Simplifying Complex Fractions
Simplifying a complex fraction means converting a fraction that has a fraction in its denominator into a simpler form. The technique is to multiply the entire complex fraction by a special form of 1 — the reciprocal of the denominator fraction divided by itself — which makes the denominator equal to 1 and leaves a clean simple fraction. For example, to simplify (3/4) / (2/5): multiply by (5/2) / (5/2), giving (3/4 × 5/2) / 1 = 15/8. This is equivalent to multiplying by the reciprocal of the denominator. Covered in Saxon Math, Course 2, this skill is essential for 7th grade rational number fluency.
Key Concepts
Property To simplify a complex fraction, multiply it by a clever form of 1. This special form of 1 is built by using the reciprocal of the denominator over itself, which makes the new denominator equal to 1.
Examples To solve, multiply by the reciprocal of the denominator: $ \frac{\frac{3}{5}}{\frac{2}{3}} \cdot \frac{\frac{3}{2}}{\frac{3}{2}} = \frac{\frac{9}{10}}{1} = \frac{9}{10} $ First, convert to improper fractions, then multiply: $ \frac{15}{7\frac{1}{3}} = \frac{\frac{15}{1}}{\frac{22}{3}} \cdot \frac{\frac{3}{22}}{\frac{3}{22}} = \frac{\frac{45}{22}}{1} = 2\frac{1}{22} $ Flip the bottom fraction and multiply: $ \frac{\frac{1}{4}}{\frac{5}{8}} \cdot \frac{\frac{8}{5}}{\frac{8}{5}} = \frac{\frac{8}{20}}{1} = \frac{2}{5} $.
Explanation The ultimate trick is to make the denominator vanish by turning it into 1! Find the reciprocal of the bottom fraction (just flip it upside down) and multiply both the top and bottom of the big fraction by this flipped version. The bottom becomes 1, and the top becomes your new, simplified answer!
Common Questions
What is a complex fraction?
A complex fraction is a fraction that has another fraction in its numerator, denominator, or both. For example, (3/4) / (5/6) is a complex fraction.
How do you simplify a complex fraction?
Multiply the numerator fraction by the reciprocal of the denominator fraction. For (3/4) / (5/6): multiply 3/4 by 6/5 to get 18/20, which simplifies to 9/10.
Why does multiplying by the reciprocal of the denominator work?
Any number divided by itself equals 1. By multiplying both the numerator and denominator of the complex fraction by the reciprocal of the denominator fraction, the denominator becomes 1, leaving just the simplified numerator.
What is a reciprocal and how is it used?
The reciprocal of a fraction is made by flipping it: the reciprocal of 3/4 is 4/3. Multiplying a fraction by its reciprocal always gives 1. In complex fractions, using the reciprocal of the denominator turns the denominator into 1.
How does simplifying complex fractions relate to dividing fractions?
Simplifying a complex fraction is the same as dividing fractions. (a/b) / (c/d) = (a/b) × (d/c), which is the standard ‘multiply by the reciprocal’ rule for fraction division.
When do students learn complex fractions?
Complex fractions are covered in 7th grade math as an extension of fraction operations, particularly division of fractions.
Which textbook covers simplifying complex fractions?
Saxon Math, Course 2 covers simplifying complex fractions.