10. In a survey, people were asked to name their favourite fruit pie. The pie chart shows the outcome. The angles shown are exact with no rounding.
What is the smallest number of people who could have been surveyed?
A 45
B 60
C 80
D 90
E 180
SolUTION
D

Suppose that $p$ people were surveyed.
The total of the angles is $360^{\circ}$. Therefore the proportion of the people who said that their favourite is apple pie is $\frac{140}{360}=\frac{7}{18}$. Hence the number who chose apple pie was $\frac{7}{18} p$. This is an integer. Therefore $p$ is a multiple of 18 .
Similarly, as the proportion who said their favourite is cherry pie is $\frac{108}{360}=\frac{3}{10}$, we deduce that $p$ is a multiple of 10 .
Likewise, as $\frac{72}{360}=\frac{1}{5}$ and $\frac{40}{360}=\frac{1}{9}$, we know that $p$ is also a multiple of 5 and of 9 .
Therefore the smallest possible value of $p$ is the least common multiple of $18,10,5$ and 9 , which is 90 . Hence the smallest number of people who could have been surveyed is 90 .