MATHEMATICS · FREE ONLINE TOOL

The Ratio Calculator that shows its work

Leave one value blank and the solver finds it by cross multiplication. Feed it 16 : 12 or 45 : 60 : 75 and Euclid's algorithm reduces it in front of you. Hand it an amount and a ratio, and it splits the sum into shares that add back exactly. Every line of the working is written out beneath the screen — live, in your browser, no sign-up.

CROSS-MULTIPLICATION SOLVER 2 & 3-TERM SIMPLIFY SPLIT AMOUNTS EXACTLY DECIMAL-EXACT ARITHMETIC WORKS OFFLINE
UPDATED — 2025 · WORKS OFFLINE
01

The instrument

SOLVE cross-multiplies its way to the missing value, SIMPLIFY runs Euclid's algorithm down to lowest terms, SHARE splits any amount into parts that add back exactly — and the panel beneath the screen writes out every move as it happens.

SWS · RT-01
A
:
B
=
C
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D
LEAVE EXACTLY ONE BOX BLANK — THAT ONE IS THE UNKNOWN · FILL ALL FOUR TO VERIFY A PROPORTION
TRY
TERM 1
:
TERM 2
:
TERM 3 · OPTIONAL
DECIMALS AND FRACTIONS LIKE 3/4 ARE FINE — THEY ARE CLEARED EXACTLY, NEVER ROUNDED
TRY
AMOUNT
in
PART 1
:
PART 2
:
PART 3 · OPTIONAL
THE AMOUNT IS SPLIT INTO EXACT SHARES THAT ADD BACK TO THE TOTAL — MONEY, MARKS, LAND, ANYTHING
TRY
THE WORK — LIVESOLVING AS YOU TYPE
EXACT RATIONAL CORE·EUCLID GCD ENGINE·REPEATING-DECIMAL DISPLAY·100% CLIENT-SIDE
02

How to use it

Three modes, one exact core. Sixty seconds from blank page to a fully written proof — here is the whole workflow.

STEP 01

Pick the mode

SOLVE finds a missing value in a proportion, SIMPLIFY reduces a ratio to lowest terms, SHARE splits an amount. The three tabs sit at the top of the instrument.

→ SOLVE · SIMPLIFY · SHARE
STEP 02

Feed it what you know

In SOLVE, leave exactly one box blank — that's your unknown. Decimals like 1.5 and fractions like 3/4 both parse exactly; no rounding anywhere in the pipeline.

→ 3 : 4 = ? : 12
STEP 03

It solves as you type

There is no equals button. The screen re-solves on every keystroke, and the panel beneath it rewrites the complete proof.

→ no · equals · button
STEP 04

Read it every way

The big line is the answer; the rows below carry the completed proportion, simplest form, unit value, part-to-whole fractions and percentages — repeating decimals overlined, exact.

→ 3 : 4 = 0.75 = 75 %
STEP 05

Copy the working

Every step is written the way a teacher wants it: cross-multiplication lines, Euclid's algorithm, share checks that rebuild the total.

→ 3·12 = 4·x → x = 9
STEP 06

Learn from the chips

One tap on any example chip loads a worked scenario — recipe scaling, map ratios, Pythagorean triples, fair splits.

→ 45:60:75 → 3:4:5
~/tools/ratio-calculatorWORKED SESSION
▸ session.log
solve/ 3:4 = x:12x = 9
solve/ 2:250 = 5:xx = 625
simplify/ 45:60:75gcd 15 → 3:4:5
simplify/ 1.5:2×10 → 3:4
share/ 120 in 2:348 | 72
share/ 84 in 3:4:521 | 28 | 35
verify/ 9:12 ≡ 3:436 = 36 ✓
decimal/ 1:60.16̄ exact
ALL ARITHMETIC INTEGER-EXACT
03

Questions, answered

The ratio questions that actually come up — short answers, no fluff.

Q01How do I solve a ratio with a missing value?
Write it as a proportion and cross-multiply. For 3:4 = x:12, read it as 3/4 = x/12, so 3 × 12 = 4 × x, giving 36 = 4x and x = 9. The SOLVE mode does this for a blank in any of the four positions and writes each line out as it goes.
Q02How do I simplify a ratio to lowest terms?
Divide every term by the greatest common divisor. For 16:12, Euclid's algorithm gives gcd = 4, so 16÷4 : 12÷4 = 4:3. The SIMPLIFY tab runs Euclid line by line so you can copy the working straight into your notes.
Q03How do I simplify a ratio with three terms?
The same rule, applied to all terms at once — the gcd of three numbers is gcd(gcd(a, b), c). For 45:60:75 that is gcd(15, 75) = 15, giving 3:4:5, the classic Pythagorean triple. The third box in SIMPLIFY mode is optional: leave it empty for two-term ratios.
Q04How do I divide an amount in a given ratio?
Add the terms to find the total parts, divide the amount by that total to get one part, then multiply by each term. $120 in 2:3 gives 5 parts, one part = 24, and shares of 48 and 72. The SHARE mode finishes by checking that the shares add back to the amount.
Q05What is the difference between part-to-part and part-to-whole ratios?
2:3 is a part-to-part comparison — two quantities set against each other. The whole is 2 + 3 = 5 parts, so the same information part-to-whole is 2/5 and 3/5, or 40% and 60%. The rows under the screen show both readings.
Q06How do I convert a ratio to a fraction, decimal or percentage?
Part-to-part, a:b is simply the fraction a/b: 3:4 = 3/4 = 0.75 = 75%, with repeating decimals shown exactly under an overline. Part-to-whole puts the total underneath instead: 3:4 becomes 3/7 and 4/7, about 42.9% and 57.1%.
Q07Can a ratio contain decimals, fractions or units?
Yes. Clear decimals by scaling: 1.5:2 → ×10 → 15:20 → 3:4. Units are fine as long as both sides use the same one — 250 g flour to 500 g water is just 1:2. Only a zero in the second term of a ratio used as a fraction is forbidden.
Q08What is the golden ratio?
Two quantities are in the golden ratio when the whole is to the larger part as the larger is to the smaller: (a+b):a = a:b. It equals φ ≈ 1.6180339…, and 1:φ is the only ratio that reproduces itself under that split.
Q09Is it free, and does my input leave the page?
Completely free — no sign-up, no locked features. All arithmetic runs locally in your browser; nothing is recorded or transmitted, and the page keeps working offline once it has loaded.