Proportional Reasoning–2 Class 8 Maths Ganita Prakash Part 2 Chapter 3 NCERT Solutions

Proportional Reasoning–2 Class 8 Maths Ganita Prakash Part 2 Chapter 3 NCERT Solutions Looking for the Proportional Reasoning–2 Class 8 Maths Ganita Prakash Part 2 Chapter 3 NCERT Solutions? At schoollearners.in/, we provide easy-to-understand, step-by-step NCERT solutions prepared according to the latest CBSE syllabus. These solutions help students understand every concept clearly, improve problem-solving skills, and score better in exams.

Figure it Out (Page 60)

1. A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?
Solution:
Warm-up/cool-down : batting : bowling : fielding = 3 : 4 : 3 : 5.
Total time of session = 150 minutes
Sum of ratios = 3 + 4 + 3 + 5 = 15
Warm-up/cool-down time = 150 × $\frac{3}{15}$ = 30 minutes
Batting time = 150 × $\frac{4}{15}$ = 40 minutes
Bowling time = 150 × $\frac{3}{15}$ = 30 minutes
Fielding time = 150 × $\frac{5}{15}$ = 50 minutes

2. A school library has books in different languages in the following ratio — no.of Odiya books : no.of Hindi books : no.of English books :: 3 : 2 : 1. If the library has 288 Odiya books, how many Hindi and English books does it have?
Solution:
Ratio of books = Odiya : Hindi : English = 3 : 2 : 1
Number of Odiya books = 288
3 parts = 288 books
So, 1 part = $\frac{288}{3}$ = 96 books
Number of Hindi books = 2 × 96 = 192
Number of English books = 1 × 96 = 96

3. I have100 coins in the ratio — no. of ₹10 coins : no. of ₹5 coins : no. of ₹2 coins : no. of ₹1coins :: 4 : 3 : 2 : 1. How much money do I have in coins?
Solution:
Total coins = 100
Let the number of coins be in the ratio
₹10 : ₹5 : ₹2 : ₹1 = 4 : 3 : 2 : 1
Sum of ratios = 4 + 3 + 2 + 1 = 10
₹10 coins = 100 × $\frac{4}{10}$ = 40 coins
₹5 coins = 100 × $\frac{3}{10}$ = 30 coins
₹2 coins = 100 × $\frac{2}{10}$ = 20 coins
₹1coins = 100 × $\frac{1}{10}$ = 10 coins
Total money = (40 × ₹10) + (30 × ₹5) + (20 × ₹2) + (10 × ₹1)
= 400 + 150 + 40 + 10
= ₹600

4. Construct a triangle with sidelengths in the ratio 3 : 4 : 5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?
Solution:
No, all triangles with sidelengths in the ratio 3 : 4 : 5 are not congruent.
This is because triangles with the same ratio of sides are similar, but their actual side lengths may be different (for example, 3, 4, 5 and 6, 8, 10).

5. Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?
Solution:
Let the sides be x, 3x, 5x.
To form a triangle, the sum of any two sides must be greater than the third side.
x + 3x = 4x < 5x
This violates the triangle inequality.
Therefore, a triangle with sidelengths in the ratio 1: 3: 5 cannot be constructed.

Figure it Out (Page 62)

1. 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:
Total number of people = 360
People liking summer season = 90
People liking rainy season = 120
People liking winter season = 360 − (90 + 120) = 150
Summer season’s angle = $\frac{90}{360}$ × 360 = 90°
Rainy season’s angle = $\frac{120}{360}$ × 360 = 120°
Winter season’s angle = $\frac{150}{360}$ × 360 = 150°

2. Draw a pie chart based on the following information about viewers ̓ favourite type of TV channel: Entertainment — 50%, Sports — 25%, News — 15%, Information — 10%.
Solution:
Entertainment = 50%
Sports = 25%
News = 15%
Information = 10%
Entertainment’s angle = $\frac{50}{100}$ × 360 = 5 × 36 = 180°
Sport’s angle = $\frac{25}{100}$ × 360 = 90°
News’ angle = $\frac{15}{100}$ × 360 = 54°
Information’s angle = $\frac{10}{100}$ × 360 = 36°

