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How to Multiply and Divide Fractions Step by Step

Grade: 6-7 | Topic: Arithmetic

What You Will Learn​

After reading this guide, you will be able to multiply any two fractions confidently using the straight-across method with cross-cancellation. You will also master dividing fractions by flipping the divisor and multiplying. By the end, mixed numbers, whole numbers, and multi-step problems will all feel manageable.

Theory​

Multiplying fractions -- the straight-across rule​

Multiplying fractions is one of the most straightforward operations in arithmetic. You simply multiply the numerators together and the denominators together:

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

For example:

23×45=2×43×5=815\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}

Notice that you do not need a common denominator. This is a key difference from adding and subtracting fractions.

Cross-cancellation -- simplify before you multiply​

Cross-cancellation lets you divide any numerator and any denominator by a common factor before multiplying. This keeps the numbers small and often means the answer is already in lowest terms.

38×49\frac{3}{8} \times \frac{4}{9}

Here, 3 and 9 share a factor of 3, and 4 and 8 share a factor of 4:

3182×4193=1×12×3=16\frac{\cancel{3}^{1}}{\cancel{8}_{2}} \times \frac{\cancel{4}^{1}}{\cancel{9}_{3}} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6}

Without cross-cancellation you would get 1272\frac{12}{72} and still need to reduce. Cross-cancellation saves time and reduces errors.

Multiplying with whole numbers​

A whole number can be written as a fraction with a denominator of 1. For example, 5=515 = \frac{5}{1}. Then apply the same rule:

37×5=37×51=157=217\frac{3}{7} \times 5 = \frac{3}{7} \times \frac{5}{1} = \frac{15}{7} = 2\frac{1}{7}

Multiplying mixed numbers​

To multiply mixed numbers, first convert each one to an improper fraction, then multiply as usual:

213×112=73×322\frac{1}{3} \times 1\frac{1}{2} = \frac{7}{3} \times \frac{3}{2}

Cross-cancel the 3s:

73×32=72=312\frac{7}{\cancel{3}} \times \frac{\cancel{3}}{2} = \frac{7}{2} = 3\frac{1}{2}

Dividing fractions -- Keep, Change, Flip​

Dividing by a fraction means multiplying by its reciprocal (the fraction flipped upside down). Many students remember this as Keep, Change, Flip:

  1. Keep the first fraction as it is.
  2. Change the division sign to multiplication.
  3. Flip the second fraction (take its reciprocal).

ab÷cd=ab×dc=a×db×c\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}

For example:

56÷23=56×32=1512=54=114\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}

This works because dividing by 23\frac{2}{3} asks "how many groups of 23\frac{2}{3} fit into 56\frac{5}{6}?" Multiplying by the reciprocal answers that question.

Dividing mixed numbers​

Just like with multiplication, convert mixed numbers to improper fractions first, then apply Keep, Change, Flip:

312÷114=72÷54=72×45=2810=145=2453\frac{1}{2} \div 1\frac{1}{4} = \frac{7}{2} \div \frac{5}{4} = \frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = \frac{14}{5} = 2\frac{4}{5}

Worked Examples​

Example 1: Basic fraction multiplication (easy)​

Problem: Multiply 25×37\dfrac{2}{5} \times \dfrac{3}{7}.

Step 1: Multiply numerators and denominators straight across. 2×35×7=635\frac{2 \times 3}{5 \times 7} = \frac{6}{35}

Step 2: Check for simplification. The GCD of 6 and 35 is 1, so the fraction is already in lowest terms.

Answer: 635\dfrac{6}{35}

Example 2: Multiplication with cross-cancellation (medium)​

Problem: Multiply 815×512\dfrac{8}{15} \times \dfrac{5}{12}.

Step 1: Look for common factors between any numerator and any denominator.

  • 8 and 12 share a factor of 4: 8→28 \to 2, 12→312 \to 3.
  • 5 and 15 share a factor of 5: 5→15 \to 1, 15→315 \to 3.

Step 2: Multiply the simplified values. 23×13=29\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}

Answer: 29\dfrac{2}{9}

Example 3: Multiplying mixed numbers (medium)​

Problem: Multiply 123×2141\dfrac{2}{3} \times 2\dfrac{1}{4}.

Step 1: Convert to improper fractions. 123=53,214=941\frac{2}{3} = \frac{5}{3}, \quad 2\frac{1}{4} = \frac{9}{4}

Step 2: Cross-cancel. 3 and 9 share a factor of 3: 9→39 \to 3, 3→13 \to 1. 51×34=154\frac{5}{1} \times \frac{3}{4} = \frac{15}{4}

Step 3: Convert back to a mixed number. 154=334\frac{15}{4} = 3\frac{3}{4}

Answer: 3343\dfrac{3}{4}

Example 4: Basic fraction division (medium)​

Problem: Divide 78÷34\dfrac{7}{8} \div \dfrac{3}{4}.

