## Save Rate

#### Problem

During a football match in a tournament a goalkeepers save rate was given as 33%. After saving the next shot on target it rose to 40%. How many more shots on target does he need to save to raise his save rate to 50%?

#### Solution

A save rate of 33% means 1 in 3, 2 in 6, 3 in 9, etc. (assuming 33% has been rounded down from 33.333...%).

i.e. ^{1}/_{3} = ^{2}/_{6} = ^{3}/_{9} = ^{4}/_{12} = ...

If the next shot on target was a save, both the numerator and denominator will have increased by 1.

40% = ^{2}/_{5} = ^{4}/_{10} = ^{6}/_{15} = ...

We can see that ^{3 + 1}/_{9 + 1} = ^{4}/_{10}, so the goalkeeper must currently have a save rate of 4 out of 10 shots on target.

If all subsequent shots on target are saved,^{4}/_{10} ^{5}/_{11} ^{6}/_{12} = ^{1}/_{2} = 50%.

So it will necessary to save the next two shots on target.

What happens if one of the next two shots on target are not saved?