3. Prepare a pie chart that shows the favourite subjects of the students in your class. You can collect the data on the number of students for each subject shown in the table (each student should choose only one subject). Then write these numbers in the table and construct a pie chart:

Solution:

Language = $\frac{6}{36} \times 360 = 60^{\circ}$
Arts Education = $\frac{4}{36} \times 360 = 40^{\circ}$
Vocational Education = $\frac{2}{36} \times 360 = 20^{\circ}$
Social Science = $\frac{8}{36} \times 360 = 80^{\circ}$
Physical Education = $\frac{4}{36} \times 360 = 40^{\circ}$
Maths = $\frac{6}{36} \times 360 = 60^{\circ}$
Science = $\frac{6}{36} \times 360 = 60^{\circ}$

Figure it Out (Page 65)

1. Which of these are in inverse proportion?

Solution:
(i) x₁ = 40, x₂ = 80, x₃ = 25, x₄ = 16
y₁ = 20, y₂ = 10, y₃ = 32, y₄ = 50
x₁y₁ = 40 × 20. = 800
x₂y₂ = 80 × 10 = 800
x₃y₃ = 25 × 32 = 800
x₄y₄ = 16 × 50 = 800
Since the product is constant, x and y are in inverse proportion.

(ii) x₁ = 40, x₂ = 80, x₃ = 25, x₄ = 16
y₁ = 20, y₂ = 10, y₃ = 12.5, y₄ = 8
x₁y₁ = 40 × 20. = 800
x₂y₂ = 80 × 10 = 800
x₃y₃ = 25 × 12.5 = 312.5
x₄y₄ = 16 × 8 = 128
Since the product is not constant, x and y are not in inverse proportion.

(iii) x₁ = 30, x₂ = 90, x₃ = 150, x₄ = 10
y₁ = 15, y₂ = 5, y₃ = 3, y₄ = 45
x₁y₁ = 30 × 15. = 450
x₂y₂ = 90 × 5 = 450
x₃y₃ = 150 × 3 = 450
x₄y₄ = 10 × 45 = 450
Since the product is constant, x and y are in inverse proportion.

2. Fill in the empty cells if x and y are in inverse proportion.

Solution:

Since x and y are in inverse proportion, their product (x × y) remains constant.
12 × y₂ = 16 × 9
12 × y₂ = 144
y₂ = $\frac{144}{12}$ = 12

x₃ × 48 = 16 × 9
x₃ × 48 = 144
x₃ = $\frac{144}{48}$ = 3

36 × y₄ = 16 × 9
36 × y₄ = 144
y₄ = $\frac{144}{36}$ = 4

Figure it Out (Page 67 – 68)

1. Which of the following pairs of quantities are in inverse proportion?
(i) The number of taps filling a water tank and the time taken to fill it.
(ii) The number of painters hired and the days needed to paint a wall of fixed size.
(iii) The distance a car can travel and the amount of petrol in the tank.
(iv) The speed of a cyclist and the time taken to cover a fixed route.
(v) The length of cloth bought and the price paid at a fixed rate per metre.
(vi) The number of pages in a book and the time required to read it at a fixed reading speed.
Solution:
(i) More taps → less time to fill the tank
Fewer taps → more time to fill the tank
So, the quantities are inversely proportional.

(ii) More painters → fewer days to paint the wall
Fewer painters → more days to paint the wall
So, the quantities are inversely proportional.

(iii) More petrol → more distance covered
Less petrol → less distance covered
So, the quantities are directly proportional.

(iv) Higher speed → less time taken
Lower speed → more time taken
So, the quantities are inversely proportional.

(v) More cloth → higher price
Less cloth → lower price
So, the quantities are directly proportional.

(vi) More pages → more time to read
Fewer pages → less time to read
So, the quantities are directly proportional.

