Multiplying and Dividing Fractions

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Multiplying and dividing fractions is often easier than adding them. To multiply, multiply the numerators and multiply the denominators. To divide, flip the second fraction and multiply. For example, \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2} and \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2. When you are done, simplify the answer if the top and bottom share a factor.

Before this lesson: You should know how to simplify fractions and how to multiply and divide integers.

What Multiplying Fractions Means

Multiplying by a fraction means taking part of something. So \frac{2}{3} \times \frac{3}{4} means “two thirds of three fourths.”

In the picture, the orange columns show \frac{3}{4} of the square. The blue rows show \frac{2}{3}. The overlap is 6 of the 12 boxes, which is \frac{1}{2} of the square.

A square split into 3 rows and 4 columns, making 12 equal boxes. Two of the 3 rows are shaded blue to show 2/3. Three of the 4 columns are shaded orange to show 3/4. The boxes shaded both colors make 6 of the 12 boxes, so 2/3 × 3/4 = 6/12 = 1/2.
The overlap of 2/3 and 3/4 covers 6 of the 12 boxes, so the product is 6/12, or 1/2.

What Dividing Fractions Means

Dividing asks, “How many of these fit into that?” So \frac{1}{2} \div \frac{1}{4} asks how many fourths fit into one half. The answer is 2.

Two bars of the same length. The top bar shows 1/2 shaded. The bottom bar is split into fourths, and 2 fourths fit exactly under the shaded half, so 1/2 ÷ 1/4 = 2.
Dividing asks how many 1/4 pieces fit in 1/2. Two fit, so 1/2 ÷ 1/4 = 2.

Rules for Multiplying and Dividing Fractions

Multiplying

Multiply straight across: tops times tops, and bottoms times bottoms. You don’t need a common denominator.

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Dividing

The reciprocal of a fraction is the fraction flipped upside down. The reciprocal of \frac{2}{5} is \frac{5}{2}. To divide, multiply by the reciprocal of the second fraction.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Many students remember this as keep, change, flip: keep the first fraction, change ÷ to ×, and flip the second fraction. Here a, b, c and d are whole numbers, and none of the bottom numbers is 0.

Simplify before you multiply

If a top number and a bottom number share a factor, you can divide them both by it before you multiply. This keeps the numbers small. It works even when the two numbers are in different fractions.

Worked Examples

Example 1: Multiply two fractions

Find \frac{2}{5} \times \frac{3}{7}.

Step 1. Multiply the numerators: 2 × 3 = 6.

Step 2. Multiply the denominators: 5 × 7 = 35.

Answer: \frac{2}{5} \times \frac{3}{7} = \frac{6}{35}. It is already in simplest form.

Example 2: Simplify first

Find \frac{4}{9} \times \frac{3}{8}.

Step 1. The 4 on top and the 8 on the bottom share 4. Divide both by 4 to get 1 and 2.

Step 2. The 3 on top and the 9 on the bottom share 3. Divide both by 3 to get 1 and 3.

\frac{4}{9} \times \frac{3}{8} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

Answer: \frac{1}{6}. If you multiply first, you get \frac{12}{72}, which also simplifies to \frac{1}{6}.

Example 3: A whole number times a fraction

Find 6 \times \frac{2}{3}.

Step 1. Write the whole number as a fraction: 6 = \frac{6}{1}.

Step 2. Multiply: \frac{6}{1} \times \frac{2}{3} = \frac{12}{3}.

Answer: \frac{12}{3} = 4.

Example 4: Divide two fractions

Find \frac{3}{4} \div \frac{2}{5}.

Step 1. Keep \frac{3}{4}. Change ÷ to ×. Flip \frac{2}{5} to \frac{5}{2}.

\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}

Answer: \frac{15}{8}, which is the same as 1\frac{7}{8}.

Example 5: Word problem

A ribbon is \frac{3}{4} of a yard long. You cut it into pieces that are each \frac{1}{8} of a yard long. How many pieces do you get?

Step 1. “How many pieces fit” means divide: \frac{3}{4} \div \frac{1}{8}.

Step 2. Keep, change, flip.

\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6

Answer: You get 6 pieces.

