - 1)(x + 1)} = \frac{5x + 5 - 3x + 3}{(x - 1)(x + 1)} = \frac{2x + 8}{(x - 1)(x + 1)}
\]

Factor numerator if possible: \(2x + 8 = 2(x + 4)\)

Denominator is difference of squares: \(x^2 - 1\)

So final answer:
\[
\frac{2(x + 4)}{x^2 - 1}
\]

---

Problem ②:
\[
\frac{\frac{1}{x} - \frac{1}{x - 1}}{1 - \frac{1}{x}}
\]

First, simplify numerator and denominator separately.

Numerator: \(\frac{1}{x} - \frac{1}{x - 1}\)

Common denominator: \(x(x - 1)\)

\[
= \frac{(x - 1) - x}{x(x - 1)} = \frac{-1}{x(x - 1)}
\]

Denominator: \(1 - \frac{1}{x} = \frac{x - 1}{x}\)

Now divide the two:

\[
\frac{\frac{-1}{x(x - 1)}}{\frac{x - 1}{x}} = \frac{-1}{x(x - 1)} \cdot \frac{x}{x - 1} = \frac{-1 \cdot x}{x(x - 1)(x - 1)} = \frac{-1}{(x - 1)^2}
\]

(Cancel one \(x\) from top and bottom.)

Final answer:
\[
-\frac{1}{(x - 1)^2}
\]

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Problem ③:
\[
\frac{5 - \frac{2}{5}}{6 + \frac{1}{3}}
\]

Simplify numerator and denominator.

Numerator: \(5 - \frac{2}{5} = \frac{25}{5} - \frac{2}{5} = \frac{23}{5}\)

Denominator: \(6 + \frac{1}{3} = \frac{18}{3} + \frac{1}{3} = \frac{19}{3}\)

Now divide:

\[
\frac{23/5}{19/3} = \frac{23}{5} \cdot \frac{3}{19} = \frac{69}{95}
\]

Check if reducible: 69 = 3×23, 95 = 5×19 → no common factors.

Final answer:
\[
\frac{69}{95}
\]

---

Problem ④:
\[
\frac{x + \frac{2d}{3ac}}{x + \frac{3d}{3ac}}
\]

Note: \(\frac{3d}{3ac} = \frac{d}{ac}\), so rewrite:

\[
\frac{x + \frac{2d}{3ac}}{x + \frac{d}{ac}}
\]

Get common denominators in numerator and denominator.

Numerator: \(x + \frac{2d}{3ac} = \frac{3acx + 2d}{3ac}\)

Denominator: \(x + \frac{d}{ac} = \frac{acx + d}{ac}\)

Now divide:

\[
\frac{\frac{3acx + 2d}{3ac}}{\frac{acx + d}{ac}} = \frac{3acx + 2d}{3ac} \cdot \frac{ac}{acx + d} = \frac{(3acx + 2d) \cdot ac}{3ac \cdot (acx + d)}
\]

Cancel \(ac\) top and bottom:

\[
= \frac{3acx + 2d}{3(acx + d)}
\]

Can’t simplify further unless factoring helps — but it doesn’t here.

Final answer:
\[
\frac{3acx + 2d}{3(acx + d)}
\]

---

Problem ⑤:
\[
\frac{x^2\left(\frac{2}{x^2} + \frac{1}{x}\right)}{x^2\left(\frac{4}{x^2} + \frac{1}{x}\right)}
\]

Notice \(x^2\) is multiplied in both numerator and denominator → cancel them!

Left with:

\[
\frac{\frac{2}{x^2} + \frac{1}{x}}{\frac{4}{x^2} + \frac{1}{x}}
\]

Multiply numerator and denominator by \(x^2\) to eliminate small fractions:

Numerator: \(x^2 \cdot \left(\frac{2}{x^2} + \frac{1}{x}\right) = 2 + x\)

Denominator: \(x^2 \cdot \left(\frac{4}{x^2} + \frac{1}{x}\right) = 4 + x\)

So simplified:

\[
\frac{x + 2}{x + 4}
\]

Final answer:
\[
\frac{x + 2}{x + 4}
\]

---

Problem ⑥:
\[
\frac{\frac{1}{2} + \frac{1}{3} + \frac{1}{4}}{3 - \frac{4}{5}}
\]

Numerator: Find LCD of 2,3,4 → 12

\[
\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12}
\]

Denominator: \(3 - \frac{4}{5} = \frac{15}{5} - \frac{4}{5} = \frac{11}{5}\)

Now divide:

\[
\frac{13/12}{11/5} = \frac{13}{12} \cdot \frac{5}{11} = \frac{65}{132}
\]

Check reduction: 65 = 5×13, 132 = 12×11 = 2²×3×11 → no common factors.

Final answer:
\[
\frac{65}{132}
\]

---

Problem ⑦:
\[
\frac{\frac{4}{5}}{\frac{1}{5} + \frac{2}{3}}
\]

Denominator: LCD of 5 and 3 is 15

\[
\frac{1}{5} + \frac{2}{3} = \frac{3}{15} + \frac{10}{15} = \frac{13}{15}
\]

Now divide:

\[
\frac{4/5}{13/15} = \frac{4}{5} \cdot \frac{15}{13} = \frac{4 \cdot 15}{5 \cdot 13} = \frac{60}{65}
\]

Reduce: divide numerator and denominator by 5 → \(\frac{12}{13}\)

Final answer:
\[
\frac{12}{13}
\]

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Problem ⑧:
\[
\frac{\frac{1}{4x} + \frac{2}{3x}}{\frac{1}{4x} + \frac{2}{3x}}
\]

Wait — numerator and denominator are identical!

So this equals 1, as long as the expression is defined (i.e., \(x \ne 0\)).

Final answer:
\[
1
\]

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Final Answer:
① $\frac{2(x + 4)}{x^2 - 1}$
② $-\frac{1}{(x - 1)^2}$
③ $\frac{69}{95}$
④ $\frac{3acx + 2d}{3(acx + d)}$
⑤ $\frac{x + 2}{x + 4}$
⑥ $\frac{65}{132}$
⑦ $\frac{12}{13}$
⑧ $1$