Class 8 Maths Ganita Prakash Chapter 7 Solutions (NCERT 2026–27) – Proportional Reasoning

These Class 8 Maths Ganita Prakash Chapter 7 solutions cover Proportional Reasoning from the new NCF-2023 textbook (Part I, Reprint 2026–27). In the book this chapter is titled “Proportional Reasoning-1” because it is the Part I half of the topic; Part II continues it as “Proportional Reasoning-2.” Every “Figure it Out” question is solved step by step, with the values cross-checked against the NCERT answer key, so you can master ratios, proportion and the Rule of Three.

Class: 8 Subject: Mathematics Book: Ganita Prakash (Part I) Chapter: 7 Exercises: 4 “Figure it Out” sets Session: 2026–27
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Chapter 7 Overview

Chapter 7 of Ganita Prakash, Proportional Reasoning-1, starts from a simple everyday observation — why some resized images look natural while others look stretched — and builds it into the powerful idea of proportion. You learn what a ratio a : b means, how to reduce a ratio to its simplest form using the HCF, and how to test whether two ratios are proportional (a : b :: c : d) using cross multiplication (ad = bc). The chapter then applies this to real problems through the ancient Indian Trāirāśika or Rule of Three, to sharing a quantity in a given ratio, and to unit conversions. The Class 8 Maths Ganita Prakash Chapter 7 solutions below solve every exercise so each method is crystal clear.

Key Concepts & Definitions

Ratio (a : b): a comparison of two quantities of the same kind — for every ‘a’ units of the first quantity there are ‘b’ units of the second. The numbers a and b are the terms of the ratio.

Simplest form: a ratio with its terms divided by their HCF (e.g. 60 : 40 = 3 : 2).

Proportion: two ratios are proportional (written a : b :: c : d) when they are equal in their simplest forms, i.e. their terms change by the same factor.

Cross multiplication: a : b :: c : d is true exactly when ad = bc. This lets us find an unknown fourth term.

Rule of Three (Trāirāśika): given three of the four terms in a proportion, the fourth is found by Āryabhaṭa’s rule — multiply the phala by the ichchhā and divide by the pramāṇa.

Sharing in a ratio: a quantity x divided in the ratio m : n gives parts of size m × x/(m+n) and n × x/(m+n).

Direct proportion: as one quantity increases, the other increases by the same factor (more students ⇒ proportionally more rice). Not every situation is proportional — e.g. higher speed gives less travel time.

Important Formulas (Chapter 7)

Simplest form: a : b = (a÷HCF) : (b÷HCF)

Proportion test: a : b :: c : d  ⇔  a × d = b × c (cross multiplication)

Missing term (Rule of Three): if a : b :: c : d, then d = (b × c) ÷ a

Sharing x in the ratio m : n: first part = m × x/(m+n), second part = n × x/(m+n)

Useful conversions: 1 m = 3.281 ft • 1 acre = 43,560 sq ft • 1 hectare = 10,000 m2 = 2.471 acres • 1 L = 1000 mL = 1000 cc • 1 tonne = 1000 kg

Figure it Out (Page 165)

1. Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21    (ii) 8 : 3 :: 24 : 6 (iii) 7 : 12 :: 12 : 7    (iv) 21 : 6 :: 35 : 10 (v) 12 : 18 :: 28 : 12    (vi) 24 : 8 :: 9 : 3

SOLUTION Use cross multiplication: a : b :: c : d is true when a×d = b×c. (i) 4×21 = 84 and 7×12 = 84 → equal, TRUE. (ii) 8×6 = 48 and 3×24 = 72 → not equal, false. (iii) 7×7 = 49 and 12×12 = 144 → not equal, false. (iv) 21×10 = 210 and 6×35 = 210 → equal, TRUE. (v) 12×12 = 144 and 18×28 = 504 → not equal, false. (vi) 24×3 = 72 and 8×9 = 72 → equal, TRUE. ∴ The true statements are (i), (iv) and (vi).

2. Give 3 ratios that are proportional to 4 : 9.

SOLUTION Multiply both terms by the same number to keep the ratio proportional. 4×2 : 9×2 = 8 : 18;   4×3 : 9×3 = 12 : 27;   4×4 : 9×4 = 16 : 36. (Any ratios obtained by multiplying both terms by the same factor are correct.)

