To simplify a numerical expression with factors, write each number as a product of smaller numbers, then cancel what appears in both numerator and denominator. It works for long products and quotients, fractions in lowest terms, and questions on the highest common factor (HCF) and lowest common multiple (LCM).
The skill sits early in number sense and exact arithmetic and comes back in algebraic fractions, surds and ratio.
Why do factors make a calculation shorter?
Multiplying big numbers first and dividing afterwards creates large intermediate values. Factors let you remove the shared part before you multiply anything, so the numbers stay small and the answer is exact.
A prime factor is a factor that is a prime number, such as 2, 3, 5 or 7. Every whole number above 1 can be written as a product of primes in exactly one way, for example 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
How to simplify step by step
- Write every number in the expression as a product of primes, using a factor tree or repeated division.
- Line up the numerator and the denominator, using powers where a prime repeats.
- Cancel each prime that appears on both sides. Subtract the powers: 2⁴ on top and 2² on the bottom leaves 2².
- Multiply what is left to get the simplified value.
- Check with one rough estimate to make sure the size of the answer is sensible.
Cancelling is only valid for factors of the whole numerator and the whole denominator. A number sitting inside an addition or subtraction is not a factor of it.
Worked example
Simplify (48 × 35) / (14 × 30).
Step 1, prime factors: 48 = 2⁴ × 3, 35 = 5 × 7, 14 = 2 × 7, 30 = 2 × 3 × 5.
Step 2, line up: the numerator is 2⁴ × 3 × 5 × 7. The denominator is 2 × 7 × 2 × 3 × 5 = 2² × 3 × 5 × 7.
Step 3, cancel: the 3, the 5 and the 7 appear on both sides and cancel. Of the twos, 2⁴ over 2² leaves 2².
Step 4, multiply: 2² = 4.
Check: 48 × 35 = 1680 and 14 × 30 = 420, and 1680 ÷ 420 = 4. Both routes agree, and the factor route never needed the big products.
The mistake to watch for
A common slip is to cancel a number that is only part of a sum.
Mistaken answer: (20 + 8) / 4 = 20 + 2 = 22
The student divided only the 8 by 4 and left the 20 untouched.
The numerator is a sum, so 4 must divide the whole of it. The correct working is (20 + 8) / 4 = 28 / 4 = 7.
Or factor first: (4 × 5 + 4 × 2) / 4 = 4 × (5 + 2) / 4 = 5 + 2 = 7. Factoring the numerator is what makes cancelling legal.
Check yourself
Try these without a calculator, then open each answer.
1. Simplify (72 × 25) / (30 × 36).
Show answer
72 = 2³ × 3², 25 = 5², 30 = 2 × 3 × 5, 36 = 2² × 3². The numerator is 2³ × 3² × 5². The denominator is 2³ × 3³ × 5. Cancelling leaves 5 on top and 3 on the bottom.
5/3. Check: 72 × 25 = 1800, 30 × 36 = 1080, and 1800/1080 = 5/3.
2. Write 84 and 126 as products of primes, then find their HCF and LCM.
Show answer
84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. HCF uses the lower power of each shared prime: 2 × 3 × 7 = 42. LCM uses the higher power: 2² × 3² × 7 = 252.
Check: 42 × 252 = 10 584 and 84 × 126 = 10 584.
3. Simplify (15 + 35) / 5.
Show answer
The numerator is a sum, so divide the whole of it: 50 / 5 = 10. Or factor first: 5 × (3 + 7) / 5 = 3 + 7 = 10. Cancelling only the 35 would give 15 + 7 = 22, which is wrong.
Where this leads next
Once cancelling is reliable, move on to working with fractions without decimal rounding, then test the module with the number sense practice set. The non-calculator working trainer lets you check a cancelling step with exact values.
Students often follow each cancelling step in class but hesitate when a sum sits inside the expression. A teacher in online one-to-one Mathematics tuition can point to the exact line where that hesitation starts.