Equivalent Ratio Calculator

Generate a table of ratios equal to the one you start with, in both directions, or test two ratios for equivalence by cross multiplication. Two, three and four-term ratios are all supported.

Formula a : b = na : nb

Try an example
Method

One rule, applied to every term

Equivalence is the easiest idea in ratio work and the one people most often break by accident — usually by changing one term and not the other.

Pick any non-zero multiplier

It can be a whole number, a fraction or a decimal. Whole numbers scale the ratio up; fractions scale it down. Zero is excluded because it would collapse every term to nothing, and the resulting 0 : 0 is not a ratio at all.

Apply it to every term without exception

From 2 : 3, multiplying by 4 means 2 × 4 and 3 × 4, giving 8 : 12. Adding 4 to each term instead gives 6 : 7, which is not equivalent — 0.857 rather than 0.667. Only multiplication and division preserve a ratio; addition does not.

To scale down, divide by a common factor

8 : 12 divided by 4 returns 2 : 3. Dividing by something that is not a common factor still gives an equivalent ratio, but with fractional terms — 8 : 12 divided by 5 is 1.6 : 2.4, correct but awkward.

Verify by cross multiplying

2 × 12 = 24 and 3 × 8 = 24. Equal cross products mean equivalent ratios, and this test works no matter how the two ratios were arrived at.

Or reduce both and compare

Every ratio has exactly one simplest form, so two ratios are equivalent precisely when they reduce to the same one. This is slower than cross multiplying but easier to see, and it gives you the canonical form as a by-product.

a : b = ka : kb   (k ≠ 0)

Generating equivalents. Scale every term by the same k. The quotient a/b is untouched, which is what "equivalent" means.

a : b ≡ c : d  ⟺  ad = bc

Testing equivalence. One multiplication each side, then compare. See the cross multiplication calculator for the same test with an unknown term.

Adding breaks the ratio

A recipe for 2 : 3 flour to water does not become 3 : 4 by adding one cup of each — you have changed the mixture. It becomes 4 : 6 by doubling. This is the single most common ratio mistake, and it is easy to spot because the quotient moves: 0.667 versus 0.75.

Worked examples

Generating and testing

Generating up

List four ratios equivalent to 2 : 3.

× 2
4 : 6
× 3
6 : 9
× 5
10 : 15
× 10
20 : 30
All give
0.6667 ✓

4:6, 6:9, 10:15, 20:30

Testing equivalence

Are 3 : 4 and 9 : 12 equivalent?

Cross product 1
3 × 12 = 36
Cross product 2
4 × 9 = 36
Compare
36 = 36
Scale factor
9 ÷ 3 = 3, 12 ÷ 4 = 3
Both reduce to
3 : 4

Equivalent ✓ — factor of 3

Not equivalent

Are 2 : 3 and 4 : 9 equivalent?

Cross product 1
2 × 9 = 18
Cross product 2
3 × 4 = 12
Compare
18 ≠ 12
Factors differ
4÷2 = 2 but 9÷3 = 3
Values
0.6667 vs 0.4444

Not equivalent ✕

Three terms

Is 1 : 2 : 3 equivalent to 5 : 10 : 15?

First terms
5 ÷ 1 = 5
Second terms
10 ÷ 2 = 5
Third terms
15 ÷ 3 = 5
All match
one factor of 5 throughout
Both reduce to
1 : 2 : 3

Equivalent ✓

Reference

Equivalent-ratio families

Each row is one family: infinitely many ratios that all reduce to the same simplest form.

Simplest × 2 × 3 × 5 × 10 Value
1 : 22 : 43 : 65 : 1010 : 200.5
2 : 34 : 66 : 910 : 1520 : 300.6667
3 : 46 : 89 : 1215 : 2030 : 400.75
3 : 56 : 109 : 1515 : 2530 : 500.6
4 : 38 : 612 : 920 : 1540 : 301.3333
5 : 810 : 1615 : 2425 : 4050 : 800.625
16 : 932 : 1848 : 2780 : 45160 : 901.7778
1 : 2 : 32 : 4 : 63 : 6 : 95 : 10 : 1510 : 20 : 30—
2 : 3 : 54 : 6 : 106 : 9 : 1510 : 15 : 2520 : 30 : 50—
Background

Equivalence is what makes ratios useful

A ratio would be of limited value if it only applied at one size. The whole point of writing a recipe as 2 : 3 rather than "400 g and 600 g" is that the ratio holds at any scale, so the same instruction works for two people or twenty. Equivalence is precisely that property, and generating equivalents is how you get from the abstract relationship to the concrete quantities you need.

