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.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 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.