Class 8 Mathematics · Part 2 · Chapter 3

3. Proportional Reasoning-2

Solutions to all Figure it Out and in-text questions. CBSE Class 8 Mathematics, NCERT Ganita Prakash Part 2.

3.1–3.2 Ratios and Maps

1Convert 60,00,000 cm to kilometres.

60,00,000 cm = 60,000 m (÷ 100) = 60 km (÷ 1000).

2Using the map (RF 1 : 60,00,000), find the geographical distance between Bengaluru and Chennai, and between Mangaluru and Chennai.

Measure the distance on the map with a ruler and multiply by 60 km (since 1 cm = 60 km).

Bengaluru–Chennai: about 4.8 cm on a map of this scale → 4.8 × 60 ≈ 290 km

Mangaluru–Chennai: about 9.8 cm → 9.8 × 60 ≈ 590 km

The map in the book is printed smaller (“Map not to scale”), so your ruler reading on the printed page may be smaller. Using the latitude–longitude grid, the actual straight-line distances are about 290 km and 590 km.

3.3–3.4 Ratios with More than 2 Terms

3Example 3 check: total purple paint for 4 L red, 6 L blue, 10 L white.

Total = 4 + 6 + 10 = 20 litres

Figure it Out (Page 60)

1Warm-up/cool-down : batting : bowling : fielding = 3 : 4 : 3 : 5. Each session is 150 minutes. Find the time for each activity.

Total parts = 15. One part = 150 ÷ 15 = 10 min.

Warm-up/cool-down = 30 min, batting = 40 min, bowling = 30 min, fielding = 50 min

2Odiya : Hindi : English books = 3 : 2 : 1. There are 288 Odiya books. Find the Hindi and English books.

3 parts = 288, so 1 part = 96.

Hindi = 2 × 96 = 192, English = 96

3100 coins: ₹10 : ₹5 : ₹2 : ₹1 coins = 4 : 3 : 2 : 1. How much money is there?

One part = 100 ÷ 10 = 10 coins. So 40, 30, 20 and 10 coins.

Money = 40 × 10 + 30 × 5 + 20 × 2 + 10 × 1 = 400 + 150 + 40 + 10 = ₹600

4Construct a triangle with sides in the ratio 3 : 4 : 5. Will all such triangles be congruent?

Draw, for example, sides 3 cm, 4 cm and 5 cm (or 6 cm, 8 cm and 10 cm).

They will not all be congruent. They have the same shape (angles are the same), but different sizes. Only triangles with the same side lengths are congruent.

5Can you construct a triangle with sides in the ratio 1 : 3 : 5?

No. If the sides are k, 3k and 5k, then k + 3k = 4k is less than 5k. The sum of two sides of a triangle must be greater than the third side.

Figure it Out (Page 62) — Pie Charts

1360 people: 90 liked summer, 120 liked rainy, the rest liked winter. Draw a pie chart.

Winter = 360 − 90 − 120 = 150. Since there are 360 people, each person = 1°.

Summer = 90°, Rainy = 120°, Winter = 150°

Favourite season

2Entertainment 50%, Sports 25%, News 15%, Information 10%. Draw a pie chart.

Angle = percentage × 3.6°.

Entertainment = 180°, Sports = 90°, News = 54°, Information = 36°

Favourite TV channel type

3Prepare a pie chart of the favourite subjects of students in your class.

Sample (40 students): Language 8, Maths 10, Science 6, Social Science 4, Arts Education 6, Vocational Education 2, Physical Education 4.

Angle = (number of students ÷ 40) × 360° = number × 9°. So 72°, 90°, 54°, 36°, 54°, 18°, 36°.

Use your own class data. The angles must add up to 360°.

3.6 Inverse Proportions

4Puneeth’s father: 3 hours at 30 km/h. Can we write 30 : 60 :: 3 : x? Will time increase or decrease as speed increases?

No. As speed increases, time decreases. They are inversely proportional.

30 × 3 = 60 × x, so x = 1.5 hours

5Check whether speed and time change by the same factor for the other modes.

Walk → motorcycle: speed × 6 (5 to 30), time ÷ 6 (18 to 3). Bicycle → car: speed × 4, time ÷ 4 (6 to 1.5).

The product speed × time is always 90 km.

Figure it Out (Page 65)

1Which of these are in inverse proportion?

(i) 40 × 20 = 80 × 10 = 25 × 32 = 16 × 50 = 800. Inverse

(ii) 40 × 20 = 800, but 25 × 12.5 = 312.5. Not inverse (here x ÷ y = 2, so it is direct).

(iii) 30 × 15 = 90 × 5 = 150 × 3 = 10 × 45 = 450. Inverse

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

xy = 16 × 9 = 144.

x	16	12	3	36
y	9	12	48	4

6Ram takes 1 hour and Shyam 1.5 hours. Is the quantity of work and the time taken directly or inversely proportional?

Directly proportional. More work takes more time at the same rate. Together they take 35 hour = 36 minutes.

Figure it Out (Page 67)

1Which pairs are in inverse proportion?

(i) Taps and time to fill a tank — inverse

(ii) Painters and days to paint a fixed wall — inverse

(iii) Distance travelled and petrol — direct

(iv) Speed of a cyclist and time for a fixed route — inverse

(v) Length of cloth and price — direct

(vi) Pages and reading time — direct

2If 24 pencils cost ₹120, how much will 20 pencils cost?

Direct proportion. One pencil = ₹5. 20 pencils = ₹100

3Water lasts 6 days for 20 families. If 10 more families move in, how long will it last? What assumptions are needed?

Inverse proportion: 20 × 6 = 30 × y, so y = 4 days

We assume every family uses the same amount of water each day and no water is added.

4Fill in the average hours each living being sleeps in a day (15, 2.5, 20, 8, 3.5, 13, 10.5, 18).

The blue part of each ring shows the hours of sleep out of 24.

Living being	Sleep (hours)
Giraffe	2.5
Elephant	3.5
Human (child)	8
Dog	10.5
Cat	13
Squirrel	15
Python	18
Bat	20

5Pie chart of modes of transport (Walk 90°, Bus 120°, Two-wheeler 60°, Cycle 60°, Car).

Car = 360° − 90° − 120° − 60° − 60° = 30°.

(i) Most common: Bus (120°).

(ii) Fraction by car = 30360 = 112

(iii) 112 of total = 18, so total = 216 children. No slice shows taxis, so 0 children use taxis.

(iv) Cycle and two-wheeler (60° each).

6Three workers paint a fence in 4 days. If one more joins, how many days will it take? Assumptions?

3 × 4 = 4 × y, so y = 3 days

We assume all workers work at the same speed and for the same hours.

7It takes 6 hours to fill 2 tanks with a pump. How long for 5 tanks?

Direct proportion. One tank = 3 hours. 5 tanks = 15 hours

8Chairs in 25 rows of 12. How many rows with 20 chairs in each row?

Total chairs = 300. Rows = 300 ÷ 20 = 15

98 periods of 45 minutes each. How long is each period if there are 9 periods?

Total time = 360 minutes. Each period = 360 ÷ 9 = 40 minutes

10A small pump fills a tank in 3 hours and a large pump in 2 hours. How long together?

In 1 hour: 13 + 12 = 56 of the tank.

Time = 65 hours = 1 hour 12 minutes

1142 machines produce toys in 63 days. How many machines are needed for 54 days?

Inverse proportion: 42 × 63 = x × 54. x = 2646 ÷ 54 = 49 machines

12A car takes 2 hours at 60 km/h. How long at 80 km/h?

60 × 2 = 80 × t. t = 120 ÷ 80 = 1.5 hours

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