Simplifying a fraction means dividing the top and the bottom by the same number until nothing divides them both. 18/24 becomes 3/4, and the two are the same number — 0.75 either way. The whole job is finding what to divide by. This is how to spot it by eye in the cases that come up most, what to do when nothing obvious works, and why it is worth the effort rather than leaving the fraction as it is.
A fraction is a division waiting to happen. 18/24 means 18 ÷ 24, and scaling both numbers by the same amount does not change a ratio — three slices of a twelve-slice pizza is the same amount as one slice of a four-slice one.
Formally, dividing top and bottom by 6 is the same as dividing the fraction by 6/6, and 6/6 is 1. Dividing by 1 changes nothing, which is why the value survives and only the notation changes.
The same logic runs the other way. Multiplying top and bottom by 2 turns 3/4 into 6/8, which is how you build equivalent fractions when you need a common denominator.
Three checks catch the overwhelming majority of school fractions in about a second each.
Both even? Then 2 divides both. Halve and look again — often twice or three times.
Both end in 0 or 5? Then 5 divides both.
Do the digits of each add to a multiple of 3? Then 3 divides both. For 18 the digits give 1 + 8 = 9, for 24 they give 2 + 4 = 6, so both divide by 3.
A few more are worth having. A number divides by 4 if its last two digits do; by 6 if it passes both the 2 and the 3 checks; by 9 if its digits add to a multiple of 9; and by 10 if it ends in 0.
You do not have to find the biggest common factor in one go. 18/24 halved is 9/12, which halves no further, but both divide by 3 to give 3/4. Several small steps land in exactly the same place as one big one, and they are easier to do in your head.
When nothing obvious works, break both numbers into primes and see what they share.
Take 84/126. Neither divisibility rule leaps out, so: 84 = 2 × 2 × 3 × 7, and 126 = 2 × 3 × 3 × 7. They share one 2, one 3 and one 7, which multiply to 42. Dividing both by 42 gives 2/3.
This always works and it is never wrong, which makes it the fallback worth knowing. It is also where the times tables earn their keep — spotting that 126 is 2 × 63 and 63 is 7 × 9 is the whole calculation.
The Fraction Simplifier shows the prime factorisation of both numbers alongside the answer, so you can see which factors were shared.
18/24
Both even, so halve: 9/12. Both divide by 3: 3/4. Nothing divides 3 and 4, so that is lowest terms. As a decimal, 0.75; as a percentage, 75%.
45/60
Both end in 0 or 5, so divide by 5: 9/12. Both divide by 3: 3/4. Same answer as the last one, which is the point — 18/24 and 45/60 are the same number written two ways.
7/21
Neither is even and only one ends in a 5-friendly digit, but 21 is 3 × 7, so 7 divides both: 1/3. Recognising 21 as 3 × 7 is the entire step.
84/126
Both even: 42/63. 42 is not divisible by anything obvious that 63 also shares until you notice both divide by 21: 2/3. Or go straight to primes, as above.
17/23
Both are prime, so nothing divides either except 1 and itself. The fraction is already in lowest terms and there is nothing to do.
Because unsimplified fractions make every later step harder. Adding 18/24 to 5/8 means finding a common denominator for 24 and 8; adding 3/4 to 5/8 is a much smaller problem, and the numbers you carry through the rest of the working stay small.
It also makes answers comparable. Two people arriving at 6/8 and 3/4 have the same answer and it does not look like it, which is exactly the sort of thing that causes an argument or a lost mark.
And it makes the size obvious. 3/4 is instantly recognisable as just under one; 84/126 is not, until you turn it into 2/3.
These are three notations for the same thing, and converting is always the same move. Divide the top by the bottom for the decimal, then multiply by 100 for the percentage. 3 ÷ 4 = 0.75, so 3/4 is 75%.
Some fractions give a decimal that never ends. 1/3 is 0.333…, because 3 does not divide neatly into any power of ten. The rule is exact: a fraction in lowest terms terminates as a decimal only when its denominator is built from nothing but 2s and 5s.
So anything over 8, 20, 25 or 40 will terminate, and anything over 3, 6, 7, 9 or 11 will repeat. That is a quick check on whether a decimal answer that looks messy is actually wrong.
Divide the numerator and the denominator by the same number until nothing greater than 1 divides both. 18/24 divides by 6 to give 3/4.
Check whether both numbers are even, whether both end in 0 or 5, and whether the digits of each add to a multiple of 3. Those three checks cover most fractions. For the rest, break both into prime factors.
A fraction is in lowest terms when the only number dividing both the top and the bottom is 1. 3/4 is in lowest terms; 6/8 is not.
No. If the numerator and denominator share no factor other than 1 — as in 17/23 — the fraction is already as simple as it gets.
Divide the top by the bottom to get a decimal, then multiply by 100. 3 ÷ 4 = 0.75, so 3/4 is 75%.
Any fraction whose denominator, in lowest terms, has a prime factor other than 2 or 5. 1/3 and 1/7 repeat; 1/8 and 3/20 do not.
Three divisibility checks solve most fractions in a second, and prime factors solve the rest with certainty. Try one on the [Fraction Simplifier](/tools/fraction-simplifier) to see the factors laid out, then drill the recognition on [Fraction Match](/games/fraction-match).