How this instrument works
A ratio compares two quantities that share a unit, and 12:18 means exactly what 2:3 means — for every 2 units of the first there are 3 of the second, whether you're counting 12 and 18 items or the plainest pair that still says it. Reducing to lowest terms means finding the greatest common divisor of the two numbers, the largest whole number that divides both evenly, and dividing both terms by it in one stroke.
The reason a single division always finishes the job traces back to how the GCD is defined: if g is the greatest common divisor of a and b, then a and b are each a whole-number multiple of g, so a⁄g and b⁄g come out as whole numbers too, and those two results cannot share any further common factor — if they did, g times that factor would itself divide both a and b, making it a larger common divisor than g, which contradicts g being the greatest. That one-line argument is exactly the proof that reducing 12⁄18 to 2⁄3 as a fraction and reducing 12:18 to 2:3 as a ratio are the same operation wearing different notation.
Two edges are worth knowing. When the two numbers share no factor beyond 1 — 5 and 7, say — their GCD is 1, dividing by it changes nothing, and the ratio was already as reduced as it could get. And a number ratioed against itself, like 100:100, always collapses to 1:1, since any number is its own greatest common divisor. Stretching the same idea to three numbers instead of two — the companion Ratio of 3 Numbers sheet — isn't a straightforward extra column: a shared GCD across three terms only falls out after chaining two pairwise reductions, the GCD of the first pair fed back in against the third number, rather than one GCD covering all three terms at once.
- Enter the first quantity into Number 1 — say 12 parts of something.
- Enter the second quantity into Number 2 — say 18 parts it's compared against.
- Read GCD used to simplify: the largest whole number dividing both entries exactly.
- Read Simplified ratio: first term and Simplified ratio: second term — the same comparison in its plainest whole-number form.
- Multiply the two simplified terms back by the GCD to check the arithmetic returns your original pair.
Worked example — simplifying a 12:18 ratio
A recipe calls for 12 cups of flour to 18 cups of water — Number 1 is 12 and Number 2 is 18. The sheet finds GCD used to simplify: gcd(12, 18) = 6, since 6 is the largest whole number that divides both 12 and 18 without a remainder. Dividing each term by that 6 gives Simplified ratio: first term = 12 ⁄ 6 = 2 and Simplified ratio: second term = 18 ⁄ 6 = 3.
The result, 2:3, is the same proportion as 12:18 at the smallest scale that still uses whole numbers — for every 2 cups of flour there are 3 cups of water, in a batch a sixth the size. Checking by hand confirms it: 2 × 6 = 12 and 3 × 6 = 18, so multiplying the simplified ratio back up by the GCD returns exactly the pair that was entered.
Questions
What does it mean to simplify a ratio to lowest terms?
It means dividing both numbers by their greatest common divisor so no whole number bigger than 1 divides both remaining terms. 12:18 and 2:3 describe the identical proportion; 2:3 is just the version with the smallest possible whole-number terms, reached by dividing 12 and 18 by their GCD of 6.
How is simplifying a ratio different from simplifying a fraction?
It isn't, mechanically — a ratio a:b and a fraction a⁄b reduce by the identical operation, dividing both parts by their shared GCD. The difference is presentation: a fraction reads as one number, 12⁄18 = 0.666…, while a ratio keeps both quantities visible side by side, useful when the individual amounts still matter, like cups of flour and water.
What happens if the two numbers share no common factor?
Their GCD comes out as 1, and dividing each term by 1 changes nothing — the ratio was already in lowest terms. Try Number 1 = 5 and Number 2 = 7: gcd(5, 7) = 1, so Simplified ratio: first term and second term return 5 and 7, identical to what went in.
Why does a ratio of a number with itself always reduce to 1:1?
Because a number is always its own greatest common divisor. Entering 100 and 100 gives gcd(100, 100) = 100, and dividing both terms by that 100 leaves 1 and 1 — every self-ratio collapses to plain equality no matter how large the starting numbers are.
How is this different from finding the ratio of three numbers?
This sheet finds one GCD and divides twice. Three numbers need an extra pass: find the GCD of the first two, then find the GCD of that result and the third, since one greatest common divisor shared by all three terms only emerges after that chaining step — it isn't just the same pairwise formula with a third column bolted on.
Do Number 1 and Number 2 have to be whole numbers?
The greatest common divisor is defined for integers, so both entries should be whole numbers for 'lowest terms' to mean what it usually means. A ratio built from decimals, like 2.5:3.75, needs converting to a whole-number scale first — multiply both sides by 4 to get 10:15 — before this GCD-based reduction applies.