An efficient fraction method is the shortest correct route: cancel before you multiply, use a common denominator only for adding and subtracting, and turn division into multiplication by the reciprocal. Choosing the route before you write anything is what separates a five-line answer from a two-page one.
This skill belongs to non-calculator strategy and builds on the exact arithmetic in number sense and exact arithmetic.
How do you pick the method?
Look at the operation first, then at the numbers.
- Multiplying: look for a common factor between any numerator and any denominator, and cancel it before multiplying.
- Dividing: rewrite as multiplication by the reciprocal of the second fraction, then cancel.
- Adding or subtracting: find a common denominator, ideally the lowest. Change each fraction, then combine the numerators only.
- Mixed numbers: convert to improper fractions before multiplying or dividing.
The reciprocal step only ever applies to the fraction you are dividing by. The first fraction stays exactly as it is.
Worked example
Work out 3/4 ÷ 2/5, giving the answer as a mixed number.
Step 1, choose the method: this is division, so multiply by the reciprocal of 2/5.
Step 2, rewrite: 3/4 ÷ 2/5 = 3/4 × 5/2.
Step 3, look for cancelling: 3 and 2 share no factor, and neither do 5 and 4. Multiply straight across.
Step 4, multiply: (3 × 5) / (4 × 2) = 15/8.
Step 5, convert: 15/8 = 1 7/8.
Step 6, check: 3/4 = 0.75 and 2/5 = 0.4, and 0.75 ÷ 0.4 = 1.875. That matches 1 7/8.
A second example where cancelling pays off: 8/15 × 5/12. Cancel 8 with 12 to get 2 and 3, and cancel 5 with 15 to get 1 and 3. This gives (2 × 1) / (3 × 3) = 2/9.
Multiplying first would give 40/180, which needs a long simplification.
The mistake to watch for
A common error is to multiply by the original second fraction instead of its reciprocal.
Mistaken working: 3/4 ÷ 2/5 = 3/4 × 2/5 = 6/20 = 3/10
The division sign was changed to multiplication, but the second fraction was never flipped.
The answer 3/10 is smaller than 3/4, yet dividing by a fraction less than 1 must make the answer larger. A quick size check catches this. The correct working flips 2/5 to 5/2, giving 15/8.
Check yourself
Try these without a calculator, then open each answer.
1. Work out 7/9 × 3/14, cancelling first.
Show answer
Cancel 7 with 14 to get 1 and 2. Cancel 3 with 9 to get 1 and 3. The product is (1 × 1) / (3 × 2) = 1/6.
2. Work out 2/3 ÷ 4/9.
Show answer
2/3 × 9/4. Cancel 9 with 3 to get 3 and 1, and cancel 2 with 4 to get 1 and 2. This gives 3/2, which is 1 1/2.
3. Work out 7/10 − 1/4.
Show answer
The lowest common denominator is 20. 7/10 = 14/20 and 1/4 = 5/20. So 14/20 − 5/20 = 9/20.
Where this leads next
Next, see why keeping exact values beats rounding early. When you are ready, test the whole cluster with the non-calculator strategy practice set. The non-calculator working trainer lets you test step choices, and the percentage-base explorer does the same for chained percentage changes.
Students who know every rule but still choose the long route often need someone to watch the first line of each question. That is the kind of habit our teachers look for in online one-to-one Mathematics tuition.