Working together

Here's a problem that catches almost everyone out. Ram cuts a pile of vegetables in 1 hour; Shyam takes 1.5 hours. Working together, how long will they take?

The wrong instinct is to average the times, or to add them. Neither is right. The trick is to stop thinking about times and start thinking about rates: how much of the job each person does in one hour.

Call the whole job 1 unit of work.

• Ram finishes it in 1 hour, so in an hour he does 1 unit.
• Shyam takes 1.5 hours, so in an hour he does 1/1.5 = 2/3 of a unit.

Rates add, even though times don't. Together they do 1 + 2/3 = 5/3 units per hour.

Now turn that back into a time. Work and time are directly proportional (twice the work takes twice as long), so 5/3 units take 1 hour and 1 unit takes x: 5/3 : 1 :: 1 : x, so x = 3/5 of an hour, which is 36 minutes. Sensible: quicker than Ram alone, as it must be.

The chapter ends with the two kinds of proportion side by side:

And for sharing: a ratio a : b : c : … means 'for every a of the first there are b of the second, c of the third…', and dividing x in the ratio p : q : r : … gives parts x × p/(p + q + r + …), and so on.

Deciding which of the two a problem is deserves more thought than the arithmetic that follows.