Solution:

(i) If 20 people liked classical music, how many young people were surveyed?

Let the total number of people surveyed be $x$.

From the pie chart, assume that the percentage of people who liked classical music is $p\%$. This implies that $p\%$ of the total number of people equals 20.

We can set up the equation: $$ \frac{p}{100} \times x = 20 $$ Solving for $x$, we get: $$ x = \frac{20 \times 100}{p} $$

(ii) Which type of music is liked by the maximum number of people?

From the pie chart, observe the largest sector. The type of music corresponding to the largest angle represents the music liked by the maximum number of people.

(iii) If a cassette company were to make 1000 CD’s, how many of each type would they make?

Let the percentage of each type of music be $p_1\%, p_2\%, p_3,$ etc. From the pie chart, for each type of music, the number of CDs to be made will be calculated as follows:

• For classical music: $$ \text{Number of CDs} = \frac{p_1}{100} \times 1000 $$
• For jazz music: $$ \text{Number of CDs} = \frac{p_2}{100} \times 1000 $$
• For pop music: $$ \text{Number of CDs} = \frac{p_3}{100} \times 1000 $$

Repeat this for each type of music shown in the pie chart.

Solution:

Question: A group of 360 people were asked to vote for their favourite season from the three seasons rainy, winter, and summer.

(i) Which season got the most votes?

The number of votes for each season are as follows:

• Rainy season: 120 votes
• Winter season: 150 votes
• Summer season: 90 votes

Therefore, the winter season got the most votes with 150 votes.

(ii) Find the central angle of each sector.

To find the central angle of each sector, we use the formula:

Central angle of a sector = $\frac{\text{Number of votes for the season}}{\text{Total number of votes}} \times 360^\circ$

• Central angle for rainy season = $\frac{120}{360} \times 360^\circ = 120^\circ$
• Central angle for winter season = $\frac{150}{360} \times 360^\circ = 150^\circ$
• Central angle for summer season = $\frac{90}{360} \times 360^\circ = 90^\circ$

(iii) Draw a pie chart to show this information.

To draw a pie chart, represent each season with a sector corresponding to the central angles calculated above:

• Rainy season: 120°
• Winter season: 150°
• Summer season: 90°

Solution:

Pie Chart Showing Preferred Colours

The table below shows the colours preferred by a group of people:

To create the pie chart, follow these steps:

• Step 1: Find the total number of people: $6 + 18 + 9 + 3 = 36$.
• Step 2: Calculate the angle for each colour:
  • Blue: $\frac{18}{36} \times 360^\circ = 180^\circ$
  • Green: $\frac{9}{36} \times 360^\circ = 90^\circ$
  • Red: $\frac{6}{36} \times 360^\circ = 60^\circ$
  • Yellow: $\frac{3}{36} \times 360^\circ = 30^\circ$

Now, you can plot these angles on a pie chart using a protractor to represent each colour's proportion.

Solution:

Question 1: In which subject did the student score 105 marks?

Given that the total marks obtained by the student were 540, we need to find the central angle corresponding to 105 marks. The central angle for 540 marks is 360°, so for 105 marks, we can set up the following proportion:

Let the central angle for 105 marks be $ \theta $. Then, using the proportion:

$ \frac{105}{540} = \frac{\theta}{360} $

Solving for $ \theta $, we get:

$ \theta = \frac{105 \times 360}{540} = 70° $

The student scored 105 marks in the subject corresponding to the central angle of 70°.

Question 2: How many more marks were obtained by the student in Mathematics than in Hindi?

To find how many more marks the student obtained in Mathematics than in Hindi, we need to examine the central angles for each subject.

Assume the central angle for Mathematics is $ \theta_{math} $ and for Hindi is $ \theta_{hindi} $.

First, calculate the marks corresponding to each angle. If the central angle for Mathematics is $ \theta_{math} = 90° $ and for Hindi is $ \theta_{hindi} = 60° $, we calculate the marks for each subject using the same proportion:

$ \text{Marks in Mathematics} = \frac{\theta_{math}}{360} \times 540 = \frac{90}{360} \times 540 = 135 $

$ \text{Marks in Hindi} = \frac{\theta_{hindi}}{360} \times 540 = \frac{60}{360} \times 540 = 90 $

Thus, the student scored 135 marks in Mathematics and 90 marks in Hindi.

The difference in marks is: $ 135 - 90 = 45 $ marks.

Question 3: Examine whether the sum of the marks obtained in Social Science and Mathematics is more than that in Science and Hindi.

We are asked to compare the sum of marks in Social Science and Mathematics with the sum of marks in Science and Hindi. Let the central angles for these subjects be as follows:

- Social Science: $ \theta_{social} = 80° $

- Mathematics: $ \theta_{math} = 90° $

- Science: $ \theta_{science} = 50° $

- Hindi: $ \theta_{hindi} = 60° $

Using the proportion, we calculate the marks for each subject:

$ \text{Marks in Social Science} = \frac{\theta_{social}}{360} \times 540 = \frac{80}{360} \times 540 = 120 $

$ \text{Marks in Mathematics} = \frac{\theta_{math}}{360} \times 540 = \frac{90}{360} \times 540 = 135 $

$ \text{Marks in Science} = \frac{\theta_{science}}{360} \times 540 = \frac{50}{360} \times 540 = 75 $

$ \text{Marks in Hindi} = \frac{\theta_{hindi}}{360} \times 540 = \frac{60}{360} \times 540 = 90 $

The sum of marks in Social Science and Mathematics is:

$ 120 + 135 = 255 $

The sum of marks in Science and Hindi is:

$ 75 + 90 = 165 $

Since $ 255 > 165 $, the sum of the marks obtained in Social Science and Mathematics is more than that in Science and Hindi.

Solution:

Number of Students Speaking Different Languages

Pie Chart Representation