2. If 24 pencils cost ₹120, how much will 20 such pencils cost?
Solution:
The pencils and their cost are in direct proportion.
Let y is the cost of pencils, then
24 : 20 :: 120 : y
24 × y = 20 × 120
y = $\frac{20 \times 120}{24}$ = $\frac{2400}{24}$ = 100.
Thus, 20 pencils will cost ₹100.

3. A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?
Solution:
The number of families and the number of days are in inverse proportion.
Assumptions:
(i) The tank is not refilled during this period.
(ii) Each family uses the same amount of water per day.
(iii) Water consumption remains constant.
(iv) There is no wastage or leakage of water.
Let the number of days be y, then
20 × 6 = (20 + 10) × y
120 = 30y
y = $\frac{120}{30}$ = 4.
Therefore, the water will last for 4 days.

4. Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list : 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.

Solution:

5. The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions.

(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?
Solution:
Walk = 90°
Bus = 120°
Cycle = 60°
Two-wheeler = 60°
Car = 360° − (90 + 120 + 60 + 60) = 30°
(i) The largest angle is 120°, which corresponds to the bus.
(ii) Fraction of children travelling by car = $\frac{30}{360}$ = $\frac{1}{12}$
(iii) Let the number of children be y, then
$\frac{1}{12}$ × y = 18
y = 18 × 12 = 216
(iv) Cycle and two-wheeler (both 60°).

6. Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?
Solution:
The number of workers and the number of days are in inverse proportion.
Assumptions:
(i) All workers work at the same rate.
(ii) They work for the same number of hours each day.
(iii) Total work remains fixed.
Let the number of days be y.
3 × 4 = 4 × y
12 = 4y
y = $\frac{12}{4}$ = 3
Therefore, they will take 3 days to finish the work.

7. It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?
Solution:
Here, work is directly proportional to time.
Let the time taken be y hours, then
2 : 5 :: 6 : y
2 × y = 5 × 6
2y = 30
y = $\frac{30}{2}$ = 15
Therefore, it will take 15 hours to fill 5 tanks.

8. A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?
Solution:
The number of rows and chairs in each row are inversely proportional.
Let the number of rows be y, then
25 × 12 = 20 × y
300 = 20y
y = $\frac{300}{20}$ = 15
Therefore, the new arrangement has 15 chairs.

9. A school has 8 periods a day, each of 45 minutes duration. How long is each period, if the school has 9 periods a day, assuming that the number of school hours per day stays the same?
Solution:
The number of periods and their duration are inversely proportional.
Let the duration of each period be y minutes.
8 × 45 = 9 × y
360 = 9y
y = $\frac{360}{9}$ = 40
Therefore, each period is 40 minutes long.

10. A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?

Solution:
Let the capacity of the tank be x litres.
Water filled by the small pump in 1 hour = $\frac{x}{3}$ litres
Water filled by the small pump in 1 hour = $\frac{x}{2}$ litres
Water filled by both pumps in 1 hour =
= $\frac{x}{3}$ + $\frac{x}{2}$
= $\frac{2 x + 3 x}{6}$ = $\frac{5 x}{6}$ litres
Time taken by both the pumps to fill the tank =
= (1 ÷ $\frac{5 x}{6}$ ) × x
= 1 × $\frac{6}{5 x}$ × x
= $\frac{6}{5}$ hours = 1 $\frac{1}{5}$ hours.
Therefore, the tank will take 1 $\frac{1}{5}$ hours to fill.

11. 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?

Solution:
The number of machines and the number of days are in inverse proportion.
Let the required number of machines be y.
42 × 63 = 54 × y
2646 = 54y
y = $\frac{2646}{54}$ = 49
Therefore, 49 machines are required.

12. A car takes 2 hours to reach a destination, travelling at a speed of 60 km/h. How long will the car take if it travels at a speed of 80 km/h?
Solution:
The speed of the car and time are inversely proportional.
Let the time taken at 80 km/h be t hours.
60 × 2 = 80 × t
120 = 80t
t = $\frac{120}{80}$ = 1.5 hours
Therefore, the car will take 1.5 hours.

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