In inverse proportion doubling one quantity halves the other
More workers, fewer days
A team is building a wall. 2 workers take 12 days to build it. If 4 workers share the same job, working at the same pace, they finish in only 6 days. Twice as many workers take half as many days.
The product stays the same
Count the work in worker-days: one worker working for one day is one worker-day. 2 workers for 12 days is 2 × 12 = 24 worker-days, and 4 workers for 6 days is 4 × 6 = 24 worker-days. The wall needs 24 worker-days of work however the team is made up.
So the number of days is always 24 ÷ the number of workers. 6 workers take 24 ÷ 6 = 4 days, 3 workers take 24 ÷ 3 = 8 days, and 1 worker alone takes 24 days. Multiply the workers by any number and the days are divided by that same number.
When two quantities change like this, so that their product stays the same, they are in inverse proportion. Here the number of days is inversely proportional to the number of workers.
2 workers for 12 days: a row for each worker and a square for each day, 2 × 12 = 24 worker-days. The gold squares are their first 6 days.
4 workers for 6 days: 4 × 6 = 24 worker-days. The gold rows are the same first 6 days of the first 2 workers, and the 2 new workers do the other 12 worker-days in those same 6 days.
Days needed by 1, 2, 3, 4 and 6 workers: 24, 12, 8, 6 and 4. Each number of days times its number of workers is 24.
Direct and inverse, side by side
In direct proportion, doubling one quantity doubles the other, and the quotient stays the same. In inverse proportion, doubling one quantity halves the other, and the product stays the same.
In the drawing below, the left graph is y = 2x, where y is always 2 times x. The right graph is , where x × y is always 8, drawn as a rectangle x wide and y tall. Start at x = 2 and double it.
Double x from 2 to 4 and compare what y does on each side
Drag x from 2 to 4. On the left, y doubles from 4 to 8. On the right, y halves from 4 to 2, and the rectangle keeps its area of 8.
The usual mistakes
Thinking that "more of one, less of the other" is enough. If 2 workers took 12 days and 4 workers took 10 days, there would be more workers and fewer days, but 2 × 12 = 24 and 4 × 10 = 40. The products are not equal, so that is not inverse proportion. Check that the product stays the same.
Treating it as direct proportion. More workers do not take more days: 4 workers do not take 2 × 12 = 24 days. They share the same 24 worker-days, so they take 24 ÷ 4 = 6 days.
Adding or taking away a day instead of dividing. 3 workers take 24 ÷ 3 = 8 days. The answer comes from the 24 worker-days, not from the difference between 2 and 3 workers.
Food that has to last
The same idea works for anything shared at a steady rate. A store of food feeds a number of people for a number of days, and the product, counted in person-days, is the size of the store. In the next problem, part of the store is eaten before some of the people leave, so first find what is left.
Rate and work · Food Stored for a Camp, and Some Campers Leave Early
QuestionA youth camp stored enough food to feed 40 campers for 24 days, and each camper eats the same amount each day. After 6 days, 10 campers had to leave the camp because of bad weather. (a) For how many more days will the food that is left feed the campers who stay? (b) How many days longer did the food last in all than was first planned?
1.After 6 days, the food left would have fed the 40 campers for 24 − 6 = 18 more days.
After 6 days, the food left would feed 40 campers for 18 more days. 2.That is 40 × 18 = 720 rations, one ration for each camper for each day.
That is 40 × 18 = 720 rations, one for each camper for each day. 3.After 10 campers leave, 40 − 10 = 30 campers stay, and they eat 30 rations a day.
Now 30 campers share the 720 rations, 30 rations a day. 4.(a) The 720 rations last them 720 ÷ 30 = 24 more days.
(a) 720 ÷ 30 = 24 more days. 5.(b) The plan had 18 days of food left, and the food now lasts 24 more days, so it lasts 24 − 18 = 6 days longer. Check: 6 + 24 = 30 days in all, against the 24 days planned.
(b) The plan had 18 more days; 24 is 24 − 18 = 6 days longer.
Answer: (a) 24 more days; (b) 6 days longer
Watch out for
- Sharing all 960 rations among the 30 campers: 960 ÷ 30 = 32 days. The 40 campers had already eaten 240 rations in the first 6 days, so only 720 rations are left to share.
- Giving 24 days as the answer to (b). The 24 days are the days the food lasts after day 6, and the plan had 18 days left at that point, so the food lasts only 24 − 18 = 6 days longer.
Speed and time over a fixed distance
Over the same distance, speed and time are in inverse proportion: speed × time = distance, which does not change. Driving at a faster speed takes less time. In the next problem the two speeds are in the ratio 3 : 4, so the two times are in the ratio 4 : 3.
Speed · The Same Road Driven There and Back at Two Speeds
QuestionMr. Lee drove from Town P to Town Q at an average speed of 60 km/h. He drove back along the same road at 80 km/h, and the trip back took 30 minutes less than the trip there. (a) How long did the trip from Town P to Town Q take? (b) What is the distance between Town P and Town Q?
1.The road is the same both ways, so speed × time is the same both ways. The speeds are in the ratio 60 : 80 = 3 : 4, so the times are the other way around, 4 : 3.
Same road both ways. Speeds 60 : 80 = 3 : 4, so the times are the other way around, 4 : 3. 2.Draw the trip there as 4 units of time and the trip back as 3 units. The trip back takes 1 unit less.
The trip there is 4 units of time and the trip back 3 units: 1 unit less. 3.The trip back took 30 minutes less, so 1 unit is 30 minutes.
The trip back took 30 minutes less, so 1 unit is 30 minutes. 4.(a) The trip there took 4 × 30 = 120 minutes, which is 2 hours.
(a) The trip there: 4 × 30 = 120 minutes, which is 2 hours. 5.(b) The distance is 60 × 2 = 120 km. Check: the trip back took 3 × 30 = 90 minutes, which is 112 hours, and 80 × 112 = 120 km.
(b) 60 × 2 = 120 km. The trip back agrees: 112 hours at 80 km/h is 120 km.
Answer: (a) 2 hours; (b) 120 km
Watch out for
- Giving the times the same ratio as the speeds, 3 : 4. The trip back is faster, so it must take less time, not more: the times are the other way around, 4 : 3.
- Using 30 as the unit in hours: 4 × 30 = 120 hours, and 60 × 120 = 7200 km. The 30 is minutes, so 4 units are 120 minutes, which is 2 hours.
Your turn
1.4 workers take 3 days. How long do 3 workers take?