Inverse proportion

Everything so far has been direct proportion: one quantity goes up, and the other goes up by the same factor. 5 workers move 4500 bricks a day, so for 18000 bricks, 4500 : 18000 :: 5 : x gives x = 20 workers. More bricks, more workers.

But Part 1 left a problem unfinished. Puneeth's father rides Lucknow to Kanpur in 3 hours at 30 km/h. At 60 km/h, how long? Writing 30 : 60 :: 3 : x would predict a longer journey for going faster: plainly wrong.

Look at a table for the whole route:

As the speed rises, the time falls.

By how much? A bicycle is 3 times faster than walking (15 ÷ 5 = 3), and the time drops 3 times (18 ÷ 6 = 3). The same factor, but in the opposite direction.

So the two quantities are proportional; they just move opposite ways. Such relationships are called inverse proportions. (So the first question to ask in any problem: when one goes up, does the other go up or down?)

Here's the clean way to spot one. Multiply speed by time in every row: 5 × 18 = 90, 15 × 6 = 90, 30 × 3 = 90, 60 × 1.5 = 90. The product is constant, and of course it is: it's the distance from Lucknow to Kanpur, which doesn't change.

Definition: x and y vary in inverse proportion if xy = k for some constant k. So x₁y₁ = x₂y₂ = k, and rearranging gives x₁/x₂ = y₂/y₁ (note the swap).

That one equation solves a whole family of problems:

Halve the workers and the days double; double the pumps and the hours halve. (5 workers taking 10 days means 1 worker would take 50.)