Inverse proportion: when one goes up, the other comes down

The rule of three, again

In direct proportion, a : b :: c : d and the fourth value is d = b × c ÷ a. If 5 workers move 4500 bricks in a day, then 18000 bricks need 18000 × 5 ÷ 4500 = 20 workers: four times the bricks, four times the workers. Both quantities grow together.

A different pattern

Puneeth’s father rides from Lucknow to Kanpur on a motorcycle at 30 km/h and takes 3 hours. How long would a car at 60 km/h take? Writing 30 : 60 :: 3 : x would say the faster car takes longer, which cannot be right. Look at the whole table:

	Walk	Bicycle	Motorcycle	Car
Speed (km/h)	5	15	30	60
Time (hours)	18	6	3	1.5

The bicycle is 3 times as fast as walking, and the trip takes a third of the time: 18 ÷ 3 = 6. The car is twice as fast as the motorcycle, and takes half the time. The two quantities change by the same factor, but in opposite directions. Such quantities are in inverse proportion.

The product stays the same

Multiply speed by time for each column: 5 × 18, 15 × 6, 30 × 3 and 60 × 1.5 are all 90. That 90 km is the distance from Lucknow to Kanpur, and it does not change. In general, x and y are in inverse proportion when x × y = k for a constant k. So for any two pairs of values, x₁ × y₁ = x₂ × y₂, which can also be written x₁ ÷ x₂ = y₂ ÷ y₁. For walking and the car: 5 ÷ 60 = 0.0833…, and 1.5 ÷ 18 = 0.0833… too.

Put another way: multiply one quantity by n and the other is multiplied by 1/n.

Testing a table

To test a table, multiply each pair. In x = 40, 80, 25, 16 with y = 20, 10, 32, 50, every product is 800, so x and y are in inverse proportion. Change the y values to 20, 10, 12.5 and 8 and the products are 800, 800, 312.5 and 128, so that table is not; there x ÷ y is always 2, which makes it direct proportion. To fill a table, find k from a complete pair: with x = 16 and y = 9, k = 144, so x = 12 gives y = 12, x = 36 gives y = 4, and y = 48 needs x = 3.

Solving problems

The same idea, x₁ × y₁ = x₂ × y₂, solves many problems:

• Road laying: 20 workers take 4 days, so 10 workers take 20 × 4 ÷ 10 = 8 days.
• Filling a tank: 2 pumps take 18 hours, so 4 pumps take 2 × 18 ÷ 4 = 9 hours.
• Food stocks: food for 80 students for 15 days lasts 100 students 80 × 15 ÷ 100 = 12 days.
• Rows of chairs: 25 rows of 12 is 300 chairs, so rows of 20 make 15 rows.

Each answer quietly assumes that everyone works, pumps or eats at the same steady rate. If a new worker is slower, or one family uses more water, the simple answer is only an estimate.

Direct or inverse?

Before solving, ask which way the quantities move. More taps or more painters mean less time: inverse. More petrol means more distance, more cloth means a bigger bill, more pages mean more reading time: direct. Speed and time on a fixed route: inverse.

Keep going with pie charts and ratios with more than two terms, or see the chapter notes and the NCERT solutions.