Class 8 Maths Ganita Prakash Chapter 10 Solutions (NCERT 2026–27) – Proportional Reasoning-2

These Class 8 Maths Ganita Prakash Chapter 10 solutions cover Proportional Reasoning-2 from the new NCERT textbook (2026–27). This is Chapter 3 of Ganita Prakash Part II (the 10th chapter of the Class 8 course); it continues “Proportional Reasoning” from Part I, extending it to ratios with many terms, dividing a whole in a ratio, pie charts and inverse proportions. Every “Figure it Out” question is solved step by step, with each proportion and pie-chart angle verified.

Class: 8 Subject: Mathematics Book: Ganita Prakash (Part II) Chapter: 10 (Part II Ch 3) Exercises: 3 × Figure it Out Session: 2026–27
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Chapter 10 Overview

Chapter 3 of Ganita Prakash Part II, Proportional Reasoning-2, builds on the ratio and proportion ideas from Part I. It begins with a quick recap of when two ratios are proportional (a × d = b × c), then looks at ratios in maps (Representative Fraction), ratios with more than two terms, and how to divide a whole in a given ratio. The chapter then introduces pie charts — turning each part of the data into a sector angle out of 360° — and finally inverse proportions, where two quantities satisfy x × y = k (one goes up by a factor n as the other goes down by 1/n). The solutions below work through all three “Figure it Out” sets step by step.

Key Concepts & Definitions

Proportional ratios: two ratios a : b and c : d are proportional if a × d = b × c (cross-multiplication).

Representative Fraction (RF): the ratio of a distance on a map to the actual distance on the ground (e.g. RF 1 : 60,00,000 means 1 cm on the map = 60,00,000 cm = 60 km on the ground).

Ratios with many terms: a : b : c : d … — for every a units of the first quantity there are b, c, d… units of the others.

Dividing a whole in a ratio: share each part as (total) × (its term) ÷ (sum of all terms).

Pie chart: a circle in which each category gets a sector whose angle is proportional to its value; all the sector angles add up to 360°.

Direct proportion: both quantities change by the same factor; their quotient x/y stays constant.

Inverse proportion: as one quantity is multiplied by n, the other is multiplied by 1/n; their product x × y = k stays constant.

Important Formulas (Chapter 10)

Proportion of two ratios: a : b :: c : d  ⇒  a × d = b × c.

Dividing a whole x in the ratio p : q : r … : each part = x × (its term) ÷ (p + q + r + …).

Pie-chart sector angle = (value ÷ total) × 360°.

Direct proportion: x1/y1 = x2/y2 = k.

Inverse proportion: x × y = k, i.e. x1y1 = x2y2 (so x1/x2 = y2/y1).

Figure it Out — Ratios with Many Terms (after Example 5, 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 Sum of ratio terms = 3 + 4 + 3 + 5 = 15. One part = 150 ÷ 15 = 10 minutes. Warm-up/cool-down = 3 × 10 = 30 min; Batting = 4 × 10 = 40 min; Bowling = 3 × 10 = 30 min; Fielding = 5 × 10 = 50 min. Check: 30 + 40 + 30 + 50 = 150 min. ✓

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 Odiya books = 3 parts = 288, so 1 part = 288 ÷ 3 = 96 books. Hindi books = 2 parts = 2 × 96 = 192; English books = 1 part = 96.

3. I have 100 coins in the ratio — no. of ₹10 coins : no. of ₹5 coins : no. of ₹2 coins : no. of ₹1 coins :: 4 : 3 : 2 : 1. How much money do I have in coins?

SOLUTION Sum of terms = 4 + 3 + 2 + 1 = 10. With 100 coins, 1 part = 100 ÷ 10 = 10 coins.
CoinNo. of coinsValue
₹104 × 10 = 4040 × 10 = ₹400
₹53 × 10 = 3030 × 5 = ₹150
₹22 × 10 = 2020 × 2 = ₹40
₹11 × 10 = 1010 × 1 = ₹10
Total money = 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 The ratio 3 : 4 : 5 satisfies the triangle inequality (3 + 4 = 7 > 5), so a triangle can be drawn — for example with sides 3 cm, 4 cm, 5 cm (a right triangle, since 32 + 42 = 52). No, the triangles need not be congruent. Sides 3, 4, 5 and sides 6, 8, 10 both have the ratio 3 : 4 : 5 but are different sizes. Same ratio makes them similar (same shape), but congruence needs the same actual lengths.

5. Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?

SOLUTION No. For any triangle, the sum of two sides must be greater than the third (triangle inequality). Here 1 + 3 = 4, which is less than 5. The two shorter sides cannot meet to close the triangle, so a triangle with sides in the ratio 1 : 3 : 5 cannot be constructed.

