Proportional Reasoning - 2 | FIO

Question 6

A group of 360 people were asked to vote for their favourite season from the three seasons — rainy, winter and summer. 90 liked the summer season, 120 liked the rainy season, and the rest liked the winter. Draw a pie chart to show this information.

Solution

We need to find the central angle for each season to draw a pie chart.

Step 1 — Find people who liked winter

First, let us find how many people liked the winter season. We know the total number of people surveyed. We also know how many liked summer and rainy seasons.

Let $N_{\text{total}}$ be the total number of people. Let $N_{\text{summer}}$ be the number of people who liked summer. Let $N_{\text{rainy}}$ be the number of people who liked rainy. Let $N_{\text{winter}}$ be the number of people who liked winter.

$N_{\text{total}} = 360$

$N_{\text{summer}} = 90$

$N_{\text{rainy}} = 120$

We can find $N_{\text{winter}}$ by subtracting the others from the total.

$N_{\text{winter}} = N_{\text{total}} - (N_{\text{summer}} + N_{\text{rainy}})$

$N_{\text{winter}} = 360 - (90 + 120)$

$N_{\text{winter}} = 360 - 210$

$\boxed{N_{\text{winter}} = 150 \text{ people}}$

Step 2 — Calculate central angles

To draw a pie chart, we need to find the central angle for each season. The central angle for a category is its proportion of the total, multiplied by 360 degrees.

Let $\theta_{\text{season}}$ be the central angle for a season.

For the summer season:

$\theta_{\text{summer}} = \left(\frac{N_{\text{summer}}}{N_{\text{total}}}\right) \times 360^\circ$

$\theta_{\text{summer}} = \left(\frac{90}{360}\right) \times 360^\circ$

$\boxed{\theta_{\text{summer}} = 90^\circ}$

For the rainy season:

$\theta_{\text{rainy}} = \left(\frac{N_{\text{rainy}}}{N_{\text{total}}}\right) \times 360^\circ$

$\theta_{\text{rainy}} = \left(\frac{120}{360}\right) \times 360^\circ$

$\boxed{\theta_{\text{rainy}} = 120^\circ}$

For the winter season:

$\theta_{\text{winter}} = \left(\frac{N_{\text{winter}}}{N_{\text{total}}}\right) \times 360^\circ$

$\theta_{\text{winter}} = \left(\frac{150}{360}\right) \times 360^\circ$

$\boxed{\theta_{\text{winter}} = 150^\circ}$

We can check our work by adding the angles. $90^\circ + 120^\circ + 150^\circ = 360^\circ$ The sum is $360^\circ$ , which is correct for a full circle.

Diagram 1

Answer

(i) The central angle for the summer season is $\mathbf{90^\circ}$ . (ii) The central angle for the rainy season is $\mathbf{120^\circ}$ . (iii) The central angle for the winter season is $\mathbf{150^\circ}$ .

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Q10

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Q11

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(vi) The number of pages in a book and the time required to read it at a fixed reading speed.

Q12

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Q13

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Q14

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Q15

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(i) What is the most common mode of transport?

(ii) What fraction of children travel by car?

(iii) If 18 children travel by car, how many children took part in the survey? How many children use taxis to travel to school?

(iv) By which two modes of transport are equal numbers of children travelling?

Q16

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Q17

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Q18

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Q19

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Q20

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Q21

A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?

Q22

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