by Tiwari Academy Team
NCERT Solutions for Class 8 Maths Ganita Prakash Part 2 Chapter 3, Proportional Reasoning-2, cover every Figure it Out question with clear, step-by-step working. Building on Proportional Reasoning-1 from Part 1, the chapter shows how to use the scale on a map to find actual distances.
NCERT Class 8 Maths Ganita Prakash Part 2 Chapter 3 Solutions
The chapter 3 shows how to work with ratios of more than two terms, such as cement, sand and gravel in concrete. Next come dividing a whole in a given ratio and turning ratios into angles to draw pie charts. The chapter ends with direct and inverse proportion, using problems on speed and time, workers and days, and pumps filling a tank.
Get here the Solutions and Answers of Figure it Out at Page 60.
Page 60 – Figure it Out
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?
Answer:
Sum of the terms of the ratio = 3 + 4 + 3 + 5 = 15.
Time for 1 part = 150 รท 15 = 10 minutes.
Time for warm-up/cool-down = 3 ร 10 = 30 minutes.
Time for batting = 4 ร 10 = 40 minutes.
Time for bowling = 3 ร 10 = 30 minutes.
Time for fielding = 5 ร 10 = 50 minutes.
Therefore, 30 minutes are spent on warm-up/cool-down, 40 minutes on batting, 30 minutes on bowling and 50 minutes on fielding.
2. A school library has books in different languages in the following ratio โ 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?
Answer:
3 parts represent 288 Odiya books.
So, 1 part = 288 รท 3 = 96 books.
Number of Hindi books = 2 ร 96 = 192.
Number of English books = 1 ร 96 = 96.
Therefore, the library has 192 Hindi books and 96 English books.
3. I have 100 coins in the ratio – no. of โน10 coins : no. of โน5 coins : no. of โน2 coins : no. of โน1 coins :: 4 : 3 : 2 : 1. How much money do I have in coins?
Answer:
Sum of the terms of the ratio = 4 + 3 + 2 + 1 = 10.
Number of coins in 1 part = 100 รท 10 = 10.
Number of โน10 coins = 4 ร 10 = 40.
Number of โน5 coins = 3 ร 10 = 30.
Number of โน2 coins = 2 ร 10 = 20.
Number of โน1 coins = 1 ร 10 = 10.
Total money = (40 ร โน10) + (30 ร โน5) + (20 ร โน2) + (10 ร โน1)
= โน400 + โน150 + โน40 + โน10
= โน600.
Therefore, I have โน600 in coins.
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?
Answer:
We can take the sidelengths as 3 cm, 4 cm, and 5 cm.
- Step 1: Draw AB = 4 cm.
- Step 2: With A as the centre and a radius of 3 cm, draw an arc.
- Step 3: With B as the centre and a radius of 5 cm, draw another arc to meet the first arc at C.
- Step 4: Join AC and BC.
Thus, AC : AB : BC = 3 : 4 : 5 and โณABC is the required triangle.
All triangles with this ratio of side lengths need not be congruent.
For example, a triangle with side lengths 6 cm, 8 cm and 10 cm also has the ratio 3 : 4 : 5.
But its sides are twice as long as the sides of the triangle with side lengths 3 cm, 4 cm and 5 cm.
The two triangles have different sizes, so they are not congruent.
5. Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?
Answer:
No, we cannot construct such a triangle.
In a triangle, the sum of any two side lengths must be greater than the third side length.
But 1 + 3 = 4, which is less than 5.
Multiplying all three terms by the same positive number does not change this inequality.
Therefore, no triangle can have its side lengths in the ratio 1 : 3 : 5.
Get here the Solutions and Answers of Figure it Out at Page 62.
Page 62 – Figure it Out
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.
Answer:
Number of people who liked winter = 360 โ (90 + 120)
= 360 โ 210
= 150.
Angle of a sector = (Number of people รท Total number of people) ร 360ยฐ.
Summer: (90 รท 360) ร 360ยฐ = 90ยฐ.
Rainy: (120 รท 360) ร 360ยฐ = 120ยฐ.
Winter: (150 รท 360) ร 360ยฐ = 150ยฐ.
