Ratio Calculator

What Is a Ratio?

Four things are worth separating before any arithmetic happens: what a ratio actually asserts, the difference between comparing parts with each other and with the total, how ratios relate to fractions and percentages, and the several ways the same ratio gets written down.

A Ratio Compares Two Quantities by Division

Wolfram MathWorld states it in a single line: "The ratio of two numbers r and s is written r/s, where r is the numerator and s is the denominator." That is the whole idea. A ratio is a quotient wearing a colon, and every property it has follows from division.

The consequence that matters most is invariance. Multiply both terms by the same number, or divide both by the same number, and the quotient does not move — so 2 : 3, 4 : 6, 6 : 9, 8 : 12, 10 : 15, 20 : 30 and 200 : 300 are the same ratio in seven costumes. Enter any of them in simplify mode and the headline comes back as 2 : 3 every time. For the six scaled-up pairs the second line names the divisor it removed — 2, 3, 4, 5, 10 and 100 respectively — while 2 : 3 entered as itself has nothing to remove, so that line reads "2 : 3 is already in lowest terms".

When both quantities carry the same unit, the units cancel and the ratio is a pure number: 24 cm to 36 cm is 2 : 3, with no centimeters left in the answer. When the units differ and do not cancel — miles against hours, dollars against square feet — the result is a rate, and the colon notation is the wrong tool for it.

Part to Part, Part to Whole, and Why 1 : 4 Is Five Parts

A colon ratio compares one part with another part, not with the total. Read 1 : 4 as one measure of the first thing alongside four of the second, which puts five parts in the container: the first quantity is one fifth of the mixture, or 20%, and the second is four fifths, or 80%.

The 25% answer comes from dividing by 4 rather than by 5, and it is a live error rather than a classroom curiosity — cleaning concentrates, plant food and lab reagents are all labeled in parts. Mixing at 1 : 4 with 250 mL of concentrate takes 1,000 mL of water and fills 1,250 mL, not 1,000.

This calculator works strictly part-to-part. Nothing in the panel is a share of the whole: the line labeled "As a fraction" prints A ÷ B, which for 1 : 4 is 0.25 — the first part measured against the second, not against the total. Converting to shares is two steps of your own arithmetic: add the terms, then divide each by that sum.

Ratio, Fraction, Percentage and Rate: Six Readings of One Division

Every one of them is a division. What separates them is which two things go into it and how the answer is dressed. The 2 : 3 case, run through this calculator and then finished by hand, shows the family resemblance:

Notice that two different divisions sit inside one panel. A ÷ B gives 0.6667 and B ÷ A gives 1.5, and both describe 2 : 3 correctly. Quoting the wrong one is the easiest way to invert an answer, which is why the calculator labels them separately rather than printing a single decimal.

The last two rows are a different question — how big each part is relative to the total, rather than relative to each other. That conversion, in both directions, is the job of the Percentage Calculator.

Four Ways to Write the Same Ratio

The notation changes with the trade, while the arithmetic does not:

Odds use the colon too, and they mean something narrower. Odds of 3 : 1 against describe three losing outcomes for every winning one, so there are four outcomes in total and the probability of the win is 1/4. That is a part-to-part ratio being read as a share of the whole, which is the same five-parts trap in a different costume.

• Colon form, 2 : 3. The default in recipes, mixes, maps and screen sizes, and the form this calculator takes in its A and B boxes.
• Word form, "2 to 3". Common in spoken instructions and in school textbooks, and identical in meaning.
• Fraction form, 2/3. Written vertically a ratio is indistinguishable from a fraction, which is exactly why both reduce by the same greatest-common-divisor step.
• Normalized form, 1 : n or n : 1. Scaled so that one side is a single unit. Map scales are written 1 : n, gear and two-stroke fuel ratios are usually written n : 1, and the panel supplies the 1 : n version on every simplify run.

Because odds and probability trade places constantly — and because bookmakers, statisticians and epidemiologists each write them differently — that conversion has its own tool in the Odds Calculator.

