#### Hide answer & extension

If there is not a point when you had scored exactly 75%, then there must be a shot which you score which took your average over 75%. In other words, there must be an $a$ and a $b$ so that:

$$\frac{a}{b}<\frac{3}{4}<\frac{a+1}{b+1}$$

Rearranging these inequalities gives:

$$4a<3b$$
$$3b+3<4a+4$$

Taking 3 from the bottom inequality gives:

$$4a<3b$$
$$3b<4a+1$$

Combining the inequalities gives:

$$4a<3b<4a+1$$

Therefore \(3b\) must be an integer which is between \(4a\) and \(4a+1\). But \(4a\) and \(4a+1\) are consecutive numbers so there is no other integer between them.

This means that this situation is impossible and there must have been a time when you had scored exactly 75% of your shots.

#### Extension

You spend another afternoon practising throwing a basketball through a hoop.

One hour into the afternoon, you have scored more than 75% of your shots. At the end of the afternoon, you have score less than 75% of your shots.

Is there a point in the afternoon when you had scored exactly 75% of your shots?