Practice Maths

L26 — Dividing Fractions

Key Terms

reciprocal
The result of flipping a fraction. The reciprocal of 34 is 43. The reciprocal of 5 is 15.
KCF (Keep, Change, Flip)
The method for dividing fractions: Keep the first fraction unchanged, Change ÷ to ×, Flip (take the reciprocal of) the second fraction. Then multiply.
dividend
The number being divided (the first number in a division).
divisor
The number you divide by (the second number in a division).

Dividing Fractions: KCF

To divide by a fraction: Keep the first fraction, Change ÷ to ×, Flip the second fraction.

34 ÷ 12 = 34 × 21 = 64 = 32 = 112

Worked Example

How many 14-cup servings can you get from 32 cups of juice?

Step 1 — Set up: 32 ÷ 14

Step 2 — Apply KCF: 32 × 41

Step 3 — Multiply and simplify: 122 = 6 servings

Only flip the SECOND fraction. Many students accidentally flip both fractions, which gives a wrong answer. In KCF, the first fraction is kept exactly as it is.

What Does "Dividing by a Fraction" Actually Mean?

Division means “how many times does this fit into that?” Consider 34 ÷ 14: “How many quarter-pieces fit into three-quarter-pieces?” Clearly 3. And 34 ÷ 12: “How many half-pieces fit into three-quarters?” One half fills 2 of the 4 quarters, so 112 halves fit. KCF confirms: 34 × 21 = 64 = 112.

Keep-Change-Flip (KCF) — And WHY It Works

Dividing by a number is the same as multiplying by its reciprocal. This is true for ALL numbers: 12 ÷ 4 is the same as 12 × 14 = 3. With fractions, the reciprocal is found by flipping the fraction.

Mathematically: ab ÷ cd = ab × dc. This is a real mathematical truth, not just a trick!

Only the SECOND fraction gets flipped. The first fraction stays exactly as it is. If you flip both, you get the wrong answer every time.

Dividing by a Fraction Makes the Answer Bigger

When you divide by a fraction less than 1, the result is larger than what you started with. How many half-sandwiches fit into 4 whole sandwiches? 8! Because 4 ÷ 12 = 4 × 2 = 8. More pieces, because each piece is smaller.

Rope-cutting: 4 m of rope divided into 13-metre pieces gives 4 ÷ 13 = 4 × 3 = 12 pieces.

Mixed Numbers: Convert First

If a division problem involves mixed numbers, convert them to improper fractions first, then apply KCF. For example: 214 ÷ 34

  • Convert: 214 = 94
  • KCF: 94 × 43 = 3612 = 3
Always convert mixed numbers to improper fractions before applying KCF — otherwise the arithmetic goes completely wrong.

  1. Calculate

    1. 12 ÷ 14
    2. 34 ÷ 12
    3. 23 ÷ 13
    4. 56 ÷ 512

  2. Divide a whole number by a fraction

    1. 3 ÷ 12
    2. 4 ÷ 23
    3. 6 ÷ 34

  3. Calculate and simplify

    1. 45 ÷ 23
    2. 78 ÷ 74
    3. 35 ÷ 67
    4. 89 ÷ 43

  4. Cutting Rope

    How many 13 m pieces of rope can be cut from 4 m of rope?

  5. Recipe Fractions

    A recipe uses 34 cup of sugar. If you only have 38 cup, what fraction of the recipe can you make?

  6. Calculate

    1. 56 ÷ 53
    2. 910 ÷ 35
    3. 27 ÷ 47
    4. 59 ÷ 103

  7. Pizza Slices

    A pizza is cut into slices of 18 each. How many slices are in 54 of a pizza?

  8. Missing Value

    Fill in the missing value: 34 ÷ ______ = 32

  9. Fabric Cutting

    Sam has 214 m of fabric. He cuts it into pieces each 34 m long. How many pieces does he get?

  10. Water Tank Draining

    A water tank drains at 25 of a litre per minute. How many minutes will it take to drain 3 litres?

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