A spinner was created by drawing five radii from the centre of a circle. The first four radii divide the circle into four equal wedges. The fifth radius divides one of the wedges into two parts, one having twice the area of the other. The five wedges are labelled as pictured with the wedge labeled by 2 having twice the area of the wedge labeled by 1 . Determine the probability of spinning an odd number.

![](https://cdn.mathpix.com/cropped/2024_04_17_d49dd011068fc0fd28d1g-1.jpg?height=333&width=320&top_left_y=1685&top_left_x=1466)
Solution 1

The odd numbers on the spinner are 1,3 , and 5 .

The wedges labelled by 3 and 5 each take up $\frac{1}{4}$ of the spinner and so each will be spun with a probability of $\frac{1}{4}$.

If we let the probability of spinning 1 be $x$, then we have that the probability of spinning 2 is $2 x$.

The probability of spinning either 1 or 2 is $\frac{1}{4}$, which means $x+2 x=\frac{1}{4}$ or $3 x=\frac{1}{4}$, so $x=\frac{1}{12}$. Therefore, the probability of spinning an odd number is $\frac{1}{4}+\frac{1}{4}+\frac{1}{12}=\frac{7}{12}$.

## Solution 2

The radii defining the sector containing the wedges labelled by 1 and 2 meet at an angle of $90^{\circ}$. Let $y^{\circ}$ be the angle made by the radii defining the wedge labelled by 1 . Then $y+2 y=90$, so $y=30$.

The angles made by the radii defining the wedges labelled by 3 and 5 are both $90^{\circ}$. Since there are $360^{\circ}$ in a circle, this means the probability of spinning an odd number is

$$
\frac{30^{\circ}+90^{\circ}+90^{\circ}}{360^{\circ}}=\frac{210^{\circ}}{360^{\circ}}=\frac{7}{12}
$$

ANSWER: $\frac{7}{12}$