Step 1: Keep, Change, Flip. 78÷34=78×43\frac{7}{8} \div \frac{3}{4} = \frac{7}{8} \times \frac{4}{3}

Step 2: Cross-cancel. 4 and 8 share a factor of 4: 4→14 \to 1, 8→28 \to 2. 72×13=76\frac{7}{2} \times \frac{1}{3} = \frac{7}{6}

Step 3: Convert to a mixed number. 76=116\frac{7}{6} = 1\frac{1}{6}

Answer: 1161\dfrac{1}{6}

Example 5: Dividing mixed numbers (challenging)​

Problem: Divide 412÷1134\dfrac{1}{2} \div 1\dfrac{1}{3}.

Step 1: Convert to improper fractions. 412=92,113=434\frac{1}{2} = \frac{9}{2}, \quad 1\frac{1}{3} = \frac{4}{3}

Step 2: Keep, Change, Flip. 92×34\frac{9}{2} \times \frac{3}{4}

Step 3: Multiply straight across (no common factors to cancel). 9×32×4=278\frac{9 \times 3}{2 \times 4} = \frac{27}{8}

Step 4: Convert to a mixed number. 278=338\frac{27}{8} = 3\frac{3}{8}

Answer: 3383\dfrac{3}{8}

Common Mistakes​

Mistake 1: Looking for a common denominator when multiplying

❌ 23×14=812×312=24144\frac{2}{3} \times \frac{1}{4} = \frac{8}{12} \times \frac{3}{12} = \frac{24}{144}

✅ 23×14=2×13×4=212=16\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}

Why this matters: Common denominators are for addition and subtraction only. When multiplying, you multiply straight across. Converting to a common denominator first makes the numbers unnecessarily large and leads to errors.

Mistake 2: Forgetting to flip the second fraction when dividing

❌ 35÷27=3×25×7=635\frac{3}{5} \div \frac{2}{7} = \frac{3 \times 2}{5 \times 7} = \frac{6}{35}

✅ 35÷27=35×72=2110=2110\frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10} = 2\frac{1}{10}

Why this matters: Division and multiplication give completely different results. You must take the reciprocal of the divisor first. Remember: Keep, Change, Flip.

Mistake 3: Not converting mixed numbers before multiplying

❌ 213×4=2432\frac{1}{3} \times 4 = 2\frac{4}{3} (multiplying only the fraction part)

✅ 213×4=73×41=283=9132\frac{1}{3} \times 4 = \frac{7}{3} \times \frac{4}{1} = \frac{28}{3} = 9\frac{1}{3}

Why this matters: The whole-number part and the fraction part are not separate pieces you can multiply independently. Always convert to an improper fraction first to ensure the entire value is multiplied.

Practice Problems​

Try these on your own before checking the answers:

  1. 34×25\dfrac{3}{4} \times \dfrac{2}{5}
  2. 67×1415\dfrac{6}{7} \times \dfrac{14}{15}
  3. 212×1352\dfrac{1}{2} \times 1\dfrac{3}{5}
  4. 59÷1027\dfrac{5}{9} \div \dfrac{10}{27}
  5. 313÷2123\dfrac{1}{3} \div 2\dfrac{1}{2}
Click to see answers
  1. 3×24×5=620=310\frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10}
  2. Cross-cancel (6 and 15 by 3, 14 and 7 by 7): 21×25=45\frac{2}{1} \times \frac{2}{5} = \frac{4}{5}
  3. Convert: 52×85\frac{5}{2} \times \frac{8}{5}. Cross-cancel the 5s: 12×81=82=4\frac{1}{2} \times \frac{8}{1} = \frac{8}{2} = 4
  4. Keep, Change, Flip: 59×2710\frac{5}{9} \times \frac{27}{10}. Cross-cancel (5 and 10 by 5, 9 and 27 by 9): 11×32=32=112\frac{1}{1} \times \frac{3}{2} = \frac{3}{2} = 1\frac{1}{2}
  5. Convert: 103÷52=103×25\frac{10}{3} \div \frac{5}{2} = \frac{10}{3} \times \frac{2}{5}. Cross-cancel (10 and 5 by 5): 23×21=43=113\frac{2}{3} \times \frac{2}{1} = \frac{4}{3} = 1\frac{1}{3}

Summary​

  • To multiply fractions, multiply numerators together and denominators together: ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}.
  • Use cross-cancellation before multiplying to keep numbers small and avoid simplifying large products.
  • To divide fractions, use Keep, Change, Flip: keep the first fraction, change division to multiplication, flip the second fraction.
  • Always convert mixed numbers to improper fractions before multiplying or dividing.
  • Simplify your final answer and convert improper fractions to mixed numbers when appropriate.

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