Multiplying of Proper and Mixed Fractions

Multiplying fractions is one of the easiest fraction operations! Just multiply the numerators together and the denominators together. Below, you’ll find detailed explanations, examples, and a calculator to practice with.

Result

\[ \frac{3}{10}\]

Simple Principle

Multiply the numerators, multiply the denominators, and simplify the result—that’s all there is to it!

Practical Skills

Multiplying fractions is a key skill in math, cooking, DIY projects, and many other areas of life.

What Is Fraction Multiplication?

Multiplying fractions involves combining two fractions by multiplying their respective parts. It’s one of the simplest fraction operations because it doesn’t require a common denominator.

Formula for multiplying fractions:

\( \huge \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)

Where:

Think of fraction multiplication like this:

Multiplying \( \large \frac{1}{2} \times \frac{1}{3} \) means taking half of a third, which is \( \large \frac{1}{6} \) of the whole.

\( \large \frac{1}{2} \times \frac{1}{3} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6} \)

The result of multiplying fractions is usually smaller than either fraction, unlike multiplying whole numbers.

5 Simple Steps to Multiply Fractions

1

Write the fractions side by side with a multiplication sign between them

2

Multiply the numerators together (top numbers)

3

Multiply the denominators together (bottom numbers)

4

Write the new fraction using the multiplied numerator and denominator

5

Simplify the result if possible, or convert it to a mixed number

Practical Examples

Examples of Multiplying Fractions

Example 1: Simple Fraction Multiplication

\[ \frac{2}{3} \times \frac{3}{4} \]
  1. Multiply the numerators: \( 2 \times 3 = 6 \)
  2. Multiply the denominators: \( 3 \times 4 = 12 \)
  3. Write the result: \( \large \frac{6}{12} \)
  4. Simplify (divide numerator and denominator by 6): \( \large \frac{6}{12} = \frac{1}{2} \)

Example 2: Simplifying Before Multiplying

\[ \frac{4}{5} \times \frac{10}{12} \]
  1. First simplify \( \large \frac{10}{12} = \frac{5}{6} \)
  2. Now multiply: \( \large \frac{4}{5} \times \frac{5}{6} \)
  3. Simplify before multiplying: \( \large \frac{4 \times 5}{5 \times 6} = \frac{4 \times \cancel{5}}{\cancel{5} \times 6} = \frac{4}{6} \)
  4. Simplify the final result: \( \large \frac{4}{6} = \frac{2}{3} \)

Example 3: Multiplying with a Mixed Number

\[ 2\frac{1}{2} \times \frac{2}{3} \]
  1. Convert the mixed number to an improper fraction: \( \large 2\frac{1}{2} = \frac{5}{2} \)
  2. Multiply: \( \large \frac{5}{2} \times \frac{2}{3} \)
  3. Multiply the numerators: \( 5 \times 2 = 10 \)
  4. Multiply the denominators: \( 2 \times 3 = 6 \)
  5. Result: \( \large \frac{10}{6} = \frac{5}{3} = 1\frac{2}{3} \)

Practical Tips and Tricks

Simplify Early

Simplifying before multiplying is easier than simplifying afterward. Look for numbers that can be canceled "crosswise."

Multiplying by Whole Numbers

Treat a whole number as a fraction with a denominator of 1, e.g., \( \large 5 = \frac{5}{1} \).

Visualize Multiplication

Picture a fraction as part of a rectangle. Multiplying fractions means finding a part of a part.

Convert Mixed Numbers

Always convert mixed numbers to improper fractions before multiplying.

Common Mistakes to Avoid

Don’t add numerators and denominators!

Wrong: \( \large \frac{2}{3} \times \frac{1}{4} = \frac{2+1}{3+4} = \frac{3}{7} \) ❌

Right: \( \large \frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6} \) ✓

Multiplying Only Numerators

Remember to multiply both numerators and denominators.

Forgetting to Simplify

Always check if the result can be simplified by finding a common factor.

Incorrect Mixed Number Conversion

To convert \( \large 2\frac{3}{4} \) to an improper fraction, calculate \( 2 \times 4 + 3 = 11 \), so \( \large \frac{11}{4} \).

Improper Simplification

You can only simplify a numerator with a denominator, not numerators or denominators with each other.

Practice Exercises—Test Yourself!

Try These on Your Own:

Exercise 1: Multiply Fractions

  • \( \large \frac{2}{3} \times \frac{3}{4} \)
  • \( \large \frac{5}{7} \times \frac{2}{8} \)
  • \( \large \frac{4}{5} \times \frac{7}{10} \)

Exercise 2: Multiply and Simplify

  • \( \large \frac{3}{9} \times \frac{8}{12} \)
  • \( \large \frac{6}{11} \times \frac{5}{13} \)
  • \( \large \frac{10}{15} \times \frac{6}{8} \)

Exercise 3: Multiply with Mixed Numbers

  • \( \large 1\frac{1}{2} \times \frac{2}{3} \)
  • \( \large 2\frac{3}{4} \times 1\frac{1}{5} \)
  • \( \large 3 \times \frac{5}{6} \)

Key Takeaways

Practice with our calculator to sharpen your fraction multiplication skills!

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