Figure it Out — Pie Charts (after Step 7, 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 Winter = 360 − (90 + 120) = 360 − 210 = 150 people. Each sector angle = (number of people ÷ 360) × 360° — here the total is 360, so each angle equals the number of people in degrees.
SeasonPeopleSector angle = (people ÷ 360) × 360°
Summer90(90 ÷ 360) × 360° = 90°
Rainy120(120 ÷ 360) × 360° = 120°
Winter150(150 ÷ 360) × 360° = 150°
Check: 90° + 120° + 150° = 360°. ✓ Draw a circle and mark sectors of 90° (summer), 120° (rainy) and 150° (winter).

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 Sector angle = (percentage ÷ 100) × 360° for each category.
Channel typePercentageSector angle = (% ÷ 100) × 360°
Entertainment50%0.50 × 360° = 180°
Sports25%0.25 × 360° = 90°
News15%0.15 × 360° = 54°
Information10%0.10 × 360° = 36°
Check: 180° + 90° + 54° + 36° = 360°. ✓ Draw the sectors of these angles and label them.

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: Subjects: Language, Arts Education, Vocational Education, Social Science, Physical Education, Maths, Science.

SOLUTION This is a data-collection activity, so the exact numbers depend on your own class. The method is the same for every set of values: (i) Add the number of students for all seven subjects to get the total. (ii) For each subject, sector angle = (number choosing it ÷ total) × 360°. (iii) Make sure the seven angles add up to 360°, then draw and label the sectors. Worked example (sample data, total = 40): Language 10 → (10/40)×360° = 90°; Arts 4 → 36°; Vocational 2 → 18°; Social Science 6 → 54°; Physical Education 4 → 36°; Maths 8 → 72°; Science 6 → 54°. Total = 90° + 36° + 18° + 54° + 36° + 72° + 54° = 360°. ✓ Use your own class data in place of these numbers.

Figure it Out — Inverse Proportion (Tables) (after Example 2, page 65)

1. Which of these are in inverse proportion? (i) x: 40, 80, 25, 16   /   y: 20, 10, 32, 50 (ii) x: 40, 80, 25, 16   /   y: 20, 10, 12.5, 8 (iii) x: 30, 90, 150, 10   /   y: 15, 5, 3, 45

SOLUTION Two quantities are in inverse proportion when the product x × y is the same for every pair. (i) 40×20 = 800, 80×10 = 800, 25×32 = 800, 16×50 = 800. All equal 800 ⇒ inverse proportion. (ii) 40×20 = 800, 80×10 = 800, 25×12.5 = 312.5, 16×8 = 128. Products differ ⇒ not inverse proportion. (iii) 30×15 = 450, 90×5 = 450, 150×3 = 450, 10×45 = 450. All equal 450 ⇒ inverse proportion.

2. Fill in the empty cells if x and y are in inverse proportion. x: 16, 12, __, 36   /   y: 9, __, 48, __

SOLUTION For inverse proportion x × y = k. From the first pair, k = 16 × 9 = 144. When x = 12: y = 144 ÷ 12 = 12. When y = 48: x = 144 ÷ 48 = 3. When x = 36: y = 144 ÷ 36 = 4.
x1612336
y912484
Check: 16×9 = 12×12 = 3×48 = 36×4 = 144. ✓

Figure it Out — Inverse Proportion (Word Problems) (after Example 6, pages 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) Inverse — more taps fill the tank in less time. (ii) Inverse — more painters finish the fixed wall in fewer days. (iii) Direct (not inverse) — more petrol lets the car travel a longer distance. (iv) Inverse — a faster cyclist covers the fixed route in less time. (v) Direct (not inverse) — more cloth costs more money. (vi) Direct (not inverse) — more pages take more time to read.

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

SOLUTION Cost is directly proportional to the number of pencils. Cost of 1 pencil = 120 ÷ 24 = ₹5. Cost of 20 pencils = 20 × 5 = ₹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 More families → the same water lasts fewer days ⇒ inverse proportion: (families) × (days) = constant. New number of families = 20 + 10 = 30. So 20 × 6 = 30 × x ⇒ x = 120 ÷ 30 = 4 days. Assumption: every family uses the same amount of water each day, and the daily use does not change.

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 This is a chart-reading activity. In each circle, the coloured (sleep) portion is a fraction of the 24-hour day; read its share and match it to a value from the list so that sleep hours = (fraction of the circle) × 24. Method: if the sleep sector is, say, half the circle, sleeping hours = (1/2) × 24 = 12 h; if it is three-quarters, hours = (3/4) × 24 = 18 h, and so on. Pick the closest value from {15, 2.5, 20, 8, 3.5, 13, 10.5, 18} for each animal. (Typical real values: giraffe ≈ 2.5 h, elephant ≈ 3.5 h, human child ≈ 10.5 h, squirrel/cat ≈ 13–15 h, bat ≈ 20 h, python ≈ 18 h.)

