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Mathematics › ⚡ Grade 6 › The Divider Ray: Dividing Fractions and Mixed Numbers
⚡ Grade 6 · Lesson 5 of 14

The Divider Ray: Dividing Fractions and Mixed Numbers

Dividing by a fraction is really asking 'how many of these fit?' — and the answer always comes from multiplying by the reciprocal.

Grade 6Middle school
The Divider Ray: Dividing Fractions and Mixed Numbers — illustration
💡
The big idea: Dividing by a fraction answers the question 'how many of this size fit into that amount?' Turning the division into multiplication by the reciprocal (flipping the divisor) always gives the same answer, and mixed numbers just need to become improper fractions first.
🎯 By the end, you'll be able to
  • Interpret fraction division as a 'how many fit' question
  • Convert mixed numbers to improper fractions before dividing
  • Divide fractions by multiplying by the reciprocal (keep-change-flip)
  • Solve real-world problems involving division of fractions and mixed numbers
📎 You should already know
  • Multiplying fractions
  • Converting mixed numbers to improper fractions

How many fit?

Dividing whole numbers often means splitting into groups: 12 ÷ 3 asks “how many groups of 3 fit into 12?” Dividing by a fraction asks the exact same kind of question, just with a smaller measuring unit. If you have ¾ of a cup of trail mix and each scoop holds 1/8 cup, ¾ ÷ 1/8 asks “how many 1/8-cup scoops fit into ¾ cup?”

🔑 Divide by a fraction = multiply by its reciprocal
To divide by a fraction, keep the first fraction, change ÷ to ×, and flip (find the reciprocal of) the second fraction. This is often remembered as keep-change-flip.
\[ \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \]
Dividing by c/d is the same as multiplying by its reciprocal, d/c.
🎮 Divider Ray LIVE
Divide by a fraction by asking 'how many of these fit?' — see why you multiply by the reciprocal.

Why flipping and multiplying works

Think about ¾ ÷ 1/8 again. Since 1 whole cup holds 8 scoops of 1/8 cup, ¾ of a cup holds ¾ of those 8 scoops: ¾ × 8 = 6 scoops. Notice that 8 is the reciprocal of 1/8 — flipping the divisor and multiplying gives exactly the “how many fit” answer.

Mixed numbers need converting first

Before dividing a mixed number, rewrite it as an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the same denominator. Only then apply keep-change-flip.

📝 Worked example: How many 1/8-cup scoops fit into 3/4 cup of trail mix?
  1. Set up the division: 3/4 ÷ 1/8.
  2. Keep the first fraction, change ÷ to ×, and flip the second: 3/4 × 8/1.
  3. Multiply: (3 × 8) ÷ (4 × 1) = 24/4.
✓ <strong>6 scoops</strong> fit into 3/4 cup.
📝 Worked example: A ribbon is 2 &frac12; metres long. How many pieces 3/4 of a metre long can be cut from it?
  1. Convert the mixed number: 2 ½ = 5/2.
  2. Divide: 5/2 ÷ 3/4 = 5/2 × 4/3 (keep-change-flip).
  3. Multiply: (5 × 4) ÷ (2 × 3) = 20/6 = 10/3.
✓ 10/3 pieces fit, which is <strong>3 and 1/3 pieces</strong> — 3 full 3/4-metre pieces with a bit of ribbon left over.
⚠️ Flip the divisor only, and convert mixed numbers first
Two common mistakes: flipping the first fraction instead of the second, and dividing mixed numbers directly without converting to improper fractions first. Always keep the first fraction as it is, and always convert any mixed number before you begin.
✨ Dividing by less than 1 makes the answer bigger
It feels backwards, but dividing by a fraction smaller than 1 gives an answer larger than the original number — because small scoops mean more of them fit. Compare: 3/4 ÷ 1/8 = 6, which is bigger than 3/4 itself.

Check your understanding

1. What is the first step in dividing 2/3 by 5/6 using keep-change-flip?
Keep the first fraction (2/3), change ÷ to ×, and flip the second fraction (5/6 becomes 6/5).
2. Compute 1/2 ÷ 1/4.
1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. There are 2 quarters in a half.
3. Before dividing 1 3/4 by 1/2, what must you do first?
Mixed numbers must become improper fractions before applying keep-change-flip: 1 3/4 = 7/4.
4. How many 1/3-cup scoops fit into 2 cups?
2 ÷ 1/3 = 2 × 3/1 = 6. Six 1/3-cup scoops fill 2 cups.
5. Dividing a number by a fraction smaller than 1 always...
Dividing by a fraction less than 1 means the pieces are small, so more of them fit — the quotient is larger than the original number.
✅ Key takeaways
  • Dividing by a fraction asks 'how many of that size fit into this amount?'
  • Keep-change-flip: keep the first fraction, change ÷ to ×, and flip (reciprocate) the second fraction.
  • a/b ÷ c/d = a/b × d/c, always.
  • Convert any mixed number to an improper fraction before dividing.
  • Dividing by a fraction smaller than 1 makes the answer bigger than the original number.