How Do You Calculate a Ratio?

Two operations cover almost every ratio question anyone brings to a calculator, and the Task switch at the top of the form picks between them. Reducing turns an awkward pair into its simplest honest form; solving finds a missing fourth term when two ratios have to match.

The Ratio Formula

Both modes are one line of algebra each. G below is the greatest common divisor of A and B — the largest whole number that divides both without remainder.

Simplify       A : B  =  (A ÷ G) : (B ÷ G)      where G = GCD(A, B)
1 : n form     1 : (B ÷ A)
As a fraction  A ÷ B

Solve          A : B = C : x
               A × x = B × C          (cross-multiply)
               x = B × C ÷ A
Check          A × x  equals  B × C
Scale factor   C ÷ A

Cross-multiplication is not a trick to memorize. Write the proportion as the equation A/B = C/x, multiply both sides by B and by x, and the denominators clear: A × x = B × C. Dividing by A isolates x. Every step is ordinary equation-solving, which is why the identity in the Check line has to hold for a correct answer.

The reduction step stands on its own, and if the number you actually want is the divisor rather than the reduced ratio — for lining up denominators, sharing out a quantity, or checking whether two figures are coprime — it is produced directly by the GCF Calculator.

MathWorld defines the greatest common divisor as "the largest divisor common to a and b", and notes that once it has been divided out the two remaining numbers have a GCD of 1 — they are relatively prime, which is the formal meaning of "lowest terms". This is why the calculator's second line reads "is already in lowest terms" for a pair like 16 : 9: their only common divisor is 1, so there is nothing to remove.

Step by Step With This Calculator

The form holds four controls, and one of them is used by only one of the two modes:

In simplify mode the sequence is: find the largest number that divides A and B, divide both by it, then report B ÷ A as the 1 : n form and A ÷ B as the decimal. In solve mode it is: multiply B by C, divide by A, then re-multiply to prove the identity holds.

• Task. A two-button switch that opens on "Simplify A : B". Move it to "Solve A : B = C : x" for proportion work.
• A and B. The two terms of your ratio, in the order you want them compared. Both must be greater than zero; the order is not interchangeable.
• C (for solving). The third known term. Simplify mode ignores it completely — leave a stray 9 in the box and 24 : 36 still returns the same four lines it returns with the box empty.
• The Calculate button. The tool runs on submit, not as you type, and because every box opens blank there is no result waiting when the page loads.

Reset clears the boxes and returns the panel to its empty state. Each run reads the four controls afresh, so nothing carries over from the calculation before it.

Worked Example: Simplify 24 : 36

The two placeholder figures, done by hand first and then checked against the panel. A = 24, B = 36, Task left on Simplify.

Euclid's algorithm   36 = 24 × 1 + 12
                     24 = 12 × 2 + 0     →   G = 12

Lowest terms         24 ÷ 12 : 36 ÷ 12   =   2 : 3
1 : n form           1 : (36 ÷ 24)       =   1 : 1.5
As a fraction        24 ÷ 36             =   0.6667

What the result panel returns

2 : 3 · 24 : 36 ÷ 12 (the GCD) both sides · As "1 : n" form: 1 : 1.5 · As a fraction: 0.6667 · Ratios scale: multiply BOTH sides by any number and the relationship holds

Read the middle line as two answers to two different questions. The 1 : 1.5 says that for every single unit of A there is one and a half of B, which is the phrasing recipes and mixing instructions use. The 0.6667 is 24 ÷ 36 rounded to four places — the decimal value of two thirds, and the number to reach for when you need to compare this ratio against another one numerically.

What the panel does not print is the split. Two parts plus three parts is five, so A is 40% of the total and B is 60% — arithmetic you have to do yourself, and the reason the page separates part-to-part readings from shares of the whole.

Worked Example: Solve 3 : 4 = 9 : x

Move the Task switch to "Solve A : B = C : x", then enter A = 3, B = 4 and C = 9. The question is what fourth number keeps the two ratios equal.