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. (Given angles: Walk 90°, Bus 120°, Two-wheeler 60°, Car 60°.) (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 First find the missing Cycle angle: 360° − (90° + 120° + 60° + 60°) = 360° − 330° = 30° (Cycle).
ModeSector angleFraction of children
Bus120°120/360 = 1/3
Walk90°90/360 = 1/4
Two-wheeler60°60/360 = 1/6
Car60°60/360 = 1/6
Cycle30°30/360 = 1/12
(i) The largest sector is Bus (120°), so the bus is the most common mode of transport. (ii) Car = 60°, so the fraction of children travelling by car = 60/360 = 1/6. (iii) 1/6 of the children = 18, so the total = 18 × 6 = 108 children. There is no “taxi” sector in the pie chart, so 0 children use taxis. (iv) Two-wheeler and Car both have 60°, so equal numbers travel by two-wheeler and by car (18 each).

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 More workers → fewer days ⇒ inverse proportion: (workers) × (days) = constant. New number of workers = 3 + 1 = 4. So 3 × 4 = 4 × x ⇒ x = 12 ÷ 4 = 3 days. Assumptions: all workers work at the same rate, and they all work for the same number of hours each day.

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 Time is directly proportional to the number of tanks (the pump fills at a fixed rate). Time for 1 tank = 6 ÷ 2 = 3 hours. Time for 5 tanks = 5 × 3 = 15 hours.

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 Total chairs = 25 × 12 = 300 (this stays fixed), so rows × chairs-per-row is constant ⇒ inverse proportion. 25 × 12 = (rows) × 20 ⇒ rows = 300 ÷ 20 = 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?

SOLUTION Total teaching time per day is fixed, so (periods) × (length of each) is constant ⇒ inverse proportion. 8 × 45 = 9 × x ⇒ x = 360 ÷ 9 = 40 minutes per period.

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 In 1 hour the small pump fills 1/3 of the tank and the large pump fills 1/2 of the tank. Together in 1 hour they fill 1/3 + 1/2 = 2/6 + 3/6 = 5/6 of the tank. Time to fill 1 full tank = 6/5 hours = 1.2 hours = 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?

SOLUTION Fewer days → more machines ⇒ inverse proportion: (machines) × (days) = constant. 42 × 63 = x × 54 ⇒ x = (42 × 63) ÷ 54 = 2646 ÷ 54 = 49 machines.

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 Distance is fixed, so (speed) × (time) = constant ⇒ inverse proportion. 60 × 2 = 80 × x ⇒ x = 120 ÷ 80 = 1.5 hours (1 hour 30 minutes).
Math Talk & Try This (in-text questions) Idli recipe (p.55): Would Viswanath’s (6 : 3) and Puneet’s (4 : 2) idlis taste the same? Spice mix (p.57): Puneet has only 2 red chillies — how much of the other ingredients? RF (p.56): Convert 60,00,000 cm to kilometres. Answers. Idli: cross-multiplying, 6×2 = 12 = 3×4, so 6 : 3 and 4 : 2 are proportional — the idlis would taste the same. Spice mix: Viswanath used 8 : 4 : 2 : 1; Puneet has half the chillies (2 instead of 4), so halve everything — 4 spoons coriander, 2 chillies, 1 spoon toor dal, ½ spoon fenugreek (ratio 4 : 2 : 1 : 0.5). RF: 60,00,000 cm = 60,00,000 ÷ 1,00,000 = 60 km (since 1 km = 1,00,000 cm).
Math Talk (Inverse proportion, pages 63–64) 1. Can the speed–time problem be written as 30 : 60 :: 3 : x? Will travel time increase or decrease as speed increases? 2. For the table (speed 5, 15, 30, 60; time 18, 6, 3, 1.5), does the time decrease by the same factor as the speed increases? Answer. 1. No — this is not a direct proportion. As speed increases, the travel time decreases (inverse proportion), so the correct relation is 30 × 3 = 60 × x, giving x = 1.5 hours. 2. Yes — speed × time = 90 km for every column (5×18 = 15×6 = 30×3 = 60×1.5 = 90). When speed is multiplied by a factor, time is divided by the same factor.

Common Mistakes to Avoid

Watch out for these

  • Forgetting to add all the ratio terms before dividing a whole — the “one part” value uses the total of the terms.
  • Treating an inverse relationship as a direct one (e.g. “more workers, more days”) — check whether the other quantity goes up or down.
  • In a pie chart, using (value ÷ total) but forgetting to multiply by 360°, or letting the angles not add up to 360°.
  • Mixing up the formula: direct → x/y constant; inverse → x × y constant.
  • Reading a triangle’s “same ratio of sides” as “congruent” — same ratio means similar, not necessarily equal in size.
  • Ignoring the triangle inequality — a ratio like 1 : 3 : 5 cannot form a triangle (1 + 3 < 5).

