Here’s something students often find surprising: multiplying and dividing fractions is easier than adding them. Unlike addition and subtraction, you never hunt for a common denominator or compute a least common multiple. Multiplication goes straight across; division uses a clever flip — the famous Keep, Change, Flip rule.
In this tutorial you’ll learn the straight-across multiplication method, early simplification through cross-cancellation, and the full Keep-Change-Flip framework for division, with three worked examples.
How to Multiply Fractions Directly
Multiply the tops straight across, multiply the bottoms straight across, then simplify:
a/b × c/d = (a × c)/(b × d)
Simplify Early: Cross-Cancellation
Before multiplying big numbers, look diagonally — any numerator and any denominator can be divided by their greatest common factor (GCF) ahead of time. This keeps intermediate numbers small and often eliminates end-stage simplification entirely.
If you’re working on fraction addition first, check our walkthrough on how to add fractions with unlike denominators. To see how fractions translate into comparative rate differences, explore our guide on how to calculate percentage increase and decrease. You can also verify your fractional products instantly using the CalcSolver computation engine.
Dividing Fractions: The Keep, Change, Flip Framework
You never divide across numerators and denominators directly. Instead, convert division into multiplication using the reciprocal of the divisor:
- Keep the first fraction exactly as written.
- Change the division sign (÷) into a multiplication sign (×).
- Flip the second fraction upside down to create its reciprocal.
Then multiply straight across like normal.
Why does flipping work? Dividing by a fraction asks “how many of these fit?” Multiplying by its reciprocal asks the same question in multiplication form — they’re mathematically identical, just like dividing by 2 is the same as multiplying by 1/2.
Worked Example 1: Multiplying with Cross-Cancellation
Calculate:
3/4 × 2/9
Step 1: Check diagonals for common factors
- Look at 3 and 9: both divisible by 3 → 3 ÷ 3 = 1, 9 ÷ 3 = 3.
- Look at 2 and 4: both divisible by 2 → 2 ÷ 2 = 1, 4 ÷ 2 = 2.
Step 2: Multiply the reduced numerators and denominators
1/2 × 1/3 = (1 × 1)/(2 × 3) = 1/6
Final answer: 1/6.
Worked Example 2: Dividing with Keep, Change, Flip
Solve:
5/6 ÷ 2/3
Step 1: Apply Keep, Change, Flip
- Keep the first fraction: 5/6
- Change ÷ to ×
- Flip the second fraction: 2/3 becomes 3/2
Rewrite the problem:
5/6 × 3/2
Step 2: Cross-cancel where possible
3 and 6 share a divisor of 3: 3 ÷ 3 = 1, 6 ÷ 3 = 2.
Step 3: Multiply across
5/2 × 1/2 = (5 × 1)/(2 × 2) = 5/4
Step 4: Express as a mixed number
Final answer: 5/4, or 1 1/4 (4 goes into 5 once with a remainder of 1).
Worked Example 3: Mixed Numbers
Calculate:
2 1/3 × 3/4
Step 1: Convert the mixed number to an improper fraction
2 1/3 = (2 × 3 + 1)/3 = 7/3.
Step 2: Cross-cancel before multiplying
7/3 × 3/4
The 3s cancel completely (3 ÷ 3 = 1 on both sides), leaving:
7/1 × 1/4 = 7/4
Step 3: Convert back
Final answer: 7/4 = 1 3/4.
3 Critical Mistakes to Avoid
- Flipping the wrong fraction: never invert the first fraction. Keep the first fraction unchanged and flip only the one immediately after the division sign.
- Hunting for common denominators: searching for an LCD when multiplying or dividing wastes time and inflates your numbers. Common denominators belong strictly to addition and subtraction.
- Flipping both fractions: taking the reciprocal of both sides rewrites the entire problem. Only the divisor (the second term) gets flipped.
Straight-across multiplication, diagonal cancellation, and Keep-Change-Flip for division: those three habits handle every basic and advanced fraction operation you’ll meet.