Using these angles, the required pie chart is:
2. Draw a pie chart based on the following information about viewersโ favourite type of TV channel: Entertainment โ 50%, Sports โ 25%, News โ 15%, Information โ 10%.
Answer:
Angle of a sector = (Percentage รท 100) ร 360ยฐ.
Entertainment: (50 รท 100) ร 360ยฐ = 180ยฐ.
Sports: (25 รท 100) ร 360ยฐ = 90ยฐ.
News: (15 รท 100) ร 360ยฐ = 54ยฐ.
Information: (10 รท 100) ร 360ยฐ = 36ยฐ.
Using these angles, the required pie chart is:
3. Prepare a pie chart that shows the favourite subjects of the students in your class. You can collect the data of 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:
Answer:
In my class, there are 40 students. The numbers and the pie chart is given here according to our class survey.
Each student chooses only one subject.
Angle for 1 student = 360ยฐ รท 40 = 9ยฐ.
Language: 8 ร 9ยฐ = 72ยฐ.
Arts Education: 4 ร 9ยฐ = 36ยฐ.
Vocational Education: 2 ร 9ยฐ = 18ยฐ.
Social Science: 4 ร 9ยฐ = 36ยฐ.
Physical Education: 6 ร 9ยฐ = 54ยฐ.
Maths: 10 ร 9ยฐ = 90ยฐ.
Science: 6 ร 9ยฐ = 54ยฐ.
Using these angles, the pie chart is:
Get here the Solutions and Answers of Figure it Out at Page 65.
Page 65 – Figure it Out
1. Which of these are in inverse proportion?
Answer:
For inverse proportion, the product xy must be the same for every pair of values.
(i) 40 ร 20 = 800
80 ร 10 = 800
25 ร 32 = 800
16 ร 50 = 800
All the products are equal to 800. Therefore, x and y are in inverse proportion.
(ii) 40 ร 20 = 800
80 ร 10 = 800
25 ร 12.5 = 312.5
16 ร 8 = 128
The products are not all equal. Therefore, x and y are not in inverse proportion.
(iii) 30 ร 15 = 450
90 ร 5 = 450
150 ร 3 = 450
10 ร 45 = 450
All the products are equal to 450. Therefore, x and y are in inverse proportion.
Thus, (i) and (iii) are in inverse proportion.
2. Fill in the empty cells if x and y are in inverse proportion.
Answer:
Since x and y are in inverse proportion, their product remains the same.
So, xy = 16 ร 9 = 144.
When x = 12,
y = 144 รท 12 = 12.
When y = 48,
x = 144 รท 48 = 3.
When x = 36,
y = 144 รท 36 = 4.
The completed table is:
Get here the Solutions and Answers of Figure it Out at Page 67.
Page 67 – Figure it Out
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.
Answer:
(i) The quantities are in inverse proportion. If more taps of the same flow rate are used, the tank takes less time to fill. The number of taps ร time taken remains constant.
(ii) The quantities are in inverse proportion. If more painters work at the same rate for the same number of hours each day, fewer days are needed to paint the wall. The number of painters ร number of days remains constant.
(iii) The quantities are in direct proportion, assuming the car travels the same distance per litre of petrol. More petrol allows the car to travel a greater distance.
(iv) The quantities are in inverse proportion. For a fixed route, speed ร time = distance. So, when the speed increases, the time decreases by the same factor.
(v) The quantities are in direct proportion. At a fixed price per metre, buying more cloth costs more money in the same ratio.
(vi) The quantities are in direct proportion, assuming the pages have the same amount of reading. At a fixed reading speed, more pages take more time to read.
Therefore, (i), (ii) and (iv) are in inverse proportion.
2. If 24 pencils cost โน120, how much will 20 such pencils cost?
Answer:
Cost of 1 pencil = โน120 รท 24 = โน5.
Cost of 20 pencils = โน5 ร 20 = โน100.
Therefore, 20 pencils will cost โน100.
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?
Answer:
Total number of families = 20 + 10 = 30.
The number of families and the number of days are in inverse proportion.
Let the water last for x days.
So, 20 ร 6 = 30 ร x
โ x = (20 ร 6) รท 30
โ x = 4.
