Fractions can be added, subtracted, multiplied and divided without ever turning them into decimals. You keep the exact value through every step and convert only at the end, if at all.
This skill belongs to number sense and exact arithmetic. It returns in ratio, probability, algebraic fractions and trigonometry, where non-calculator working is often expected.
Why is an exact fraction safer than a decimal?
Take 1/3. As a decimal it is 0.3333 and never finishes, so you must cut it somewhere. If you write 0.33 and multiply by 3, you get 0.99, not 1.
A fraction has no such problem: 1/3 × 3 = 1 every time. That is why exact-answer questions reward fraction working.
How to work with fractions step by step
- Turn mixed numbers into improper fractions. 2 1/4 becomes 9/4 because 2 × 4 + 1 = 9.
- Adding or subtracting: find a common denominator, rewrite both fractions, then add or subtract the numerators only.
- Multiplying: multiply numerators together and denominators together. Cancel common factors first to keep numbers small.
- Dividing: flip the second fraction and multiply.
- Simplify to lowest terms and convert back to a mixed number if the question uses them.
Worked example
Work out 1 1/3 × 2 1/4 − 5/6, giving your answer as a mixed number.
Step 1, improper fractions: 1 1/3 = 4/3 and 2 1/4 = 9/4.
Step 2, multiply first: 4/3 × 9/4 = (4 × 9) / (3 × 4). Cancel the two 4s, then 9 ÷ 3 = 3, so the result is 3.
Step 3, subtract: 3 − 5/6 = 18/6 − 5/6 = 13/6.
Step 4, mixed number: 13 ÷ 6 = 2 remainder 1, so the answer is 2 1/6.
Check: with decimals, 1.3333 × 2.25 = 3.0 and 3 − 0.8333 = 2.1667, which matches 2 1/6. The fraction route reached it exactly and with smaller numbers.
The mistake to watch for
A very common slip is to add the numerators and the denominators separately.
Mistaken answer: 2/3 + 3/4 = 5/7
The student added 2 + 3 on top and 3 + 4 on the bottom.
A quick sense check exposes it. 2/3 is about 0.67 and 3/4 is 0.75, so the sum should be over 1, but 5/7 is under 1. A sum cannot be smaller than one of its own parts.
The correct working uses a common denominator: 2/3 = 8/12 and 3/4 = 9/12, so 8/12 + 9/12 = 17/12, which is 1 5/12.
Check yourself
Try these without a calculator, then open each answer.
1. Work out 5/6 − 3/8.
Show answer
The lowest common multiple of 6 and 8 is 24. 5/6 = 20/24 and 3/8 = 9/24. So 20/24 − 9/24 = 11/24, which cannot be simplified.
2. Work out 3/4 ÷ 9/10.
Show answer
Flip the second fraction and multiply: 3/4 × 10/9 = 30/36. Divide both by 6 to get 5/6.
Check: 0.75 ÷ 0.9 = 0.8333, which is 5/6.
3. Work out 2 1/2 × 1 3/5 and give an exact answer.
Show answer
2 1/2 = 5/2 and 1 3/5 = 8/5. Then 5/2 × 8/5 = 40/10 = 4. Check: 2.5 × 1.6 = 4.
Where this leads next
With exact fractions secure, go on to comparing exact and approximate answers to decide when to keep a fraction and when to round. The number sense practice set then mixes all the skills, and the non-calculator working trainer lets you test a fraction step with exact values.
Some students can follow every fraction rule but still pick the wrong one under pressure. That is the kind of pattern a teacher in online one-to-one Mathematics tuition looks for in your written working.