Cross-multiply   3 × x = 4 × 9
                 3x    = 36
                 x     = 36 ÷ 3   =   12

Check            3 × 12 = 36   and   4 × 9 = 36
Scale factor     C ÷ A  =  9 ÷ 3  =  3

What the result panel returns

x = 12 · 3 : 4 = 9 : x → cross-multiply: x = 4 × 9 ÷ 3 · Check: 3 × 12 = 36 and 4 × 9 = 36 — equal, so the proportion holds · Scale factor between the ratios: 3×

The scale factor is the shortcut. Nine is three times three, so whatever happened to the left-hand term happened to the right-hand one: x is three times four, or 12. Try 2 : 3 = 12 : x and the factor comes back as 6×, with x = 18; try 4 : 5 = 100 : x and it is 25×, with x = 125. Whenever that factor is a tidy whole number you can finish the sum in your head and use the panel only as confirmation.

Answers are not required to be whole. 2 : 5 = 7 : x returns x = 17.5 at a scale factor of 3.5×, and 5 : 3 = 12 : x returns x = 7.2 at 2.4×. Nothing is wrong in either case: a proportion produces whatever number keeps the quotients equal, and that number is a fraction more often than not.

When A or B Is Not a Whole Number

Reduction runs only when both terms are integers. Enter 2.5 : 4 and the calculator returns "2.5 : 4 is already in lowest terms" — accurate about those two figures, because a greatest common divisor is defined for whole numbers, and unhelpful about the ratio, which does reduce.

Clear the decimal first by scaling both sides, which is always legal. Doubling 2.5 : 4 gives 5 : 8, and the panel then confirms 5 : 8 is in lowest terms. The proof that nothing changed is in the other two lines: 2.5 : 4 and 5 : 8 both report a 1 : n form of 1 : 1.6 and a fraction of 0.625. Identical readings, one tidier pair of numbers.

If the two terms arrive as fractions rather than decimals — three quarters of a cup against two thirds of a cup — clear the denominators before you use this tool, which is the kind of arithmetic handled by the Fraction Calculator.

The same move works at any depth. Multiply 0.5 : 0.25 by four to get 2 : 1, and both entries return 1 : 0.5 with a fraction of 2. As a rule of thumb, multiply both terms by 10 for each decimal place present, then reduce.

Ratio Examples, Worked Out

Every figure in the tables below came out of this calculator, mode by mode, rather than from an example book. The four groups cover the four situations people usually arrive with: a ratio that looks different but is not, a ratio that will or will not reduce, a ratio hiding inside a pair of measurements, and a ratio being scaled up to a real quantity.

Equivalent Ratios That All Reduce to 2 : 3

Seven entries, one ratio. The middle column is the divisor the panel names on its second line:

The right-hand column is the quickest equivalence test there is. Two ratios are the same ratio when their 1 : n readings match, whatever the raw numbers look like, because that reading is just B ÷ A with the rounding applied at the end.

Order is not part of that equivalence. Swap the terms and you get a genuinely different ratio: 3 : 4 reports 1 : 1.3333 with a fraction of 0.75, while 4 : 3 reports 1 : 0.75 with a fraction of 1.3333. The two numbers trade places, which is a useful sanity check when a result looks upside down.

Ratios That Will Not Reduce, and Ratios That Reduce a Long Way

The second line of the panel switches wording depending on whether a common divisor exists. Both wordings appear below, alongside the decimal each pair produces:

A pair that will not reduce is not a failure of the tool. It means the two numbers share no factor above 1 — they are relatively prime — and 16 : 9, 3 : 4, 50 : 1 and 1 : 16 are all in that state on arrival.

The size of the divisor carries no meaning of its own. A million to seven hundred and fifty thousand collapses to 4 : 3 after removing 250,000, and 8 : 6 collapses to the same 4 : 3 after removing 2. Both describe an identical shape; only the units they were measured in differed.

