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.
on the left y = 2x follows x, doubling when x doubles; on the right y = 8/x halves when x doubles, and the rectangle x × y keeps its area 8
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.
Worked example: Camp Rations / Resource Depletion with Group Changes
Question A youth camp stored enough food rations to feed 40 campers for 24 days, with each camper consuming an equal daily ration. After 6 days, 10 campers had to leave the camp due to bad weather. (a) For how many additional days will the remaining food rations last the remaining campers? (b) How many days longer did the food rations last in total compared to the original planned schedule?
1.Remaining supply at Day 6: 18 days for 40 campers.
After 6 days, 18 days of food for 40 campers remain. 2.Total remaining units = 18 × 40 = 720 units.
That is 18 × 40 = 720 camper-days. 3.Now 30 campers share these 720 units.
Now 30 campers share them. 4.Days = 720 ÷ 30 = 24 days.
Now 30 campers share them. 5.(a) 24 additional days.
(a) 720 ÷ 30 = 24 more days. 6.Originally, the food was to last 18 more days.
7.Difference = 24 − 18 = 6 days longer.
(b) The plan had 18 more days; 24 is 6 days longer. 8.(b) 6 days longer.
(b) The plan had 18 more days; 24 is 6 days longer.
Answer: (a) 24 more days; (b) 6 days longer
Common mistakes
- Dividing the total 960 camper-days by 30, forgetting that 240 camper-days were already consumed by 40 people.
- Reporting 24 days as the extra duration instead of subtracting the planned 18 remaining days.
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.
Worked example: Constant Distance (Inverse Proportion of Speed and Time)
Question Mr. Lee drove from Town P to Town Q at an average speed of 60 km/h. On his return journey along the exact same route, he increased his speed to 80 km/h. The return journey took 30 minutes less than the forward journey. Find the distance between Town P and Town Q.
1.Draw Forward Time as 4 units.
Same distance both ways. Speeds 60:80 = 3:4, so times are 4:3: the faster trip takes fewer blocks. 2.Draw Return Time as 3 units.
Return time: 3 of the same blocks. 3.Difference = 4 − 3 = 1 unit = 30 minutes = 0.5 h.
The one-block difference is the 30 minutes saved. 4.Forward Time = 4 × 0.5 = 2 hours.
Forward: 4 × 0.5 = 2 hours. 5.Return Time = 3 × 0.5 = 1.5 hours.
Return: 3 × 0.5 = 1.5 hours. 6.Calculate Distance using Forward trip: 60 km/h × 2 h = 120 km.
Distance from the forward trip: 60 × 2 = 120 km. 7.Check using Return trip: 80 km/h × 1.5 h = 120 km.
The return trip agrees: 80 × 1.5 = 120 km.
Answer: 120 km
Common mistakes
- Assuming that time is directly proportional to speed (writing time ratio as 3 : 4 instead of 4 : 3).
- Using 30 minutes directly as 30 hours, obtaining an astronomical distance (60 × 120 = 7200 km).