Equivalent Fractions & Simplifying
Different names for the same amount, and how to find the simplest one.
The explanation
1/2, 2/4 and 50/100 are the same amount written three ways. You move between them by multiplying or dividing the top and bottom by the same number.
To simplify, divide top and bottom by their GCF. For 18/24 the GCF is 6, so it becomes 3/4. You can also chip away in stages — divide by 2, then by 3 — and land in the same place.
A fraction is in lowest terms when the only number dividing both parts is 1.
Two fractions a/b and c/d are equivalent exactly when ad = bc (cross products match). The operation behind this is multiplication by 1 in disguise: a/b = (a/b)·(k/k) = ak/bk for any k ≠ 0, since k/k = 1 and multiplying by 1 preserves value.
Simplifying reverses that. Dividing numerator and denominator by GCF(a,b) produces the unique lowest-terms representative, unique because after removing the GCF the remaining parts are coprime.
This is worth over-learning because the identical move appears in algebra with variables, where 6x²/9x = 2x/3 for x ≠ 0. The restriction matters there in a way it does not with numbers: cancelling a variable factor silently assumes it is not zero, which is the origin of many lost solutions later.
Worked example
Simplify 42/56.
- Factor both: 42 = 2·3·7, 56 = 2³·7.
- Shared factors: 2 and 7, so GCF = 14.
- 42 ÷ 14 = 3, 56 ÷ 14 = 4.
Answer: 3/4
Common mistakes
- Adding the same number to top and bottom instead of multiplying. 1/2 and 2/3 are not equal.
- Cancelling across a plus sign: (3+4)/3 is not 4.