Screen Resolutions: the Ones That Are 16 : 9 and the Ones That Only Look It

Pixel dimensions are ratios that nobody bothers to reduce, which makes them a good test. Each row is a resolution entered as A and B in simplify mode:

The 1366 × 768 row explains a decades-old oddity. A true 16 : 9 panel 768 pixels tall would be 1,365.33 pixels wide, and pixels do not come in thirds, so the width was rounded up to 1,366. The resulting shape reduces to 683 : 384 and measures 1.7786 against 16 : 9's 1.7778 — close enough to sell as widescreen, and not the same number.

The 2560 × 1080 row is the same story with louder marketing. It reduces exactly to 64 : 27, a quotient of 2.3704, while an actual 21 : 9 would be 2.3333. The panels are consistent with one another and simply are not the ratio printed on the box.

Working the other way — fixing a shape and asking what height goes with a given width — is solve mode's job, and 16 : 9 = 1280 : x duly returns x = 720 at a scale factor of 80×. For frame sizes, crops and export presets specifically, the labeled version of the same math is the Aspect Ratio Calculator.

Scaling a Recipe, a Mix and a Map

Solve mode is what turns a printed ratio into a quantity you can measure. Each row was run with the Task switch on "Solve A : B = C : x":

Two of those rows are worth pausing on. The dilution row answers a question people rarely ask correctly: 1 : 4 gives the water, not the finished volume, so the bucket ends up holding five parts rather than four. The ramp row is a published requirement rather than a preference — the US Access Board's guide to the ADA standards puts a ramp's running slope at 1 : 12 maximum and caps a single run at a 30-inch rise, so 360 inches, or 30 feet, is the longest a single run can be at that slope.

The map rows show why scales are always written 1 : n. With A fixed at 1, the scale factor the panel reports is simply the distance you measured, and x arrives in the same unit you measured in — 3 cm becomes 150,000 cm because both sides of the proportion are centimeters. Convert at the end, never in the middle.

Scaling one ingredient is a single proportion; scaling a whole list of them, and rounding each result into measurements a kitchen actually owns, is repetitive enough to deserve its own tool in the Recipe Conversion Calculator.

Ratio Chart: 1 : n, Fraction and Percent of the Whole

Three of the columns below come straight from the calculator; the last two are the conversion it does not perform. Keeping them side by side is the point, because almost every misread ratio is a part-to-part number being used as a share of the whole.

Common Ratios in Five Forms

Sorted by total parts. The 1 : n and A ÷ B columns are the panel's own output for that pair; the share columns are the sum-then-divide step done by hand:

The gap between the A ÷ B column and the share columns is where ratio errors live, and the two can never agree: A ÷ B measures the first part against the second, while the share measures it against a total that includes it. For 1 : 4 the quotient is 0.25 and A's share is 20%; for 1 : 16 the quotient is 0.0625 against a share of 5.88%; for 50 : 1 the quotient is 50 against a share of 98.04%.

Whether a share lands on a tidy percentage depends only on the total-parts column. Totals of 2, 4, 5 and 25 divide 100 exactly, which is why 1 : 1, 1 : 3, 2 : 3 and 16 : 9 all give clean figures. Totals of 3, 6, 7, 11, 17 and 51 do not, which is why 2 : 1 has to be written as 66.67% and 33.33% and why 1 : 16 lands on 5.88% and 94.12%.

The 16 : 9 row is a good illustration of a ratio that has no business being read as a share. Its terms are pixels along two different edges, so "64% of the total" describes nothing physical — the number is arithmetically correct and meaningless. Shares only mean something when the parts combine into one quantity.

Dilution Ratios by Bucket Size

Concentrates are labeled part-to-part, but you buy and pour by finished volume. This table converts, per 1,000 mL of made-up liquid:

Each concentrate figure is 1,000 divided by the total parts, and each water figure is what remains, both taken from the unrounded division and rounded once at the end — which is why the two columns still add to exactly 1,000 mL on every row.

