Not every relationship is proportional

The idea

The Rule of Three applies only when both quantities change by the same factor; when one quantity decreases as the other increases, the problem cannot be modelled as a proportion.

Puneeth's father rides faster

Puneeth's father rides from Lucknow to Kanpur in 2 hours at 50 km/h. He tries 75 km/h next time and writes 50 : 2 :: 75 : ?

At 75 km/h, will the ride take more time or less time?

• More than 2 hours
• Less than 2 hours
• The same, 2 hours

Double the first, watch the second

In each situation, press Double the first quantity. Does the second quantity double too?

Same factor, or no Rule of Three

The Rule of Three applies only when both quantities change by the same factor; when one quantity decreases as the other increases, the problem cannot be modelled as a proportion.

The ride is 50 × 2 = 100 km. At 75 km/h it takes 100 ÷ 75 hours, about 1 hour 20 minutes: less time.

The proportion 50 : 2 :: 75 : ? would give 3 hours. That is wrong, because speed and time do not change by the same factor.

Notes

The Rule of Three applies only when both quantities change by the same factor; when one quantity decreases as the other increases, the problem cannot be modelled as a proportion.

Check yourself

For which pair of quantities can the Rule of Three be used?

A trip takes 6 hours at 40 km/h. How many hours will it take at 80 km/h?

Answer: 3 hours

The trip is 40 × 6 = 240 km. At 80 km/h it takes 240 ÷ 80 = 3 hours: the speed doubled, so the time halved.

Ravi says: "4 painters paint a wall in 6 days, so 8 painters take 12 days: 4 : 6 :: 8 : 12." Is he right?

12 pens cost ₹90. What do 20 pens cost at the same price each? (in ₹)

Answer: 150

12 : 90 :: 20 : d, so d = 90 × 20 ÷ 12 = 1800 ÷ 12 = ₹150.

• Speed and time taken for the same trip. When the speed doubles, the time halves. That is not the same factor.
• The number of taps and the time to fill one tank. More taps fill the tank in less time. Not the same factor.
• The number of notebooks and their cost at the same price each — correct. Yes! Double the notebooks and the cost also doubles: the same factor.
• The number of workers and the days to finish one job. More workers finish in fewer days. Not the same factor.
• Yes: 8 is twice 4, so 6 days doubles to 12 days. More painters finish sooner, so the days should not double.
• No: more painters take fewer days, so it is 3 days — correct. Yes! 4 × 6 = 24 painter-days of work. 24 ÷ 8 = 3 days. The painters doubled and the days halved.
• No: 8 painters also take 6 days. More painters do the work faster, so the days go down.