Multiply by the Reciprocal
Dividing fractions by multiplying by the reciprocal is the keep-change-flip rule: keep the first fraction, change the division sign to multiplication, and flip the second fraction. To compute (3/4) divided by (2/5), rewrite as (3/4) times (5/2) = 15/8 = 1 and 7/8. This Grade 7 math skill from Saxon Math, Course 2 converts division of fractions — which is non-intuitive — into multiplication, which students already understand, and is foundational for algebra (dividing rational expressions) and proportional reasoning.
Key Concepts
Property To find the quotient of two fractions, multiply the dividend by the reciprocal of the divisor. This rule turns a tricky division problem into a simple multiplication one. For example: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.
Examples $\frac{3}{5} \div \frac{2}{3} = \frac{3}{5} \times \frac{3}{2} = \frac{9}{10}$ $\frac{7}{8} \div \frac{1}{4} = \frac{7}{8} \times \frac{4}{1} = \frac{28}{8} = \frac{7}{2}$ $6 \div \frac{2}{3} = 6 \times \frac{3}{2} = \frac{18}{2} = 9$.
Explanation Think of it like finding how many quarters are in five dollars. First, you find how many quarters are in one dollar (the reciprocal, 4), then multiply by five. This two step thinking makes dividing fractions a piece of cake!
Common Questions
How do I divide fractions by multiplying by the reciprocal?
Keep the first fraction, change the division to multiplication, and flip the second fraction (use its reciprocal). For (3/4) divided by (2/5): (3/4) times (5/2) = 15/8.
Why does dividing by a fraction equal multiplying by its reciprocal?
Division undoes multiplication. Multiplying by a number and then multiplying by its reciprocal gets you back to where you started. Dividing by 2/5 produces the same result as multiplying by 5/2.
What is the keep-change-flip method?
Keep-change-flip is a memory aid for dividing fractions: keep the first fraction unchanged, change the division sign to multiplication, flip the second fraction to get its reciprocal.
Does keep-change-flip work for dividing mixed numbers?
Yes, but first convert mixed numbers to improper fractions, then apply keep-change-flip. For 2 and 1/2 divided by 3/4: convert to 5/2 divided by 3/4 = 5/2 times 4/3 = 20/6 = 10/3.
When do students learn to divide fractions using reciprocals?
Fraction division is introduced in Grade 5-6 and reinforced in Grade 7. Saxon Math, Course 2 covers it in Chapter 7 as a key rational number operation.
What are common mistakes when dividing fractions?
Students sometimes flip the wrong fraction — always flip the second fraction (the divisor), not the first. Also, do not flip before changing the sign from division to multiplication.
How does dividing fractions connect to algebra?
In algebra, dividing by a fraction or rational expression uses exactly the same rule. Mastering fraction division in Grade 7 makes algebraic fraction manipulation much more natural.