The distinction has legal weight for disinfectants and pesticides. The EPA's guidance on registered disinfectants is explicit that a product "should be used or applied according to the Use Directions", and names dilution as one of the things those directions cover — so where a label gives a ratio, the label's reading of it is the one that counts, not a convenient reinterpretation.

If your label instead states a finished strength as a percentage, work backwards: a 20% solution is one part in five, which is 1 : 4 part-to-part, not 1 : 5. The off-by-one between "in" and "to" is the single conversion worth double-checking every time.

Map Scales as Ratios

A map scale is a ratio with the first term pinned to 1 and no units attached, so it reads the same whether you measure in centimeters, inches or thumb widths. The ground distances below are what solve mode returns for a 3 cm measurement, converted at the end:

The 1 : 24,000 row is the only one here that lands on a round number in customary units, because 24,000 inches is exactly 2,000 feet. Every other denominator in the table divides into an awkward figure of inches, which is why round-number scales suit metric measurement and that one suits a ruler marked in feet.

A bigger denominator means a smaller-scale map covering more ground in less detail, which is the reverse of what the words suggest. On the same sheet of paper, 1 : 25,000 reaches half as far in each direction as 1 : 50,000 and therefore covers a quarter of the ground area, at twice the linear size for everything drawn on it.

How to Read Your Result

Both modes return a headline plus three supporting lines, and the lines are not interchangeable between modes. Knowing which is which turns the panel into a check on your own arithmetic rather than a number to copy.

The Four Lines Simplify Mode Returns

Entering 24 and 36 with the Task left on Simplify produces these, in order:

The label "As a fraction" is doing something narrower than it sounds. It prints a decimal, not a numerator over a denominator: for 24 : 36 the value is 0.6667, which is two thirds rounded to four places rather than the string "2/3". If you want the fraction written out, the headline has already given it to you.

• The headline, "2 : 3" — your ratio in lowest terms, and the only line most people need.
• The working, "24 : 36 ÷ 12 (the GCD) both sides". Check this first when an answer surprises you, because it names the two numbers the tool actually read and the divisor it removed. When no divisor exists the line becomes "16 : 9 is already in lowest terms" instead.
• Two readings on one line: "As "1 : n" form: 1 : 1.5 · As a fraction: 0.6667". The first is B ÷ A, the second is A ÷ B, and they are reciprocals of each other.
• A fixed reminder, "Ratios scale: multiply BOTH sides by any number and the relationship holds". This text is identical on every simplify run and carries none of your numbers.

Going the other way — taking a decimal like 0.6667 or 0.625 and recovering the fraction and lowest-terms ratio behind it — is a separate piece of arithmetic, handled by the Decimal to Fraction Calculator.

The Four Lines Solve Mode Returns

Switching the Task and entering A = 3, B = 4, C = 9 replaces all four:

The scale factor is the line to look at when a result seems implausible. A factor far from 1 means the two ratios are being compared across a big jump in size — 3 : 1 = 250 : x reports 83.3333× — and while that is perfectly valid arithmetic, it is also where a misplaced decimal point in C shows up most visibly.

• The answer, "x = 12". Unlike simplify mode, the headline here carries an equals sign, so copying it into another field means taking the number after it.
• The method with your figures substituted in: "3 : 4 = 9 : x → cross-multiply: x = 4 × 9 ÷ 3".
• The proof: "Check: 3 × 12 = 36 and 4 × 9 = 36 — equal, so the proportion holds". Both products must match; that is the definition of a proportion.
• The shortcut: "Scale factor between the ratios: 3×". This is C ÷ A, and multiplying B by it reaches x without any cross-multiplication at all.

One quirk in the Check line deserves flagging. The multiplication it prints uses the rounded value of x, but the product it reports comes from the unrounded one. Solve 3 : 7 = 5 : x and the panel prints "Check: 3 × 11.6667 = 35 and 7 × 5 = 35" — yet 3 × 11.6667 is 35.0001. The identity is exact and the displayed multiplicand is not, so the line is a confirmation that the proportion holds rather than a sum to reproduce on a pocket calculator.

