How to Multiply Fractions: Complete Step-by-Step Guide
Master fraction multiplication with confidence! Whether you're a student learning fractions for the first time or an adult refreshing your math skills, this comprehensive guide will teach you everything you need to know about multiplying fractions. Created by mathematics education experts at RevisionTown, this guide provides clear explanations, step-by-step methods, worked examples, and practical applications that make fraction multiplication easy to understand and apply across IB, AP, GCSE, IGCSE, and other curricula worldwide.
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Understanding Fraction Multiplication
Multiplying fractions is one of the fundamental operations in mathematics. Unlike adding or subtracting fractions, multiplication doesn't require common denominators, making it actually simpler in many ways.
What does multiplying fractions mean?
When you multiply fractions, you're finding a part of a part. For example, \( \frac{1}{2} \times \frac{1}{3} \) means "one-half of one-third," which gives you \( \frac{1}{6} \).
Think of it this way: If you have half a pizza and want to take a third of that half, you end up with one-sixth of the original pizza.
The Basic Rule for Multiplying Fractions
The fundamental principle of fraction multiplication is beautifully simple:
The Golden Rule:
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]
In words: Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
Step-by-step breakdown:
- Multiply the top numbers (numerators): \( a \times c \)
- Multiply the bottom numbers (denominators): \( b \times d \)
- Write the answer as a new fraction: \( \frac{a \times c}{b \times d} \)
- Simplify if possible by finding common factors
Method 1: Multiplying Proper Fractions
Let's start with the simplest case: multiplying two proper fractions (where the numerator is smaller than the denominator).
1Simple Example: Multiply \( \frac{2}{5} \times \frac{3}{4} \)
Step 1: Multiply the numerators
\[ 2 \times 3 = 6 \]
Step 2: Multiply the denominators
\[ 5 \times 4 = 20 \]
Step 3: Write as a fraction
\[ \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} \]
Step 4: Simplify (if possible)
Both 6 and 20 are divisible by 2:
\[ \frac{6}{20} = \frac{6 \div 2}{20 \div 2} = \frac{3}{10} \]
Final Answer: \( \frac{2}{5} \times \frac{3}{4} = \frac{3}{10} \)
Example 1: \( \frac{1}{2} \times \frac{2}{3} \)
Solution:
Multiply numerators: \( 1 \times 2 = 2 \)
Multiply denominators: \( 2 \times 3 = 6 \)
\[ \frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3} \]
Simplified answer: \( \frac{1}{3} \)
Example 2: \( \frac{3}{7} \times \frac{2}{5} \)
Solution:
Multiply numerators: \( 3 \times 2 = 6 \)
Multiply denominators: \( 7 \times 5 = 35 \)
\[ \frac{3}{7} \times \frac{2}{5} = \frac{6}{35} \]
Answer: \( \frac{6}{35} \) (already in simplest form)
Method 2: Cross-Cancellation (Simplifying Before Multiplying)
This advanced technique makes calculations easier by simplifying before you multiply. It's especially useful with larger numbers.
Cross-Cancellation Rule: You can divide any numerator and any denominator by their greatest common factor (GCF) before multiplying.
This technique works because:
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{a}{d} \times \frac{c}{b} \]
2Example with Cross-Cancellation: \( \frac{4}{9} \times \frac{3}{8} \)
Step 1: Look for common factors
Notice that 4 and 8 share a common factor of 4, and 3 and 9 share a common factor of 3.
Step 2: Cancel common factors
4 ÷ 4 = 1 and 8 ÷ 4 = 2
3 ÷ 3 = 1 and 9 ÷ 3 = 3
Step 3: Multiply the simplified numbers
\[ \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \]
Answer: \( \frac{4}{9} \times \frac{3}{8} = \frac{1}{6} \)
Compare this to multiplying first: \( \frac{12}{72} = \frac{1}{6} \) — same answer, but cross-cancellation was easier!
Example 3: \( \frac{6}{15} \times \frac{10}{18} \)
Using Cross-Cancellation:
- 6 and 18 both divide by 6: \( 6 \div 6 = 1 \) and \( 18 \div 6 = 3 \)
- 10 and 15 both divide by 5: \( 10 \div 5 = 2 \) and \( 15 \div 5 = 3 \)
\[ \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \]
Answer: \( \frac{2}{9} \)
Multiplying Fractions by Whole Numbers
When multiplying a fraction by a whole number, remember that any whole number can be written as a fraction with denominator 1.
Converting Whole Numbers to Fractions:
\[ n = \frac{n}{1} \]
For example: \( 5 = \frac{5}{1} \), \( 12 = \frac{12}{1} \), etc.
