Ratios and Proportional Relationships
Direct and inverse proportion
Tell direct proportion from inverse proportion, and use each to find a missing quantity.
Two quantities are in direct proportion when doubling one doubles the other: buy twice as many identical pens and you pay twice as much. They are in inverse proportion when doubling one halves the other: put twice as many workers on a job and it takes half the time. Almost every proportion question is one of these two, and getting the right one is most of the work.
One test tells them apart
In direct proportion the two quantities divide to a fixed number: cost ÷ number of pens is always the price of one pen. In inverse proportion they multiply to a fixed number: workers × days is always the total amount of work. So ask yourself: as one goes up, does the other go up with it, or down against it?
Worked example
9 taps fill a tank in 20 minutes. How long would 12 taps take?
Decide which kind it is: more taps means less time, so this is inverse proportion.
Deciding first stops you from reaching for the wrong method. If you cannot tell, try an extreme case — one tap would take ages, so time must fall as taps rise.
Try it together
Now let us work out this one together: 15 machines make 750 parts in a day. How many parts would 22 machines make in a day?
First decide direct or inverse, then find the fixed number, then use it.
1.How many parts does one machine make in a day?
Have a go
Have a go on your own: 10 cows eat a stack of hay in 8 days. How many days would 16 cows take?
Hint: More cows means fewer days, so the two numbers multiply to a fixed total.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.