Unit 7 · Lesson 4 of 5
Complex fractions
You'll be able to: turn a fraction that has fractions inside it into one simple fraction.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first, then check.
Simplify: \(\dfrac{x^2}{x+4} - \dfrac{16}{x+4}\)
Same denominator, so subtract the tops: \(\dfrac{x^2 - 16}{x+4}\).
Factor the top: \(\dfrac{(x-4)(x+4)}{x+4}\). Cancel \((x+4)\). Answer: \(x - 4\).
Factor \(x^2 + x - 12\).
Find two numbers that multiply to \(-12\) and add to \(1\): \(4\) and \(-3\).
Answer: \((x+4)(x-3)\).
Step 2 of 5The idea
A complex fraction is a fraction with smaller fractions inside its top or bottom, like \(\dfrac{1 + \frac{2}{x}}{1 - \frac{4}{x^2}}\). "Complex" here just means "built from parts". It is not harder math, only more layers.
The fastest fix: multiply the big top and the big bottom by the LCD of all the little fractions. Every little denominator cancels, and you are left with one ordinary fraction.
Method 1 (main): multiply by the LCD.
- Find the LCD of every little denominator, on top and on bottom.
- Multiply every term of the big top and the big bottom by it.
- Simplify: factor and cancel.
Method 2 (also fine): divide two fractions. Combine the top into one fraction, combine the bottom into one fraction, then multiply the top by the reciprocal of the bottom.
Multiplying the top and bottom by the same thing is allowed because it is the same as multiplying by \(1\).
Why this shows up on the placement test: complex fractions mix fraction rules, the LCD, and factoring, so one question tests a lot at once.
Step 3 of 5Worked examples
Example 1: \(\dfrac{\frac{3}{4} - \frac{1}{2}}{\frac{1}{3} + \frac{1}{6}}\)
- The little denominators are 4, 2, 3, and 6. Their LCD is 12.
- Top times 12: \(12 \cdot \frac{3}{4} - 12 \cdot \frac{1}{2} = 9 - 6 = 3\).
- Bottom times 12: \(12 \cdot \frac{1}{3} + 12 \cdot \frac{1}{6} = 4 + 2 = 6\).
- \(\dfrac{3}{6} = \dfrac{1}{2}\).
Example 2: \(\dfrac{1 + \frac{2}{x}}{1 - \frac{4}{x^2}}\)
- The little denominators are \(x\) and \(x^2\). The LCD is \(x^2\).
- Top times \(x^2\): \(x^2 \cdot 1 + x^2 \cdot \frac{2}{x} = x^2 + 2x\).
- Bottom times \(x^2\): \(x^2 \cdot 1 - x^2 \cdot \frac{4}{x^2} = x^2 - 4\).
- Factor and cancel: \(\dfrac{x(x+2)}{(x-2)(x+2)} = \dfrac{x}{x-2}\).
- Same problem with Method 2: the top is \(\dfrac{x+2}{x}\), the bottom is \(\dfrac{x^2-4}{x^2}\). Then \(\dfrac{x+2}{x} \cdot \dfrac{x^2}{(x-2)(x+2)} = \dfrac{x}{x-2}\). (Same answer. Use whichever method feels clearer to you.)
Step 4 of 5Practice
Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.
Simplify: \(\dfrac{\frac{2}{3}}{\frac{4}{9}}\)
This is \(\dfrac{2}{3} \div \dfrac{4}{9} = \dfrac{2}{3} \cdot \dfrac{9}{4}\).
Cancel: \(\dfrac{1}{1} \cdot \dfrac{3}{2}\). Answer: \(\dfrac{3}{2}\).
Simplify: \(\dfrac{\frac{1}{2} + \frac{1}{4}}{\frac{3}{8}}\)
The LCD of 2, 4, and 8 is 8.
Top times 8: \(4 + 2 = 6\). Bottom times 8: \(3\).
\(\dfrac{6}{3} = 2\).
Simplify: \(\dfrac{\frac{x}{6}}{\frac{x^2}{3}}\)
This is \(\dfrac{x}{6} \cdot \dfrac{3}{x^2} = \dfrac{3x}{6x^2}\).
Cancel 3 and one \(x\). Answer: \(\dfrac{1}{2x}\).
Simplify: \(\dfrac{1 - \frac{1}{x}}{x - \frac{1}{x}}\)
The LCD is \(x\). Multiply every term by \(x\).
Top: \(x - 1\). Bottom: \(x^2 - 1\).
Factor the bottom: \(\dfrac{x-1}{(x-1)(x+1)}\). Cancel \((x-1)\). Answer: \(\dfrac{1}{x+1}\).
Simplify: \(\dfrac{\frac{3}{x} - \frac{2}{y}}{\frac{1}{xy}}\)
The LCD is \(xy\). Multiply every term by \(xy\).
Top: \(xy \cdot \frac{3}{x} - xy \cdot \frac{2}{y} = 3y - 2x\). Bottom: \(xy \cdot \frac{1}{xy} = 1\).
Answer: \(3y - 2x\).
Simplify: \(\dfrac{\frac{2}{x+1} - 1}{\frac{1}{x+1}}\)
The LCD is \(x + 1\). Multiply every term by \(x+1\).
Top: \(2 - (x+1) = 2 - x - 1 = 1 - x\). Bottom: \(1\).
Answer: \(1 - x\).
Simplify: \(\dfrac{\frac{1}{x} - \frac{1}{3}}{x - 3}\)
The LCD is \(3x\). Top times \(3x\): \(3 - x\). Bottom times \(3x\): \(3x(x-3)\).
Write \(3 - x = -(x-3)\): \(\dfrac{-(x-3)}{3x(x-3)}\).
Cancel \((x-3)\). Answer: \(-\dfrac{1}{3x}\).
Simplify: \(\dfrac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{x} + \frac{1}{y}}\)
The LCD is \(xy\). Top times \(xy\): \(x^2 - y^2\). Bottom times \(xy\): \(y + x\).
Factor the top: \(\dfrac{(x-y)(x+y)}{x+y}\).
Cancel \((x+y)\). Answer: \(x - y\).
Step 5 of 5Wrap-up
You can now simplify complex fractions: multiply every term on the top and bottom by the LCD of the little fractions, then factor and cancel.
Read or watch next: OpenStax Intermediate Algebra 2e, 7.3 Simplify Complex Rational Expressions (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Nested fractions (short videos). Quick rules: Rational expressions cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why am I allowed to multiply the top and bottom by the LCD?" It knows this course and can explain things another way.