Two quantities are in direct proportion when their ratio stays fixed: double one and the other doubles. Scaling a recipe, converting currency at a fixed rate and working out material for a larger job are all the same operation. The constant of proportionality is the ratio itself, and once you have it, every other value follows by multiplication.
The habit worth forming is the unitary method: find the value for one unit first, then multiply up. If 4 kilograms cost 6.40, then one kilogram costs 1.60, and 7 kilograms cost 11.20. Working through the unit value keeps the arithmetic simple and makes errors obvious. This page also reports the inverse-proportion answer, which applies when the quantities multiply to a constant instead — more workers on the same job means less time each, not more.
The formula
x₁, y₁- The known pair
x₂- The new value of the first quantity
y₂- The proportional result being solved for
k- The constant of proportionality, y₁ ÷ x₁
How it works, step by step
- Enter the known pair of values.
- Enter the new value of the first quantity.
- The gauge shows the proportional result.
- The unit value below is the amount per single unit, which is worth remembering.
Worked examples
Scaling 4 kg at 6.40 up to 7 kg
The unit value is 6.40 ÷ 4 = 1.60 per kilogram, so 7 kilograms cost 11.20. The scale factor is 1.75, and applying it directly to 6.40 gives the same 11.20.
Where inverse proportion applies instead
If 4 workers take 6.4 hours, the job is 25.6 worker-hours. With 7 workers it takes 3.657 hours — less, not more. Direct proportion would wrongly predict 11.2 hours, so identifying which kind you have matters more than the arithmetic.
How to read your score
Frequently asked questions
How do I know if two quantities are in direct proportion?
Their ratio must be constant, and doubling one must double the other. A useful test is whether zero of one means zero of the other; if not, the relationship is linear but not proportional.
What is the unitary method?
Find the value for a single unit, then multiply by however many you need. It keeps the numbers small and makes mistakes easy to spot, which is why it is taught before formal ratios.
When is a relationship inversely proportional?
When the product rather than the ratio stays constant. Speed and journey time for a fixed distance, or workers and time for a fixed job. More of one means less of the other.
Can proportions handle negative values?
The arithmetic works, but the interpretation usually does not. Negative quantities normally signal that the relationship is linear with an offset rather than truly proportional.
Scaling from 4 units at 6.40
| New quantity | Direct result | Scale factor | Inverse result |
|---|---|---|---|
| 1 | 1.6000 | 0.250 | 25.6000 |
| 2 | 3.2000 | 0.500 | 12.8000 |
| 4 | 6.4000 | 1.000 | 6.4000 |
| 7 | 11.2000 | 1.750 | 3.6571 |
| 10 | 16.0000 | 2.500 | 2.5600 |
| 16 | 25.6000 | 4.000 | 1.6000 |
| 25 | 40.0000 | 6.250 | 1.0240 |
The direct column rises with the quantity while the inverse column falls; they agree only at the known pair.