Multiplying and Dividing Fractions

Grade 6 · Math · Study Guide

Learn to multiply and divide fractions and mixed numbers with confidence.

Multiplying fractions is actually simpler than adding them — no common denominator needed! Just multiply the numerators together and multiply the denominators together. So 2/3 × 4/5 = (2×4)/(3×5) = 8/15. Always simplify your answer at the end, and you can also simplify before you multiply by canceling common factors diagonally.

When multiplying mixed numbers, convert them to improper fractions first. To convert 2 ¾ to an improper fraction: multiply the whole number by the denominator and add the numerator — (2×4)+3 = 11, so 2¾ = 11/4. Then multiply as usual.

Dividing fractions uses one brilliant trick: multiply by the reciprocal. The reciprocal of a fraction flips it upside down — the reciprocal of 3/4 is 4/3. So to divide by a fraction, Keep the first fraction, Change ÷ to ×, Flip the second fraction. This is called KCF (Keep, Change, Flip). For example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

Why does flipping work? Think of 6 ÷ 2 — you're asking 'how many groups of 2 fit in 6?' With fractions, 1/2 ÷ 1/4 asks 'how many quarter-pieces fit in a half?' The answer is 2, and 1/2 × 4/1 = 2 confirms it. The flip captures that relationship perfectly.

Key terms

Reciprocal

The flipped version of a fraction — e.g. the reciprocal of 3/4 is 4/3

Improper Fraction

A fraction where the numerator is greater than or equal to the denominator, like 7/4

Mixed Number

A number made of a whole number and a fraction, like 3 ½

Simplify (reduce)

Divide numerator and denominator by their GCF to write a fraction in lowest terms

Flashcards

Multiplying fractions

Multiply numerators together, multiply denominators together — e.g. 2/3 × 3/4 = 6/12 = 1/2

Reciprocal

Flip a fraction — numerator becomes denominator. Reciprocal of 5/8 is 8/5

Dividing fractions (KCF)

Keep first fraction, Change ÷ to ×, Flip second fraction

Convert mixed number to improper fraction

Multiply whole by denominator, add numerator: 2¾ → (2×4+3)/4 = 11/4

Cross-canceling

Simplify before multiplying by canceling common factors in numerator and denominator diagonally

2/3 ÷ 4/5 =

2/3 × 5/4 = 10/12 = 5/6 (KCF: keep 2/3, change to ×, flip 4/5 to 5/4)

Practice questions

1. What is 3/4 × 8/9?

   1. 2/3
   2. 11/13
   3. 24/36
   4. 3/8
   Show answer

   Answer: 2/3

   3/4 × 8/9. Cross-cancel: 3 and 9 share factor 3 → 1/4 × 8/3. Then 4 and 8 share factor 4 → 1/1 × 2/3 = 2/3.

2. What is 5/6 ÷ 1/3?

   1. 5/2
   2. 5/18
   3. 2/5
   4. 15/6
   Show answer

   Answer: 5/2

   KCF: 5/6 × 3/1 = 15/6 = 5/2. (Reciprocal of 1/3 is 3/1.)

3. To divide fractions, you multiply by the reciprocal of the second fraction.

   Show answer

   Answer: True

   KCF rule — dividing by a fraction is mathematically identical to multiplying by its reciprocal.

4. The reciprocal of 7/2 is ___.

   Show answer

   Answer: 2/7

   Flip the fraction: 7/2 flipped becomes 2/7. Check: 7/2 × 2/7 = 14/14 = 1 ✓

5. What is 1½ × 2/3?

   1. 1
   2. 3/4
   3. 2
   4. 4/3
   Show answer

   Answer: 1

   Convert 1½ to 3/2. Then 3/2 × 2/3 = 6/6 = 1.

Fun facts

• Dividing by 1/2 is the same as doubling — that's why half a recipe of 1 cup becomes 2 cups when you ask 'how many halves are in 1 cup?'
• Ancient Egyptians only used unit fractions (numerator of 1) like 1/2, 1/3, 1/7 — they would write 3/4 as 1/2 + 1/4!

Get a personalized course on Multiplying and Dividing Fractions built around your child.