Dividing Fractions
Dividing fractions follows a counterintuitive rule: multiply by the reciprocal of the divisor. The operation 2/3 ÷ 4/5 becomes 2/3 × 5/4, yielding 10/12 or 5/6. This "invert and multiply" method transforms division into multiplication, making calculations straightforward.
Why it matters
Fraction division appears throughout cooking, construction, and manufacturing. A baker dividing 34 cup of flour into 18 cup portions needs 6 servings. Carpenters cutting 58-inch boards into 14-inch strips get 2.5 pieces per board. The concept extends to rates and ratios in algebra, where dividing 23 miles by 16 hours gives 4 miles per hour. Engineering applications include gear ratios, where dividing rotational speeds requires fraction division. Medical dosages often involve dividing fractional amounts by body weight ratios. Students encounter this skill again in calculus when working with derivatives of rational functions, and in statistics when calculating probability ratios. The invert-and-multiply rule becomes essential for solving complex equations involving fractional coefficients.
How to solve dividing fractions
Dividing Fractions
- Keep the first fraction.
- Flip the second fraction (reciprocal).
- Multiply. Simplify.
Example: 23 ÷ 45 → 23 × 54 = 1012 = 56.
Worked examples
A rope is 14 m long. You cut it into pieces 12 m each. How many pieces?
Answer: 12
- Invert and multiply → 14 x 21 = 24 — Cutting into equal pieces is division. Flip the second fraction, then multiply across.
- Simplify → 12 — Reduce to lowest terms.
- Verify → 12 ✓ — Answer.
You have 23 of a pizza. You share it equally among friends who each get 26. How many shares?
Answer: 2
- Invert and multiply → 23 x 62 = 126 — Sharing equally means dividing. Flip the second fraction, then multiply across.
- Simplify → 2 — Reduce to lowest terms.
- Verify → 2 ✓ — Answer.
A rope is 410 m long. You cut it into pieces 15 m each. How many pieces?
Answer: 2
- Invert and multiply → 410 x 51 = 2010 — Cutting into equal pieces is division. Flip the second fraction, then multiply across.
- Simplify → 2 — Reduce to lowest terms.
- Verify → 2 ✓ — Answer.
Common mistakes
- A common error is dividing numerators and denominators separately, writing 4/6 ÷ 2/3 = 2/2 = 1 instead of the correct 4/6 × 3/2 = 2.
- Another mistake involves forgetting to invert the second fraction, calculating 1/2 ÷ 1/4 as 1/2 × 1/4 = 1/8 instead of 1/2 × 4/1 = 2.
- Many incorrectly apply division rules to both fractions, writing 3/4 ÷ 1/2 = 3/4 ÷ 1/2 = 6/8 instead of using the reciprocal method.