Ratio & proportion tools
Ratio vs Proportion
A ratio compares quantities. A proportion claims that two ratios are equal — which makes it something you can test.
3:5 is a ratio. It compares two quantities and nothing more; there is no claim in it to agree or disagree with. 3:5 = 12:20 is a proportion. It asserts that two comparisons describe the same relationship, and an assertion can be checked:
Testing 3:5 = 12:20
- Multiply the outer pair: 3 × 20 = 60.
- Multiply the inner pair: 5 × 12 = 60.
- The cross-products match, so the two ratios are the same relationship and the proportion is true.
What changes when you write the equals sign
A ratio is a description of relative size. Written on its own, 3:5 tells you the first quantity is three-fifths of the second and that five parts make the whole. You can simplify it, scale it or convert it — but you cannot ask whether it is correct, because it is not making a claim.
The equals sign changes the grammar of the statement. 3:5 = 12:20 has a truth value, and that is the entire difference between the two words. Everything else — cross-multiplication, solving for an unknown, checking a scale factor — follows from having something testable.
Why cross-multiplication is a valid test
A proportion a:b = c:d is the same statement as a/b = c/d. Multiply both sides of that equation by b × d:
Derivation
- Start from a/b = c/d, with b and d non-zero.
- Multiply both sides by bd: (a/b) × bd = (c/d) × bd.
- On the left the b cancels, leaving ad. On the right the d cancels, leaving bc.
- So a/b = c/d holds exactly when ad = bc.
Because each step is reversible, the two forms are equivalent — equal cross-products are not a trick that usually works, they are the same claim written without fractions. That is also why the test is safe with decimals and awkward numbers: nothing was rounded to get there.
A proportion that fails
The reason to test rather than eyeball: 12 is four times 3, and 18 is a bit more than three times 5, which at a glance can pass for “both grew.” Growing is not enough. Both parts have to grow by the same factor.
3:5 = 12:18 tested
3 × 18 = 54.
5 × 12 = 60.
54 ≠ 60, so the statement is false.
The gap also says which direction it is wrong in: 12:18 reduces to 2:3 = 0.667, while 3:5 = 0.6. The second ratio has too much of the first quantity. To make the proportion true, the fourth value must be 20.
Solving when one value is unknown
Once you accept ad = bc, an unknown in any of the four positions is one division away. Write the proportion with x in place, cross-multiply, then isolate x.
3:5 = x:30
- Cross-multiply: 3 × 30 = 5 × x.
- That gives 90 = 5x.
- Divide both sides by 5: x = 18.
- Check by substituting back: 3 × 30 = 90 and 5 × 18 = 90. The cross-products agree.
The substitution check matters more than it looks. It catches the most frequent error in this arithmetic, which is pairing the wrong two numbers — dividing by 3 instead of 5 gives 10, and 3 × 30 = 90 while 5 × 10 = 50 immediately exposes it.
Equivalent ratio or proportion? Same algebra, different starting point
Scaling 3:5 by 6 and solving 3:5 = 18:x produce the identical answer, because both are the statement 3k:5k. What differs is which piece you were handed. If you know the factor, multiply. If you know a fourth number, the factor is recovered on the way.
| You already have | Route | Worked on 3:5 |
|---|---|---|
| The multiplier | Multiply every part by it | ×6 → 18:30 |
| One matching quantity | Divide to recover the multiplier, then multiply | 18 ÷ 3 = 6 → 5 × 6 = 30 |
| A target total | Divide the total by the sum of parts | 48 ÷ 8 = 6 → 18 and 30 |
| Two complete ratios | Cross-multiply and compare | 3 × 30 = 90, 5 × 18 = 90 → equal |
Four inputs, one underlying relationship. The multiplier is either given, derived from a part, or derived from the total.
Where the distinction stops being useful
Two limits worth knowing. First, a true proportion says the ratios match, not that either one is correct for your situation — 3:5 = 12:20 stays true even if the underlying measurement was in the wrong unit. Second, proportional reasoning assumes the relationship really is linear. Doubling every ingredient in a paint mix keeps the colour; doubling every dimension of a container multiplies its volume by eight, not by two. The arithmetic will not warn you when the model does not apply.
Which one is your task?
| What you are doing | You are working with | Tool |
|---|---|---|
| Comparing two measured quantities | A ratio | Ratio workbench |
| Reducing 12:18 to its smallest form | A ratio | Simplify ratio |
| Producing another form of 2:3 at a known scale | A ratio, scaled | Equivalent ratios |
| Filling in the missing value of 2:3 = 8:x | A proportion | Proportion calculator |
| Deciding whether 3:5 and 12:18 agree | A proportion, tested | Proportion calculator |
| Dividing a total in 3:2 | A ratio, applied | Split a total |
If there is an equals sign between two ratios — stated or implied by the question — you are in proportion territory.