This is also why proportional reasoning tasks in school so often take the form of a ratio table. Listing 2 : 3, 4 : 6, 6 : 9 and so on makes the constant multiplicative relationship visible in a way that a single pair does not, and it lets you find an answer by reading down the column rather than by setting up an equation.

Why addition does not preserve a ratio

Multiplying both terms by k changes the quotient from a/b to ka/kb, and the k cancels — nothing has moved. Adding c to both gives (a+c)/(b+c), and there is nothing to cancel; the value shifts towards 1 as c grows. That drift towards 1 is the tell: adding equal amounts to both parts of a mixture always makes it more balanced, which is a real physical change, not a rewriting.

Equivalent ratios and unit rates

Among all the equivalents of a ratio, one is often the most informative: the one whose first term is 1. 2 : 3 becomes 1 : 1.5, which says directly that there is one and a half times as much of the second quantity. That form is a unit rate, and the unit ratio calculator is built around it. At the other end of the family sits the simplest whole-number form, which the simplify ratio calculator produces.

Related tools

If one term of your second ratio is unknown rather than known, you want the cross multiplication calculator or the ratio calculator — both solve for the missing value. For applying an equivalence to physical dimensions, the scale factor calculator adds unit conversion and the area and volume factors.

Questions

Equivalent ratios — questions

Why multiplying works and adding does not, plus the two ways to test equivalence.

What are equivalent ratios?
Two ratios are equivalent when they describe the same relationship at different scales. 2 : 3, 4 : 6 and 20 : 30 are all equivalent because each is the previous one multiplied through by a constant. Formally, a : b and c : d are equivalent when a/b = c/d, or equivalently when ad = bc.
How do you find equivalent ratios?
Multiply or divide every term by the same non-zero number. From 2 : 3, multiplying by 2 gives 4 : 6, by 3 gives 6 : 9, by 10 gives 20 : 30. Because you change every term identically, the quotient is untouched. Any ratio has infinitely many equivalents in each direction, though only some divisions produce whole numbers.
How do you check whether two ratios are equivalent?
Cross multiply and compare. For 3 : 4 and 9 : 12: 3 × 12 = 36 and 4 × 9 = 36. Equal, so they are equivalent. The alternative is to reduce both to simplest form and see whether you get the same pair — both reduce to 3 : 4. Cross multiplication is quicker; simplifying is easier to see.
Is 2 : 3 the same as 3 : 2?
No. Order carries meaning, so these are different ratios — one says the second quantity is larger and the other says the first is. 2 : 3 is 0.667 and 3 : 2 is 1.5. Reversing a ratio gives its reciprocal, which is only equal to the original when both terms are equal.
Can equivalent ratios have decimals?
Yes. 1 : 1.5 is equivalent to 2 : 3, and it is the 1 : n form that map scales and gear ratios use. Dividing by the first term is what produces it. This calculator includes non-integer scalings in the table when they are useful, and always reports the 1 : n form.
How many equivalent ratios does a ratio have?
Infinitely many, since you can multiply by any non-zero number. But there is exactly one in simplest form, where the terms share no common factor. That uniqueness is what makes simplification the definitive equivalence test — see the simplify ratio calculator.
Do equivalent ratios work with three or more terms?
Yes, with the same rule: multiply or divide every term by the same number. 1 : 2 : 3 is equivalent to 2 : 4 : 6 and 5 : 10 : 15. Checking equivalence for longer ratios means confirming each corresponding pair of terms has the same scale factor, which is what this calculator does when you use three or four terms.
What is the difference between equivalent ratios and equivalent fractions?
Mathematically they are the same operation — both come from multiplying numerator and denominator (or both ratio terms) by the same number. The difference is what they describe. 2/3 as a fraction is a single quantity, two thirds of something. 2 : 3 as a ratio is a comparison of two separate quantities. The arithmetic transfers completely; the interpretation does not.