Divide Fractions Using Reciprocals
Dividing fractions using reciprocals means converting a fraction division problem to multiplication by flipping the second fraction. To compute (a/b) divided by (c/d), multiply the first fraction by the reciprocal of the second: (a/b) times (d/c). For (3/4) divided by (2/3): multiply by (3/2) to get 9/8 = 1 and 1/8. This Grade 7 math skill from Saxon Math, Course 2 uses the keep-change-flip technique and is essential for all algebraic fraction work, unit rate calculations, and any applied problem involving division of fractional quantities.
Key Concepts
Property To divide a fraction by a fraction, multiply the dividend by the reciprocal of the divisor. $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$.
Examples $\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$ $\frac{2}{5} \div \frac{2}{3} = \frac{2}{5} \times \frac{3}{2} = \frac{6}{10} = \frac{3}{5}$.
Explanation The most common method for dividing fractions is to "invert and multiply". This means you keep the first fraction the same, change the division sign to a multiplication sign, and flip the second fraction to its reciprocal. After rewriting the problem as multiplication, multiply the numerators together and the denominators together. This procedure works because dividing by a number is the same as multiplying by its reciprocal.
Common Questions
How do I divide fractions using reciprocals?
Keep the first fraction, change the division sign to multiplication, and flip the second fraction to get its reciprocal. For (5/6) divided by (2/3): multiply (5/6) by (3/2) = 15/12 = 5/4.
What is the keep-change-flip rule?
Keep the first fraction unchanged. Change the operation from division to multiplication. Flip the second fraction to get its reciprocal. Then multiply straight across.
Why does dividing by a fraction equal multiplying by its reciprocal?
Reciprocals are multiplicative inverses. Multiplying by (d/c) undoes the effect of multiplying by (c/d), so dividing by (c/d) and multiplying by (d/c) give the same result.
What is (2/5) divided by (4/15)?
Apply keep-change-flip: (2/5) times (15/4) = 30/20 = 3/2 = 1 and 1/2.
When do students learn to divide fractions?
Fraction division is introduced in Grade 5-6 and mastered in Grade 7. Saxon Math, Course 2 covers it in Chapter 7 alongside fraction multiplication.
What is a common mistake when dividing fractions?
The most common error is flipping the first fraction instead of the second. Always keep the first fraction and flip only the second (the divisor) before multiplying.
How does dividing fractions by reciprocals connect to algebra?
In algebra, dividing by a fraction or rational expression uses the same rule. Understanding this for numbers in Grade 7 makes algebraic fraction division in later courses conceptually straightforward.