Example 4: \( 5 \times \frac{2}{3} \)
Step 1: Write the whole number as a fraction
\[ 5 \times \frac{2}{3} = \frac{5}{1} \times \frac{2}{3} \]
Step 2: Multiply
\[ \frac{5 \times 2}{1 \times 3} = \frac{10}{3} \]
Step 3: Convert to mixed number (if needed)
\[ \frac{10}{3} = 3\frac{1}{3} \]
Example 5: \( \frac{3}{4} \times 8 \)
Solution:
\[ \frac{3}{4} \times \frac{8}{1} = \frac{3 \times 8}{4 \times 1} = \frac{24}{4} = 6 \]
Answer: 6
Multiplying Mixed Numbers
Mixed numbers (like \( 2\frac{1}{3} \)) must be converted to improper fractions before multiplication.
Converting Mixed Numbers to Improper Fractions:
\[ a\frac{b}{c} = \frac{(a \times c) + b}{c} \]
Example:
\[ 2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3} \]
3Example: \( 1\frac{1}{2} \times 2\frac{2}{3} \)
Step 1: Convert to improper fractions
\( 1\frac{1}{2} = \frac{3}{2} \) and \( 2\frac{2}{3} = \frac{8}{3} \)
Step 2: Multiply the improper fractions
\[ \frac{3}{2} \times \frac{8}{3} \]
Step 3: Cross-cancel if possible
The 3s cancel: \( \frac{1}{2} \times \frac{8}{1} \)
Step 4: Multiply
\[ \frac{8}{2} = 4 \]
Answer: \( 1\frac{1}{2} \times 2\frac{2}{3} = 4 \)
Example 6: \( 3\frac{1}{4} \times 1\frac{1}{3} \)
Step 1: Convert to improper fractions
\( 3\frac{1}{4} = \frac{13}{4} \) and \( 1\frac{1}{3} = \frac{4}{3} \)
Step 2: Multiply
\[ \frac{13}{4} \times \frac{4}{3} = \frac{13 \times 4}{4 \times 3} \]
Step 3: Cancel the 4s
\[ \frac{13}{3} = 4\frac{1}{3} \]
Answer: \( 4\frac{1}{3} \)
Multiplying More Than Two Fractions
The same principle applies when multiplying three or more fractions: multiply all numerators together and all denominators together.
\[ \frac{a}{b} \times \frac{c}{d} \times \frac{e}{f} = \frac{a \times c \times e}{b \times d \times f} \]
Example 7: \( \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \)
Notice the pattern: We can cross-cancel before multiplying!
- The 3 in the numerator cancels with the 3 in the denominator
- The 4 in the numerator cancels with the 4 in the denominator
\[ \frac{2}{1} \times \frac{1}{1} \times \frac{1}{5} = \frac{2}{5} \]
Answer: \( \frac{2}{5} \)
Real-World Applications of Fraction Multiplication
Understanding fraction multiplication is essential for everyday situations and academic success:
Cooking & Recipes
Problem: A recipe calls for \( \frac{3}{4} \) cup of flour, but you want to make only half the recipe.
Solution: \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \) cup
You need \( \frac{3}{8} \) cup of flour.
Shopping & Discounts
Problem: A $60 item is on sale for \( \frac{2}{3} \) of the original price.
Solution: \( 60 \times \frac{2}{3} = \frac{120}{3} = 40 \)
The sale price is $40.
Time & Distance
Problem: You drive \( \frac{3}{5} \) of a 150-mile trip.
Solution: \( 150 \times \frac{3}{5} = \frac{450}{5} = 90 \)
You've driven 90 miles.
Probability
Problem: The probability of rain is \( \frac{2}{3} \), and if it rains, there's a \( \frac{1}{2} \) chance of flooding.
Solution: \( \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \)
There's a \( \frac{1}{3} \) probability of flooding.
Common Mistakes to Avoid
Mistake 1: Trying to find a common denominator
Wrong approach: Treating multiplication like addition
Remember: You don't need common denominators for multiplication!
Correct: \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
Mistake 2: Forgetting to simplify
Incomplete: \( \frac{2}{4} \times \frac{3}{6} = \frac{6}{24} \)
Complete: \( \frac{6}{24} = \frac{1}{4} \) (simplified)
Always check if your answer can be simplified!
Mistake 3: Multiplying mixed numbers directly
Wrong: \( 2\frac{1}{2} \times 3\frac{1}{3} \neq 6\frac{1}{6} \)
Correct approach: Convert to improper fractions first!
\( 2\frac{1}{2} = \frac{5}{2} \) and \( 3\frac{1}{3} = \frac{10}{3} \)
\( \frac{5}{2} \times \frac{10}{3} = \frac{50}{6} = 8\frac{1}{3} \)
Mistake 4: Adding instead of multiplying
Wrong: \( \frac{1}{2} \times \frac{1}{3} = \frac{2}{5} \)
Correct: \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
Multiply numerators together and denominators together!
