Everyday · 3 min read
How to Multiply and Divide Fractions (Step by Step)
Good news: multiplying and dividing fractions is easier than adding them, because you don't need a common denominator. To multiply, multiply the tops together and the bottoms together. To divide, flip the second fraction upside down and then multiply. That's the whole rule. Everything below is about doing it cleanly.
Multiplying fractions
2/3 × 3/5
- Tops: 2 × 3 = 6.
- Bottoms: 3 × 5 = 15.
- 6/15, which simplifies to 2/5 (divide both by 3).
A useful way to picture it: "2/3 × 3/5" means "two-thirds of three-fifths". Multiplying by a fraction smaller than 1 makes things smaller, which is why the answer is less than both starting fractions.
Cancel first to keep numbers small
Before multiplying, you can divide any top number and any bottom number by a shared factor. It's the same as simplifying at the end, just earlier, and it keeps the arithmetic easy.
4/9 × 3/8: the 4 and the 8 share a factor of 4, and the 3 and the 9 share a factor of 3. So it becomes 1/3 × 1/2 = 1/6. Without cancelling you'd get 12/72 and have to simplify that.
Dividing fractions: keep, change, flip
3/4 ÷ 1/2
- Keep the first fraction: 3/4.
- Change ÷ to ×.
- Flip the second fraction: 1/2 becomes 2/1.
- 3/4 × 2/1 = 6/4 = 1 1/2.
Why does flipping work? Dividing by 1/2 asks "how many halves fit in this?" Two halves fit in every whole, so dividing by 1/2 is the same as multiplying by 2. The flip is just a general version of that idea.
Another division example
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
Mixed numbers: convert first
This is where most mistakes happen. You can't multiply the whole numbers and the fractions separately.
1 1/2 × 2 2/3
- 1 1/2 = 3/2 and 2 2/3 = 8/3.
- 3/2 × 8/3: cancel the 3s, then 1/2 × 8/1 = 8/2 = 4.
If you'd multiplied 1 × 2 and 1/2 × 2/3 separately, you'd get 2 1/3, which is wrong.
Whole numbers
Write the whole number over 1. 5 × 3/8 = 5/1 × 3/8 = 15/8 = 1 7/8. And 6 ÷ 3/4 = 6/1 × 4/3 = 24/3 = 8. That last one answers a real question: how many 3/4-cup servings are in 6 cups? Eight.
Real-life uses
- Scaling recipes. Making 2/3 of a recipe that needs 3/4 cup of sugar: 2/3 × 3/4 = 1/2 cup.
- Cutting materials. How many 3/8" strips from a 4 1/2" board? 9/2 ÷ 3/8 = 9/2 × 8/3 = 12 strips (before saw kerf).
- Splitting things fairly. Half of 3/5 of a budget is 3/10 of the budget.
Check any of these with the fraction calculator, which shows each step. For adding and subtracting, see how to add fractions with different denominators.
Recap
- Multiply: tops × tops, bottoms × bottoms. Cancel first if you can.
- Divide: keep, change, flip, then multiply.
- Always convert mixed numbers to improper fractions before you start.
Frequently asked questions
How do you multiply fractions?
Multiply the numerators together and the denominators together, then simplify. For example 2/3 × 3/5 = 6/15 = 2/5.
How do you divide fractions?
Keep the first fraction, change division to multiplication, and flip the second fraction. For example 3/4 ÷ 1/2 = 3/4 × 2/1 = 1 1/2.
How do you multiply mixed numbers?
Convert them to improper fractions first. 1 1/2 × 2 2/3 = 3/2 × 8/3 = 4.
Do you need a common denominator to multiply fractions?
No. Common denominators are only needed for adding and subtracting.