Dividing Fractions

Learn how to divide fractions using the 'Keep, Change, Flip' method and understand why it works.

Intermediate25 min15 problems
Practice Problems

Definition

To divide fractions, we use the Keep-Change-Flip method (also called "invert and multiply"):
  1. 1.Keep the first fraction the same
  2. 2.Change the division sign to multiplication
  3. 3.Flip the second fraction (find its reciprocal)
ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
The reciprocal of a fraction is found by flipping its numerator and denominator. For example, the reciprocal of 23\frac{2}{3} is 32\frac{3}{2}.

Try it now

What is the reciprocal of 34\frac{3}{4}?

Worked Examples

Calculate 12÷14\frac{1}{2} \div \frac{1}{4}
1

Keep the first fraction

12\frac{1}{2} stays as 12\frac{1}{2}
2

Change division to multiplication

÷\div becomes ×\times → 12×\frac{1}{2} \times
3

Flip the second fraction

14\frac{1}{4} becomes 41\frac{4}{1} → 12×41\frac{1}{2} \times \frac{4}{1}
4

Multiply numerators and denominators

1×42×1=42\frac{1 \times 4}{2 \times 1} = \frac{4}{2}
5

Simplify if possible

42=2\frac{4}{2} = 2

Common Mistakes

Flipping the wrong fraction

Why it's wrong: Students sometimes flip the first fraction instead of the second, or flip both fractions.

Correct: Only flip the SECOND fraction (the divisor). The first fraction stays exactly the same.

Forgetting to change division to multiplication

Why it's wrong: After flipping, students continue dividing instead of multiplying.

Correct: Keep-Change-Flip: you must CHANGE the operation to multiplication, then FLIP.

Not simplifying the final answer

Why it's wrong: Students stop after multiplying without checking if the fraction can be reduced.

Correct: Always check if your answer can be simplified. 68\frac{6}{8} should become 34\frac{3}{4}.

Getting confused when dividing by a whole number

Why it's wrong: Students forget that a whole number like 3 is the same as 31\frac{3}{1}.

Correct: Write whole numbers as fractions first: 3=313 = \frac{3}{1}, then apply Keep-Change-Flip.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Add, Subtract, Multiply, and Divide Fractions (Step-by-Step)

by Understand The Math

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Practice Problems

15 problems
Problem 1 of 15
Easy
What is the reciprocal of 34\frac{3}{4}?

Why It Matters

Dividing fractions helps us answer questions like:
  • Cooking: A batch of pancakes needs 34\frac{3}{4} cup of flour. How many batches can you make with 3 cups?
  • Construction: How many 14\frac{1}{4}-meter pieces can you cut from a 34\frac{3}{4}-meter board?
  • Time: If you can complete 23\frac{2}{3} of a task in one hour, how long for the whole task?
Understanding fraction division builds the foundation for algebra, ratios, and rates!

Real World Applications

Cooking and Recipes

Chefs divide fractions when scaling recipes up or down.

Example:

A cake recipe calls for 34\frac{3}{4} cup of sugar. If you want to make half the recipe, you need 34÷2=38\frac{3}{4} \div 2 = \frac{3}{8} cup.

1Try It Yourself

You have 23\frac{2}{3} cup of butter. Each cookie needs 16\frac{1}{6} cup of butter.

How many cookies can you make?

Step 1: Write the mathematical expression

Set up the division and calculate it:

Construction and Measurement

Builders divide fractions when cutting materials into equal pieces.

Example:

A 34\frac{3}{4}-meter rope needs to be cut into 18\frac{1}{8}-meter pieces. You can cut 34÷18=6\frac{3}{4} \div \frac{1}{8} = 6 pieces.

2Try It Yourself

A board is 56\frac{5}{6} meter long. You need pieces that are 13\frac{1}{3} meter each.

How many 13\frac{1}{3}-meter lengths fit into the board?

Step 1: Write the mathematical expression

Divide the board length by piece length:

Time Management

Fraction division helps calculate rates and durations.

Example:

If you read 14\frac{1}{4} of a book in 12\frac{1}{2} hour, your reading rate is 14÷12=12\frac{1}{4} \div \frac{1}{2} = \frac{1}{2} of the book per hour.

3Try It Yourself

You completed 25\frac{2}{5} of a project in 45\frac{4}{5} of an hour.

At this rate, what fraction of the project can you complete per hour?

Step 1: Write the mathematical expression

Divide work done by time:

Key Takeaways

  • 1To divide fractions, use Keep-Change-Flip: keep the first fraction, change ÷\div to ×\times, flip the second
  • 2The reciprocal of ab\frac{a}{b} is ba\frac{b}{a} (flip numerator and denominator)
  • 3Dividing by a fraction is the same as multiplying by its reciprocal
  • 4Always simplify your final answer
  • 5When dividing whole numbers by fractions, write the whole number as n1\frac{n}{1} first

Frequently Asked Questions

Division asks 'how many fit?' For 12÷14\frac{1}{2} \div \frac{1}{4} you ask how many quarters fit in a half. One whole holds 4 quarters, so a half holds 12×4=2\frac{1}{2} \times 4 = 2 quarters. That 4 is exactly the reciprocal of 14\frac{1}{4}, so dividing by a fraction gives the same result as multiplying by its reciprocal.
Division asks 'how many fit?' For 12÷14\frac{1}{2} \div \frac{1}{4} you ask how many quarters fit in a half. One whole holds 4 quarters, so a half holds 12×4=2\frac{1}{2} \times 4 = 2 quarters. That 4 is exactly the reciprocal of 14\frac{1}{4}, so dividing by a fraction gives the same result as multiplying by its reciprocal.
Write the whole number as a fraction with denominator 1. For 34÷2\frac{3}{4} \div 2: rewrite as 34÷21\frac{3}{4} \div \frac{2}{1}, then apply Keep-Change-Flip to get 34×12=38\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}.
Yes! When you divide by a fraction less than 1, the answer is larger than the first fraction. For example, 12÷14=2\frac{1}{2} \div \frac{1}{4} = 2. Think of it as 'how many small pieces fit in the larger piece?'

Glossary

Reciprocal
A fraction flipped upside down. The reciprocal of ab\frac{a}{b} is ba\frac{b}{a}. Example: reciprocal of 23\frac{2}{3} is 32\frac{3}{2}
Dividend
The number being divided (the first fraction in a division problem)
Divisor
The number you divide by (the second fraction in a division problem)
Quotient
The result of a division problem
Today's challengeOne problem, three tries

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