Practice Problems with Solutions
Practice Problem 1 (Basic)
Question: \( \frac{3}{5} \times \frac{2}{7} = ? \)
Solution:
Multiply numerators: \( 3 \times 2 = 6 \)
Multiply denominators: \( 5 \times 7 = 35 \)
\[ \frac{3}{5} \times \frac{2}{7} = \frac{6}{35} \]
Answer: \( \frac{6}{35} \)
Practice Problem 2 (With Simplification)
Question: \( \frac{4}{6} \times \frac{9}{12} = ? \)
Solution:
Using cross-cancellation:
- 4 and 12 divide by 4: gives \( \frac{1}{?} \times \frac{?}{3} \)
- 6 and 9 divide by 3: gives \( \frac{?}{2} \times \frac{3}{?} \)
\[ \frac{1}{2} \times \frac{3}{3} = \frac{3}{6} = \frac{1}{2} \]
Answer: \( \frac{1}{2} \)
Practice Problem 3 (Mixed Numbers)
Question: \( 2\frac{1}{4} \times 1\frac{2}{5} = ? \)
Solution:
Convert to improper fractions:
\( 2\frac{1}{4} = \frac{9}{4} \) and \( 1\frac{2}{5} = \frac{7}{5} \)
\[ \frac{9}{4} \times \frac{7}{5} = \frac{63}{20} = 3\frac{3}{20} \]
Answer: \( 3\frac{3}{20} \)
Practice Problem 4 (Word Problem)
Question: Sarah completed \( \frac{3}{4} \) of her homework. Of that, \( \frac{2}{3} \) was math homework. What fraction of all her homework was math?
Solution:
We need to find: \( \frac{3}{4} \times \frac{2}{3} \)
Cross-cancel the 3s:
\[ \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} \]
Answer: \( \frac{1}{2} \) of her homework was math
Quick Reference Guide
| Type | How to Multiply | Example |
|---|---|---|
| Two Proper Fractions | Multiply numerators, multiply denominators | \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \) |
| Fraction × Whole Number | Write whole number as \( \frac{n}{1} \), then multiply | \( \frac{3}{4} \times 5 = \frac{15}{4} = 3\frac{3}{4} \) |
| Mixed Numbers | Convert to improper fractions first | \( 1\frac{1}{2} \times 2\frac{1}{3} = \frac{3}{2} \times \frac{7}{3} = \frac{7}{2} = 3\frac{1}{2} \) |
| Multiple Fractions | Multiply all numerators, all denominators | \( \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{6}{24} = \frac{1}{4} \) |
Expert Tips for Mastering Fraction Multiplication
Tip 1: Look for opportunities to simplify early
Cross-cancellation before multiplying makes calculations much easier and reduces errors.
Tip 2: Check your answer with estimation
If you multiply two proper fractions, your answer should be smaller than both original fractions.
Example: \( \frac{1}{2} \times \frac{1}{3} \) should be less than \( \frac{1}{2} \) ✓ (\( \frac{1}{6} \) is correct)
Tip 3: Practice mental math with simple fractions
Memorize common multiplications like:
- \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
- \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \)
- \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \)
Tip 4: Always simplify your final answer
Finding the greatest common factor (GCF) and simplifying shows complete understanding and gives the clearest answer.
Tip 5: Use visual models when learning
Draw rectangles or circles and shade fractions to see what "a fraction of a fraction" means visually.
Why Does Fraction Multiplication Work This Way?
The Mathematical Reasoning
Fraction multiplication follows logically from the definition of fractions and multiplication:
Understanding \( \frac{1}{2} \times \frac{1}{3} \):
- \( \frac{1}{2} \) means "divide something into 2 equal parts and take 1"
- \( \frac{1}{3} \) means "divide something into 3 equal parts and take 1"
- \( \frac{1}{2} \times \frac{1}{3} \) means "divide one-third into 2 parts and take 1"
When you divide something into 3 parts, then divide those parts into 2, you have 6 total parts. Taking 1 of those 6 parts gives you \( \frac{1}{6} \).
\[ 2 \times 3 = 6 \text{ parts total} \quad \Rightarrow \quad \frac{1}{6} \]
Summary: Key Points to Remember
- ✓ Multiply numerators together and denominators together
- ✓ You don't need common denominators for multiplication
- ✓ Simplify before or after multiplying — both work!
- ✓ Convert mixed numbers to improper fractions first
- ✓ Write whole numbers as fractions with denominator 1
- ✓ Cross-cancel to make calculations easier
- ✓ Always simplify your final answer
- ✓ Check your work with estimation
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