GSEB Class 8 Maths Solutions Chapter 5 Data Handling Exercise 5.2

 

Question 1. A survey was made to find the type of music that a certain group of young people liked in a city. Adjoining pie chart shows the findings of this survey?
1. If 20 people liked classical music, how many young people were surveyed?
2. Which type of music is liked by the maximum number of people?
3. If a cassette company wants to make 1000 CD's how many of each type would they make?
Answer:
1. Let the total number of young people surveyed be \( x \).
Since 10% of these people liked classical music, and we know 20 people liked classical music:
\( 10\% \text{ of } x = 20 \)
\( \frac{10}{100} \times x = 20 \)
\( x = \frac{20 \times 100}{10} \)
\( x = 200 \)
So, 200 young people were surveyed in total.
2. From the pie chart, light music has the largest percentage at 40%. This means the maximum number of people enjoy light music.
3. Total number of CDs to make is 1000.
For semi-classical music: 20% of 1000 \( = \frac{20}{100} \times 1000 = 200 \) CDs
For classical music: 10% of 1000 \( = \frac{10}{100} \times 1000 = 100 \) CDs
For folk music: 30% of 1000 \( = \frac{30}{100} \times 1000 = 300 \) CDs
For light music: 40% of 1000 \( = \frac{40}{100} \times 1000 = 400 \) CDs
In simple words: First, we figured out the total number of people surveyed by using the classical music data. Then, we identified the most popular music type from the percentages. Finally, we calculated how many CDs of each music type to make out of 1000, based on their popularity percentages.

Exam Tip: When working with percentages in pie charts, remember that the entire circle represents 100% or the total quantity. Use this relationship to find unknown totals or individual parts.

 

Question 2. A group of 360 people were asked to vote for their favourite season from the three seasons rainy, winter and summer?
1. Which season got the most votes?
2. Find the central angle of each sector?
3. Draw a pie chart to show this information?

Answer:
1. The winter season received the most votes, with 150 votes.
2. The total number of votes is \( 90 + 120 + 150 = 360 \).
The central angle for each sector is calculated as:
Summer season: \( \frac{90}{360} \times 360^\circ = 90^\circ \)
Rainy season: \( \frac{120}{360} \times 360^\circ = 120^\circ \)
Winter season: \( \frac{150}{360} \times 360^\circ = 150^\circ \)
3. The central angles calculated above are used to draw a pie chart. Each angle represents the proportion of votes for that particular season in the circle. The pie chart would visually display these proportions.
In simple words: First, we saw which season got the most votes from the table. Then, we worked out the central angle for each season by comparing its votes to the total votes. These angles tell you how big each slice of the pie chart should be.

Exam Tip: To find the central angle for a sector in a pie chart, always divide the value of the component by the total value and then multiply by 360°.

 

Question 3. The table shows the colours preferred by a group of people?

Answer:
The total number of people is 36.
The central angle for each colour is calculated as:
(a) For blue colour: \( \frac{18}{36} \times 360^\circ = 18 \times 10^\circ = 180^\circ \)
(b) For green colour: \( \frac{9}{36} \times 360^\circ = 9 \times 10^\circ = 90^\circ \)
(c) For red colour: \( \frac{6}{36} \times 360^\circ = 6 \times 10^\circ = 60^\circ \)
(d) For yellow colour: \( \frac{3}{36} \times 360^\circ = 3 \times 10^\circ = 30^\circ \)
The sum of these angles is \( 180^\circ + 90^\circ + 60^\circ + 30^\circ = 360^\circ \), which confirms the calculations are correct for a full circle. These angles are used to represent the data in a pie chart.
In simple words: To show color preferences in a pie chart, we first found the total number of people. Then, for each color, we worked out its angle by dividing the number of people who liked it by the total people and multiplying by 360 degrees. These angles determine the size of each slice in the chart.

Exam Tip: Always double-check that the sum of all central angles equals 360° to ensure accuracy in your calculations for a pie chart.

 

Question 4. The adjoining pie chart gives the marks scored in an examination by a student in Hindi, English, Mathematics, Social Science and Science. If the total marks obtained by the student were 540, answer the following questions?
1. In which subject did the student score 105 marks?
2. How many more marks were obtained by the student in Mathematics than in Hindi?
3. Examine whether the sum of the marks obtained in Social Science and Mathematics is more than that in Science and Hindi. Hint: Just study the central angles?
Answer:
Total marks obtained by the student = 540.
Total central angle for 540 marks = \( 360^\circ \).
1. To find the subject where the student scored 105 marks, we first calculate the central angle corresponding to 105 marks.
Central angle for 105 marks \( = \frac{360^\circ}{540} \times 105 = 70^\circ \)
From the pie chart, the sector with a central angle of \( 70^\circ \) corresponds to Hindi. Therefore, the student scored 105 marks in Hindi.
2. We need to find the marks obtained in Mathematics and Hindi.
From the pie chart:
Central angle for Mathematics = \( 90^\circ \)
Central angle for Hindi = \( 70^\circ \)
Marks obtained in Mathematics \( = \frac{90^\circ}{360^\circ} \times 540 = 135 \)
Marks obtained in Hindi \( = \frac{70^\circ}{360^\circ} \times 540 = 105 \)
Difference in marks \( = 135 - 105 = 30 \).
The student obtained 30 more marks in Mathematics than in Hindi.
3. We need to compare the sum of marks for Social Science and Mathematics with the sum of marks for Science and Hindi.
Using central angles, as marks are proportional to central angles:
Sum of central angles for Social Science and Mathematics \( = 65^\circ + 90^\circ = 155^\circ \)
Sum of central angles for Science and Hindi \( = 80^\circ + 70^\circ = 150^\circ \)
Since \( 155^\circ > 150^\circ \), the sum of marks obtained in Social Science and Mathematics is greater than the sum of marks obtained in Science and Hindi.
In simple words: We first found which subject matches 105 marks by converting marks to an angle. Then, we calculated the extra marks in Math compared to Hindi. Finally, we checked if Social Science and Math combined had more marks than Science and Hindi combined, just by looking at their angles.

Exam Tip: Remember that in a pie chart, the proportion of marks for a subject is directly proportional to its central angle. This allows for direct comparison of total marks using only the angles.

 

Question 5. The number of students in a hostel, speaking different languages is given below. Display the data in a pie chart?

Answer:
Total number of students = 72.
We need to calculate the central angle for each language to represent it in a pie chart.
(a) For Hindi language: \( \frac{40}{72} \times 360^\circ = 40 \times 5^\circ = 200^\circ \)
(b) For English language: \( \frac{12}{72} \times 360^\circ = 12 \times 5^\circ = 60^\circ \)
(c) For Marathi language: \( \frac{9}{72} \times 360^\circ = 9 \times 5^\circ = 45^\circ \)
(d) For Tamil language: \( \frac{7}{72} \times 360^\circ = 7 \times 5^\circ = 35^\circ \)
(e) For Bengali language: \( \frac{4}{72} \times 360^\circ = 4 \times 5^\circ = 20^\circ \)
The sum of these central angles is \( 200^\circ + 60^\circ + 45^\circ + 35^\circ + 20^\circ = 360^\circ \). These calculated angles are used to draw the pie chart, visually displaying the proportion of students speaking each language.
In simple words: To make a pie chart of language speakers, we first sum up all the students. Then, for each language, we divide the number of students speaking it by the total, and multiply by 360 degrees. These calculated angles then help us draw the correct size for each language slice in the pie chart.

Exam Tip: When presenting data in a pie chart, always ensure that each category's central angle is correctly calculated to reflect its proportion of the total. A good practice is to label each sector clearly.