How this instrument works
A fraction is one way of writing a single value, and that value has endlessly many equally correct names: 1⁄2, 2⁄4, 6⁄12, and 50⁄100 all mark the identical point on the number line. Simplifying a fraction, or reducing it to lowest terms, means picking the one name built from the smallest whole numbers available — found by dividing both the numerator and the denominator by the largest whole number that divides them both, their greatest common divisor, in a single pass.
The need for that pass shows up mid-calculation more often than at the finish line. Add two fractions that already share a denominator and the arithmetic on top is trivial, yet the sum rarely lands pre-reduced: 5⁄18 plus 7⁄18 is correctly 12⁄18, and that pair still hides a shared factor of 6 that the addition itself never checked for. Multiplication behaves the same way, since numerators and denominators multiply independently of whatever factors they happen to have in common.
Two edges are worth knowing. An improper fraction, one where the numerator outgrows the denominator, such as 18⁄12, reduces by the identical rule straight through to 3⁄2, with no requirement to convert it into a mixed number first. And when the numerator and denominator already share no factor beyond 1, as in 5⁄9, dividing by that GCD of 1 changes nothing at all — the fraction was already standing in its simplest form before you asked.
- Enter the top number of your fraction into Numerator — the 12 in 12⁄18, for instance.
- Enter the bottom number into Denominator — the 18 that pairs with it.
- Read GCD used to simplify: the largest whole number that divides both entries exactly.
- Read Simplified numerator and Simplified denominator — the same fraction rebuilt from its smallest possible whole numbers.
- Multiply the simplified pair back by the GCD to confirm it returns your original numerator and denominator.
Worked example — reducing 12⁄18 to lowest terms
Two entries on a worksheet add to 5⁄18 plus 7⁄18. Adding fractions that already share a denominator means adding only the tops, so the sum is 12⁄18 — Numerator is 12, Denominator is 18. That pair is arithmetically correct but not yet in lowest terms: gcd(12, 18) = 6, since 6 is the largest whole number dividing both 12 and 18 without a remainder.
Dividing straight through by that 6 gives Simplified numerator = 12 ⁄ 6 = 2 and Simplified denominator = 18 ⁄ 6 = 3, so the answer worth recording is 2⁄3, not 12⁄18. Multiplying back confirms it holds together: 2 × 6 = 12 and 3 × 6 = 18, the exact pair the worksheet's addition produced before this reduction step.
Questions
What does it mean to simplify a fraction?
It means dividing both the numerator and the denominator by their greatest common divisor so the pair left behind shares no factor larger than 1. 12⁄18 and 2⁄3 name the same value; 2⁄3 is just the version built from the smallest whole numbers, reached by dividing 12 and 18 by their GCD of 6.
Why doesn't a fraction addition or multiplication come out already simplified?
Combining fractions only operates on the tops and bottoms mechanically — it never checks whether the result shares a factor with the denominator. 5⁄18 + 7⁄18 = 12⁄18 is correct arithmetic that still hides a factor of 6, which is why a simplification pass belongs after every fraction calculation, not only at the start of one.
Does simplifying change the value of the fraction?
No. Dividing the numerator and denominator by the same nonzero number leaves the value untouched, since a⁄b equals (a÷g)⁄(b÷g) for any common divisor g. 12⁄18 and 2⁄3 both equal 0.666…; simplifying only shrinks the two numbers used to write that same value down.
What if the numerator and denominator share no common factor?
Then their greatest common divisor is 1, dividing each term by 1 changes nothing, and the fraction was already in lowest terms. Enter Numerator = 5 and Denominator = 9 and the sheet returns GCD used to simplify = 1, with Simplified numerator and Simplified denominator identical to what went in.
Does this work for improper fractions, where the numerator is larger?
Yes — the same division applies whether the fraction is proper or improper. 18⁄12 has gcd(18, 12) = 6, reducing straight through to 3⁄2; nothing about the method requires the numerator to be smaller than the denominator, and the sheet does not convert the result into a mixed number.
How is this different from the site's ratio simplifier?
The division is identical — both divide two integers by their shared GCD — but a fraction names one value on the number line, while a ratio keeps two separate quantities visible side by side for comparison. Reach for this sheet to finish fraction arithmetic or tidy a fraction for reporting; use the Ratio of 2 Numbers sheet when the two numbers describe amounts you want to keep as a visible comparison, like parts in a mixture.