Complex Fractions – A Fraction Within a Fraction
A complex fraction is a fraction whose numerator, denominator, or both, are themselves fractions – essentially, one fraction being divided by another. The key skill for simplifying a complex fraction is remembering that dividing by a fraction means multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).
A closely related skill is combining separate unit fractions (like 1/x and 1/y) into a single fraction over a common denominator – a building block you'll need whenever a complex fraction's numerator or denominator is itself a sum, rather than a single term.
Simplifying a Complex Fraction
(a/b) ÷ (c/d) = (a/b) × (d/c). To combine 1/x + 1/y into one fraction, use the common denominator xy: 1/x + 1/y = (y + x) ÷ xy.
(6/5) × (10/3) = (6 × 10) ÷ (5 × 3) = 60 ÷ 15 = 4.
1/3 = 4/12 and 1/4 = 3/12, so the numerator is 4 + 3 = 7.
Real-Life Application
- Combined work rates: calculating how long two workers take together involves adding unit-rate fractions.
- Electrical circuits: combining resistors in parallel uses complex fraction calculations.
- Finance: combining fractional interest rates from multiple sources.
Key Takeaways
- A complex fraction is a fraction divided by another fraction.
- Dividing by a fraction means multiplying by its reciprocal.
- Unit fractions combine over a common denominator before further simplifying.
Practice: Complex Fractions
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