You drink half a cup of water. Then I drink half of what is left.

Then you drink half of what is left after that, and I drink half of what is left after you drink.

This continues until the cup of water is all gone.

What proportion of the cup of water did you drink by the end?

#### Click to see solution

2 / 3

**Intuitive solution**

Look at the first round. You drank half the cup, and I drank half of what is left, so you drank twice as much as me. This holds true in every single round.

So intuitively, by the end, you should have drank twice as much as me in total. That means you drank 2 / 3 of the cup and I drank 1 / 3.

**If you know how to calculate the sum of an infinite geometric series**

This problem is a great place to use the formula for the sum of an infinite geometric series: a / (1 – r), where a is the initial value and r is the ratio between successive values.

In this case, the initial value is 1 / 2 of the cup, and the ratio is 1 / 4. The ratio is 1 / 4 because in any subsequent round, you drink half as much as I did, which is 1 / 4 as much as you did in the previous round.

(1 / 2) / ( 1 – 1 / 4) = (1 / 2) / (3 / 4) = 2 / 3