Practice MCQs & Assertion–Reason

1. Two quantities x and y are in inverse proportion if:

(a) x + y is constant    (b) x − y is constant    (c) x × y is constant    (d) x ÷ y is constant

2. The total angle used for all sectors of a pie chart is:

(a) 90°    (b) 180°    (c) 270°    (d) 360°

3. In a pie chart, a category that is 25% of the data is shown by a sector of:

(a) 25°    (b) 90°    (c) 60°    (d) 45°

4. ₹600 is divided in the ratio 4 : 3 : 2 : 1. The largest share is:

(a) ₹60    (b) ₹120    (c) ₹240    (d) ₹400

5. If 6 workers finish a job in 8 days, how many days will 12 workers take (same rate)?

(a) 16    (b) 12    (c) 6    (d) 4

6. A car covers a fixed distance in 2 hours at 60 km/h. At 80 km/h it will take:

(a) 1 hour    (b) 1.5 hours    (c) 2.5 hours    (d) 3 hours

7. Which pair is in direct proportion?

(a) Speed and time for a fixed distance    (b) Number of taps and time to fill a tank    (c) Length of cloth and its cost    (d) Number of workers and days for a fixed job

8. A map has RF 1 : 60,00,000. A distance of 1 cm on the map represents:

(a) 6 km    (b) 60 km    (c) 600 km    (d) 6,000 km

9. In the table x : 16, ? and y : 9, 12 (inverse proportion), the missing x value is:

(a) 8    (b) 10    (c) 12    (d) 16

10. A triangle with sides in the ratio 1 : 3 : 5:

(a) is a right triangle    (b) is equilateral    (c) cannot be constructed    (d) is isosceles

Answer key: 1-(c), 2-(d), 3-(b), 4-(c), 5-(d), 6-(b), 7-(c), 8-(b), 9-(c), 10-(c).

For each Assertion–Reason question, choose: (A) Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion; (B) Both are true but the Reason is not the correct explanation; (C) Assertion is true but Reason is false; (D) Assertion is false but Reason is true.

A-R 1. Assertion: If 20 workers take 4 days for a job, 10 workers will take 8 days.

Reason: The number of workers and the number of days are in inverse proportion.

A-R 2. Assertion: In a pie chart, a sector of 90° represents one-fourth of the data.

Reason: The sector angle = (value ÷ total) × 360°.

A-R 3. Assertion: As the speed of a vehicle increases over a fixed distance, the travel time increases.

Reason: Speed and time over a fixed distance are inversely proportional.

A-R 4. Assertion: Two triangles with sides in the ratio 3 : 4 : 5 are always congruent.

Reason: Equal ratios of sides make triangles similar.

A-R 5. Assertion: The cost of pencils is directly proportional to the number of pencils bought.

Reason: At a fixed rate, doubling the number of pencils doubles the total cost.

Answer key: 1-(A), 2-(A), 3-(D), 4-(D), 5-(A).

Quick Revision Summary

  • Two ratios are proportional when a × d = b × c (cross-multiplication).
  • Ratios can have many terms; for every a of the first there are b, c, d… of the others.
  • To divide a whole x in ratio p : q : r…, each part = x × (its term) ÷ (sum of terms).
  • Pie-chart sector angle = (value ÷ total) × 360°; all angles add to 360°.
  • Direct proportion: x/y stays constant (both change by the same factor).
  • Inverse proportion: x × y = k; if x is multiplied by n, y is multiplied by 1/n.
  • RF on a map gives the ratio of map distance to real distance (1 : 60,00,000 → 1 cm = 60 km).

How to score full marks in this chapter

For every problem, first decide whether the quantities are directly or inversely proportional — write that line down before calculating. For pie charts, always show the angle as (value ÷ total) × 360° and verify the angles total 360°. Keep concrete units (km, ₹, days, minutes) in your working, and double-check word problems by asking “if one quantity goes up, does the other go up (direct) or down (inverse)?”

Frequently Asked Questions

What is Class 8 Maths Ganita Prakash Chapter 10 about?

Chapter 10 is Proportional Reasoning-2 (Chapter 3 of Ganita Prakash Part II). It covers proportional ratios, ratios in maps, ratios with many terms, dividing a whole in a ratio, drawing pie charts, and direct vs inverse proportion.

How do you find the sector angle for a pie chart?

Each sector angle = (value of that category ÷ total of all values) × 360°. The angles for all categories always add up to 360°.

What is the difference between direct and inverse proportion?

In direct proportion both quantities change by the same factor and x/y stays constant. In inverse proportion one increases as the other decreases, and the product x × y stays constant.

Are these Class 8 Maths Ganita Prakash Chapter 10 solutions free?

Yes. All solutions are free and follow the official NCERT Ganita Prakash (Part II) textbook for 2026–27.

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