In the diagram, $O$ is the centre of the circle, $A O B$ is a diameter, and the circle graph illustrates the favourite season of 600 students. How many of the students surveyed chose Fall as their favourite season?
(A) 100
(B) 50
(C) 360
(D) 150
(E) 75

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-073.jpg?height=349&width=417&top_left_y=880&top_left_x=1296)
Solution 1

Since $A O B$ is a diameter of the circle, then $\angle A O B=180^{\circ}$.

We are told that the angle in the "Winter" sector is a right angle (or $90^{\circ}$ ). Also, we are told that the angle in the "Spring" sector is $60^{\circ}$.

Therefore, the angle in the "Fall" sector is $180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}$.

What fraction of the complete circle is $30^{\circ}$ ?

Since the whole circle has $360^{\circ}$, then the fraction is $\frac{30^{\circ}}{360^{\circ}}=\frac{1}{12}$.

Therefore, $\frac{1}{12}$ of the students chose fall as their favourite season, or $\frac{1}{12} \times 600=50$ students in total.

## Solution 2

Since $A O B$ is a diameter of the circle, then $\frac{1}{2}$ of the students chose summer as their favourite season, or $\frac{1}{2} \times 600=300$ students in total.

Since the angle in the "Winter" sector is a right angle (or $90^{\circ}$ ), then $\frac{1}{4}$ of the students (since 4 right angles make up a complete circle) chose Winter as their favourite season, or $\frac{1}{4} \times 600=150$ students in total.

Since the angle in the "Spring" sector is $60^{\circ}$, then $\frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}$ of the students chose Spring as their favourite season, or $\frac{1}{6} \times 600=100$ students in total.

Since there were 600 students in total, then the number who chose Fall as their favourite season was $600-300-150-100=50$.

ANswer: (B)