Direct Proportion as a Formula
Write y = kx for two quantities in direct proportion, find k from one pair of values, and use the formula to predict others.
What a learner can do afterwards
- Find the constant of proportionality from one given pair
- Use the formula in both directions, from x to y and back
- Recognise direct proportion from a straight line through the origin
1 · Read
Two quantities are directly proportional when they change together at a steady rate. When x doubles, y doubles too, and when x triples, y triples too. That steady rate is the number k in the formula y equals k times x. A solution is just a pair that makes it true: with y equal to 3 times x, the pair 2 and 6 works, since 3 times 2 is 6.
One pair pins down the whole rule. Divide y by x to get k, since k never changes. With x equal to 4 and y equal to 20, k is 20 divided by 4, which is 5. With the pair 5 and 40, k is 40 divided by 5, which is 8.
Once you know k, use the formula in either direction. Multiply by x to predict y: with k equal to 5 and x equal to 9, y is 45. Divide to go back: with k equal to 6 and y equal to 54, x is 54 divided by 6, which is 9.
Every direct proportion draws a straight line through the origin. In y equals k times x, putting x equal to 0 always gives y equal to 0. If a line misses the origin, or adds something or divides by x, it is not direct proportion.
Divide to find k, multiply to predict, and check the line runs through zero.
More to read
What is a proportional relationship?
Two quantities are directly proportional when they change together at a steady rate. If y is directly proportional to x, then whenever x doubles, y doubles too. Whenever x triples, y triples too. That steady rate is the number k in the formula y = kx.
k never changes. It's like a scaling factor that connects x and y no matter what values you pick. Once you know k, you can use the formula in either direction. Plug in x to find y, or rearrange it to find x from y.
A solution to y = kx is just a pair of numbers, one for x and one for y, that make the equation true. For example, if y = 3x, then (2, 6) is a solution because 3 times 2 really does equal 6. Every point that fits the formula sits on the same straight line, and that line always passes through (0, 0).
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.