Common Mistakes

MistakeWhat it looks likeHow to fix it
Finding a common denominator to multiplyRewriting \frac{2}{5} \times \frac{3}{7} with 35 on the bottomYou only need a common denominator to add or subtract. To multiply, go straight across.
Flipping the wrong fraction\frac{3}{4} \div \frac{2}{5} = \frac{4}{3} \times \frac{2}{5}Keep the first fraction. Flip only the second one.
Forgetting to flip\frac{3}{4} \div \frac{2}{5} = \frac{6}{20}Dividing means multiplying by the reciprocal: \frac{3}{4} \times \frac{5}{2}.
Multiplying a whole number by the top and bottom6 \times \frac{2}{3} = \frac{12}{18}Write 6 as \frac{6}{1}. Only the top gets multiplied by 6.
Simplifying two tops togetherDividing the 4 and the 3 in \frac{4}{9} \times \frac{3}{8}Simplify a top with a bottom, never a top with a top.

Fractions of a Number and Negative Fractions

The word of usually means multiply. To find \frac{1}{2} of 10, multiply: \frac{1}{2} \times 10 = 5.

Negative fractions use the same sign rules as multiplying integers. Different signs give a negative answer, and the same signs give a positive answer.

-\frac{3}{4} \times \frac{2}{3} = -\frac{6}{12} = -\frac{1}{2}

To multiply or divide mixed numbers, change them to improper fractions first. The next lesson shows how.

Practice Problems

Write each answer in simplest form.

  1. \frac{1}{3} \times \frac{3}{5}
  2. \frac{5}{6} \times \frac{2}{3}
  3. \frac{7}{8} \times \frac{4}{7}
  4. 10 \times \frac{3}{5}
  5. \frac{1}{2} \div \frac{1}{3}
  6. \frac{4}{5} \div 2
  7. \frac{2}{3} \div \frac{4}{9}
  8. \frac{5}{8} \div \frac{5}{6}
  9. -\frac{3}{4} \times \frac{2}{3}
  10. A class has 24 students. Two thirds of them play a sport. How many students play a sport?
Show answers
  1. \frac{1 \times 3}{3 \times 5} = \frac{3}{15} = \frac{1}{5}
  2. \frac{5 \times 2}{6 \times 3} = \frac{10}{18} = \frac{5}{9}
  3. The 7s and the 4 and 8 simplify: \frac{1}{2} \times \frac{1}{1} = \frac{1}{2}
  4. \frac{10}{1} \times \frac{3}{5} = \frac{30}{5} = 6
  5. \frac{1}{2} \times \frac{3}{1} = \frac{3}{2}
  6. Write 2 as \frac{2}{1} and flip it: \frac{4}{5} \times \frac{1}{2} = \frac{4}{10} = \frac{2}{5}
  7. \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}
  8. \frac{5}{8} \times \frac{6}{5} = \frac{30}{40} = \frac{3}{4}
  9. The signs are different, so the answer is negative: -\frac{6}{12} = -\frac{1}{2}
  10. \frac{2}{3} \times 24 = \frac{48}{3} = 16. So 16 students play a sport.

Multiplying and Dividing Fractions FAQ

Why do you flip the second fraction when dividing?

Dividing by a number is the same as multiplying by its reciprocal. For example, dividing by 2 is the same as multiplying by \frac{1}{2}. Flipping gives you the reciprocal.

Do you need a common denominator to multiply fractions?

No. Just multiply the tops and multiply the bottoms. Common denominators are only needed to add or subtract.

What is the reciprocal of a whole number?

Write the whole number over 1, then flip it. The reciprocal of 5 is \frac{1}{5}.

Does multiplying by a fraction always make a number smaller?

It does when the fraction is less than 1. Multiplying by \frac{1}{2} cuts a number in half. Multiplying by a fraction greater than 1, like \frac{3}{2}, makes it bigger.

Summary

  • To multiply fractions, multiply the numerators and multiply the denominators.
  • To divide fractions, keep the first fraction, change ÷ to ×, and flip the second.
  • You don’t need a common denominator to multiply or divide.
  • Simplify a top with a bottom before you multiply to keep numbers small.
  • Write whole numbers over 1 before you multiply or divide.

Previous lesson: Adding and Subtracting Fractions

Next lesson: Mixed Numbers and Improper Fractions

Back to the Pre-Algebra (Math Basics) Index

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