Ratio & proportion tools
How to Scale a Ratio Up or Down
Simplifying, finding an equivalent ratio, sizing a batch and working back from one known quantity are not four techniques. They are one relationship, and four ways of discovering the same factor.
The single model: a common factor k
Write the ratio as a:b. Every ratio that expresses the same relationship has the form ka:kb for some non-zero number k. Scaling is choosing k; simplifying is choosing a k below 1 (equivalently, dividing).
Why must both terms take the same factor? Because a:b is a claim about the quotient a/b, and only a shared factor cancels: ka / kb = (k/k) × (a/b) = a/b. The k disappears, so the comparison is untouched while the quantities change size. Give the terms different factors — say 2:3 becomes 4:9 — and the quotient moves from 0.666… to 0.444…, which is a different relationship wearing the same shape.
The same cancellation argument is why adding a constant is not scaling. 2:3 plus 2 to each part is 4:5, and 4/5 = 0.8 ≠ 0.666…. Addition does not cancel.
Three ways the scale factor arrives
The arithmetic after k is known is always identical: multiply every part by k. What differs between problems is only how k is determined. These are not three rules.
A — k is chosen
Scale 2:3 by 4.
k = 4
2 × 4 = 8 and 3 × 4 = 12, so the scaled ratio is 8:12.
Check: 8/12 = 2/3. Same relationship.
B — k is derived from a known component
Original ratio 2:3. The first component must become 10.
2k = 10, so k = 10 ÷ 2 = 5.
Second component: 3 × 5 = 15.
Result: 10:15, and 10/15 = 2/3.
C — k is derived from a target total
Original ratio 2:3, target total 20.
Total ratio parts: 2 + 3 = 5.
k = 20 ÷ 5 = 4 — the value of one part.
Result: 8:12, and 8 + 12 = 20.
Case C is exactly what a split does: k is the value of one part. Splitting a total and scaling to a total are the same computation with different wording.
Scaling versus simplifying
12:18 → 2:3 divides both parts by 6. 2:3 → 8:12 multiplies both parts by 4. One uses a divisor, the other a multiplier, and both leave the quotient at 0.666…. In the k model, the first is k = 1/6 and the second is k = 4 — the direction of travel is the only difference.
Simplest form is the point at which no whole-number k below 1 remains available: the GCD argument explains how far that goes. To move in the other direction with a concrete target, use the equivalent ratio calculator.
Scaling that does not land on whole numbers
Nothing in the model requires integers. The parts may be decimals, k may be a fraction, or both, and the relationship survives regardless.
1.5 : 2 scaled to a total of 35
- Total ratio parts: 1.5 + 2 = 3.5.
- k = 35 ÷ 3.5 = 10.
- First part: 1.5 × 10 = 15. Second part: 2 × 10 = 20.
- Result 15:20, which reduces to 3:4, and 15 + 20 = 35.
Here the parts were non-integer but k was not. The reverse happens just as often.
3:5 scaled to a total of 60 — non-integer k
- Total ratio parts: 3 + 5 = 8.
- k = 60 ÷ 8 = 15/2 = 7.5 exactly.
- 3 × 15/2 = 45/2 = 22.5 and 5 × 15/2 = 75/2 = 37.5.
- Check: 22.5 + 37.5 = 60, and 22.5/37.5 = 3/5.
SnapRatios keeps k as the exact fraction 15/2 rather than the decimal 7.5, so a value such as 60 ÷ 7 stays as 60/7 instead of drifting through 8.571428571. Rounding, when it happens at all, happens once at display time — see the methodology.
A non-integer k is a mathematically ordinary result, not an error. Whether the quantities it produces are usable is a question about the thing being measured, not about the ratio: 22.5 ml is easy, 22.5 bricks is not. Do not silently round a physical quantity to make it look tidy — round deliberately and accept that the ratio then holds only approximately.
More than two parts
The rule does not change with the number of terms. Every part takes the same k, and the total parts are the sum of all of them.
2:3:5 scaled to a total of 400
- Total ratio parts: 2 + 3 + 5 = 10.
- k = 400 ÷ 10 = 40.
- 2 × 40 = 80, 3 × 40 = 120, 5 × 40 = 200.
- Verify: 80 + 120 + 200 = 400, and 80:120:200 reduces back to 2:3:5.
Pairwise quotients are all preserved at once: 80/120 = 2/3, 120/200 = 3/5, 80/200 = 2/5. A dedicated layout for three inputs lives at the three-part ratio calculator.
Reverse scaling: recovering k from one quantity
Often the scaled version is what you can see, and the factor is what you want. Because every part is k times its ratio term, a single known quantity is enough to recover k, and from k every other quantity follows.
A 4:1 relationship whose first component is 600 ml
- The first term is 4, so the first quantity is 4k.
- 4k = 600, therefore k = 600 ÷ 4 = 150.
- Second component: 1k = 150 ml.
- Cross-check the total if you know it: 4k + 1k = 5k = 750 ml.
This works from any single component, not only the first: if the smaller term were the known one, 1k = 150 would give the same k and the same 600 ml partner. It also works backwards from a total, which is case C read in reverse.
Which task, which k, which tool
| Task | What you know | What you need | How k is found | Tool |
|---|---|---|---|---|
| Simplify | One ratio | Smallest whole-number form | k = 1 ÷ GCD of all parts | Simplify ratio |
| Scale by a known factor | Ratio and the factor | Scaled quantities | k is given | Equivalent ratio |
| Find an equivalent ratio | Ratio and one new term | The matching term | k = new term ÷ old term | Equivalent ratio |
| Scale to a target total | Ratio and the total | Every quantity | k = total ÷ sum of parts | Split ratio |
| Scale from a known component | Ratio and one quantity | The other quantities | k = quantity ÷ its ratio term | Mixing ratio |
| Split a total | Ratio and the amount | Each share | k = amount ÷ sum of parts | Split ratio |
| Scale a mixture | Ratio and batch or component | Each component | k from total or component | Mixing ratio |
| Scale three or more parts | Multi-part ratio and total | Every share | k = total ÷ sum of all parts | Three-part ratio |
Every row applies the same operation. Only the route to k differs.