Therefore, the water will last for 4 days.
We assume that all 30 families start using the water at the same time, each family uses the same amount of water every day, no water is added to the tank and no water is lost through leakage.
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.
Answer:
The whole circle represents 24 hours. The purple part represents the time spent sleeping.
Top row, from left to right:
Giraffe โ 2.5 hours
Elephant โ 3.5 hours
Human โ 8 hours
Dog โ 10.5 hours
Bottom row, from left to right:
Cat โ 13 hours
Squirrel โ 15 hours
Snake โ 18 hours
Bat โ 20 hours
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?
Answer:
(i) Bus is the most common mode of transport because its sector, 120ยฐ, is the largest.
(ii) Angle of the car sector = 360ยฐ โ (90ยฐ + 120ยฐ + 60ยฐ + 60ยฐ)
= 360ยฐ โ 330ยฐ
= 30ยฐ.
Fraction of children travelling by car = 30/360 = 1/12.
Therefore, 1/12 of the children travel by car.
(iii) Let the total number of children be x.
x/12 = 18
x = 18 ร 12
x = 216.
Therefore, 216 children took part in the survey.
Taxis are not shown separately in the pie chart.
So, the number of children travelling by taxi cannot be determined from this chart.
(iv) The cycle and two-wheeler sectors are both 60ยฐ.
Therefore, equal numbers of children travel by cycle and by two-wheeler.
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?
Answer:
New number of workers = 3 + 1 = 4.
The number of workers and the number of days are in inverse proportion.
Let the time needed be x days.
So, 3 ร 4 = 4 ร x
โ x = (3 ร 4) รท 4
โ x = 3.
Therefore, 4 workers can paint the fence in 3 days.
We assume that the fourth worker joins before the work starts, all workers work at the same rate and for the same number of hours each day and the amount of work remains the same.
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?
Answer:
Time taken to fill 1 tank = 6 รท 2 = 3 hours.
Time taken to fill 5 tanks = 3 ร 5 = 15 hours.
Therefore, the pump will take 15 hours to fill 5 tanks, working at the same rate.
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?
Answer:
Total number of chairs = 25 ร 12 = 300.
Number of rows in the new arrangement = 300 รท 20 = 15.
Therefore, the new arrangement has 15 rows.
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?
Answer:
Total time for the periods in a day = 8 ร 45 = 360 minutes.
Duration of each of the 9 periods = 360 รท 9 = 40 minutes.
Therefore, each period will be 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?
Answer:
Part of the tank filled by the small pump in 1 hour = 1/3.
Part of the tank filled by the large pump in 1 hour = 1/2.
Part of the tank filled by both pumps in 1 hour = 1/3 + 1/2
= 2/6 + 3/6
= 5/6.
Time taken to fill the whole tank = 1 รท (5/6)
= 6/5 hours
= 1โ
hours.
As โ
hour = (1/5) ร 60 = 12 minutes.
Therefore, both pumps together will fill the tank in 1 hour 12 minutes.
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?
Answer:
The number of machines and the number of days are in inverse proportion, assuming that all machines work at the same rate for the same number of hours each day.
Let the required number of machines be x.
Therefore, 42 ร 63 = x ร 54
โ x = (42 ร 63) รท 54
โ x = 49.
Hence, 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?
Answer:
Distance travelled = speed ร time
= 60 ร 2
= 120 km.
Time taken at 80 km/h = distance รท speed
= 120 รท 80
= 1.5 hours
= 1 hour 30 minutes.
Therefore, the car will take 1 hour 30 minutes.
Final Tips to Master Ratio and Proportion in Class 8
Proportional reasoning is everywhere, from recipes to map scales, so these skills matter well beyond exams. Before calculating, ask whether the quantities are in direct or inverse proportion: if one doubles, does the other double or halve? The last Figure it Out set mixes both types, so watch out. For ratios with several terms, find the value of one part first, then multiply it by each term; for pie charts, check the angles total 360ยฐ. Try each problem on your own, then compare your method with these Proportional Reasoning-2 Class 8 solutions. When you are ready, move on to Chapter 4, Exploring Some Geometric Themes, or browse the other chapters in our NCERT Solutions for Class 8 Maths.
