Inverse Proportion
Use y = k/x when one quantity falls as the other rises, find k from a given pair, and recognise the curve it produces.
What a learner can do afterwards
- Tell direct from inverse proportion in a word problem
- Find k and use it, for example with workers and the time a job takes
- Describe the shape of an inverse proportion graph
1 · Read
Direct proportion grows together, while inverse proportion trades off. More taps fill the tank faster, and more workers finish the job sooner. When one quantity doubles while the other halves, the product stays fixed, and that fixed product is the signature of inverse proportion.
Four taps fill a tank in 6 hours. Double the taps to 8 and the time halves to 3 hours. The job itself never changes: 5 workers for 8 days is 40 worker-days, and 4 workers need 40 divided by 4, which is 10 days. Find the fixed total first, then divide.
The constant k pins the rule down through y equals k over x. With x equal to 3 and y equal to 10, multiply to find k equal to 30. Every other pair must multiply to 30 as well, so y is 6 when x is 5. One pair buys the whole rule.
Graphs tell the two apart at a glance. Direct proportion draws a straight line through the origin, while inverse proportion draws a curve that falls steeply near zero then flattens, never touching either axis. In words, splitting money by ratio grows together, while sharing a fixed job trades off.
Spot the trade-off, find the fixed product, then divide.
More to read
8.9 Use Direct and Inverse Variation
Direct proportion grows together, inverse proportion trades off. More taps fill the tank faster; more workers finish the job sooner. When one quantity doubles while the other halves, the product stays fixed, and that fixed product is the signature of inverse proportion.
The worker-day total never changes for one job. Five workers for 8 days is 40 worker-days, and 4 workers need 40/4 = 10 days. Finding the constant product first turns every inverse question into one division.
8.9: Use Direct and Inverse Variation
The constant k pins an inverse relationship down. With y = k/x and the pair x = 3, y = 10, multiply to find k = 30. Every other pair must multiply to 30 as well, so y = 6 when x = 5. One pair buys the whole rule.
Tables test inverse proportion through products, not ratios. If every row multiplies to the same k, the table is inverse; a single rogue product breaks it. Checking 1x6 = 6 against 2x3 = 6 passes, while 2x12 = 24 fails.
5.8 Modeling Using Variation
Graphs tell the two proportions apart at a glance. Direct proportion draws a straight line through the origin, while inverse proportion draws a curve that falls steeply near zero then flattens along the axis. The curve never touches either axis, because dividing by zero is impossible.
Telling direct from inverse in words comes first. Splitting money by a ratio grows together (direct), while sharing a fixed job across workers trades off (inverse). Name the behaviour, find k or the unit rate, then calculate.
2 · Watch
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