The underlying answer keeps its full precision either way. Exactly 35/3 is what the tool computed for that example; 11.6667 is only what it had room to show.

The Two Messages That Replace an Answer

When an input fails the guard, the panel turns amber and shows one of two sentences instead of a result:

There is no partial answer behind either message. The calculator stops before computing anything, so a blank B does not quietly become zero and produce a confident, wrong result — a failure mode worth being grateful for.

• "Enter positive values for A and B" — triggered whenever A or B is blank, zero, negative, or text the browser could not read as a number. It appears in both modes.
• "Enter a positive value for C" — solve mode only, when the third box is blank, zero or negative. A and B are already valid at that point, which is why the message names C specifically.

Zero is refused rather than treated as a special case, and that is defensible: a ratio with a zero term cannot be reduced, and one of its two readings — B ÷ A or A ÷ B — has no value at all. In solve mode a zero A would leave the formula dividing by nothing. Negative terms are refused for the same reason a negative recipe quantity is.

Limits: When This Calculator Does Not Apply

This tool reduces or solves a two-term ratio of positive numbers. Being exact about where that job ends is more useful than a longer list of things it was never meant to do.

• Two terms only. There is no third box, so a mix written 1 : 2 : 3 or 2 : 3 : 5 cannot be entered as it stands. Reduce a three-term ratio by dividing all three terms by their common divisor; 2 : 3 : 5 is already lowest and splits ten parts into 20%, 30% and 50%, while concrete's 1 : 2 : 3 splits six parts into 16.67%, 33.33% and 50%.
• Positive numbers only. Zero, negatives and anything non-numeric are rejected outright with "Enter positive values for A and B", so a ratio involving a debt or a temperature below zero has to be handled by hand.
• Reduction is integer-only. A decimal pair is reported as already in lowest terms whether or not it is; scale both terms to whole numbers first, as 2.5 : 4 becomes 5 : 8.
• Everything is rounded to four decimal places, and at the extremes that erases the number. Enter 1 : 100000 and the fraction line prints 0; enter 100000 : 1 and the 1 : n line prints 1 : 0. Both of those are 0.00001 with nowhere to go.
• No share of the whole is ever printed. The panel is entirely part-to-part, so converting 2 : 3 into 40% and 60% is your own two-step arithmetic.
• The Check line rounds what it displays but not what it multiplies, so its printed sum can be a ten-thousandth out even when the proportion is exact.
• Nothing runs on load or as you type. The boxes open empty and the formula executes when you submit, which means a half-filled form simply shows the guard message.
• The tool has no idea what your numbers mean. 3 : 4 could be parts of a mixture, a screen shape, a gear pair or odds against; the arithmetic is identical and the interpretation is entirely yours.
• Irrational ratios cannot be entered exactly. MathWorld gives the golden ratio as 1.618033988749894848204586834365638117720…, so no pair of whole numbers reproduces it. Successive Fibonacci pairs get close and never arrive: 21 : 13 reports 1.6154, 34 : 21 reports 1.619 and 55 : 34 reports 1.6176, the last of those about 0.0004 away.
• A proportion is a model, not a fact. Cross-multiplication assumes the two quantities move together in a straight line through zero — double one, double the other. Ingredients and map distances behave that way; cooking times, drug doses and paint coverage frequently do not, and solving a proportion over them returns a number that is arithmetically correct and practically wrong.

One boundary sends people elsewhere more than the rest. If the two numbers you are comparing are the same measurement at two different times, what you want is not a ratio at all but the change between them, expressed as a percentage up or down, which is the job of the Percentage Increase / Decrease Calculator.

Two things that look like limits are not. Very large figures are handled without complaint — 1,000,000 : 750,000 comes back as 4 : 3 after a divisor of 250,000 — and a non-terminating answer is still exact behind the display, since the calculator carries full precision through the arithmetic and rounds only when it prints.