3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______    12 : ______    20 : ______    27 : ______

SOLUTION First reduce 18 : 24 to its simplest form: HCF(18, 24) = 6, so 18 : 24 = 3 : 4. Every proportional ratio has its second term = (4/3) × first term. (i) First term 3: second = 4/3 × 3 = 4 → 3 : 4. (ii) First term 12: second = 4/3 × 12 = 16 → 12 : 16. (iii) First term 20: second = 4/3 × 20 = 80/3 → 20 : 80/3. (iv) First term 27: second = 4/3 × 27 = 36 → 27 : 36.

4. Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios. (Rectangles A, B, C, D, E are shown in the textbook figure.)

SOLUTION Method: Two rectangles are similar when their width-to-height ratios are equal in simplest form. Measure each rectangle’s width and height with a scale, write the ratio for each, reduce it to simplest form, and group the rectangles that share the same simplest-form ratio. Those rectangles with matching width : height ratios are similar; the ones with different ratios are not. (Exact grouping depends on the printed measurements in your copy.)

5. Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates’ drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong?

SOLUTION Yes. Keep the width-to-height ratio the same and just multiply both measurements by a factor smaller than 1 (for a smaller rectangle) or greater than 1 (for a bigger one). Classmates’ rectangles may be different sizes but all share the same shape because the ratio is unchanged. They satisfy the proportion, so they are not wrong — they are all similar to the given rectangle.

6. The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form.

SOLUTION Wall (a): in one repeating block, grey bricks = 2 + 3 + 4 = 9 and coloured bricks = 3 + 2 + 1 = 6. Ratio = 9 : 6 = 3 : 2 (dividing both by HCF 3). Wall (b): in one repeating block, grey bricks = 16 and coloured bricks = 12. Ratio = 16 : 12 = 4 : 3 (dividing both by HCF 4).

7. Let us draw some human figures. Measure your friend’s body—the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below— head : torso, torso : arms, torso : legs. Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above. Does the drawing look more realistic if the ratios are proportional? Why? Why not?

SOLUTION This is a measuring activity, so the numbers depend on the friend you measure. Record the three ratios (head : torso, torso : arms, torso : legs) in simplest form. Yes, the drawing looks more realistic when the ratios are kept proportional to the measured body, because the human figure then has the correct relative sizes of its parts — just like similar rectangles look natural while distorted ones do not.

Figure it Out (Page 170)

1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?

SOLUTION There are 52 weeks in a year, and 940 million km = 940,000,000 km. Distance per week = 940,000,000 ÷ 52 ≈ 18,076,923 km (about 1.81 crore km) in one week.

2. A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that separates two rooms. To build a wall of 10-feet, he requires approximately 1450 bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness. (The plan shows segments of 12 ft, 15 ft, 9 ft, 9 ft and 6 ft, with the inner dividing wall.)

SOLUTION Brick rate: 10 ft of wall needs 1450 bricks, so the ratio is 10 : 1450. Total length of all walls (outer walls + inner dividing wall) from the plan = 12 + 12 + 12 + 15 + 9 + 15 + 9 + 9 + 9 + 6 = 108 ft. Set up the proportion 10 : 1450 :: 108 : x. x = (1450 × 108) ÷ 10 = 156600 ÷ 10 = 15,660 bricks.

Figure it Out (Page 175)

1. Divide ₹4,500 into two parts in the ratio 2 : 3.

SOLUTION Total number of groups = 2 + 3 = 5. Size of one group = 4500 ÷ 5 = ₹900. First part = 2 × 900 = ₹1,800;   second part = 3 × 900 = ₹2,700. (Check: 1800 + 2700 = 4500.)

2. In a science lab, acid and water are mixed in the ratio of 1 : 5 to make a solution. In a bottle that has 240 mL of the solution, how much acid and water does the solution contain?

SOLUTION Total groups = 1 + 5 = 6. Size of one group = 240 ÷ 6 = 40 mL. Acid = 1 × 40 = 40 mL;   water = 5 × 40 = 200 mL. (Check: 40 + 200 = 240 mL.)

3. Blue and yellow paints are mixed in the ratio of 3 : 5 to produce green paint. To produce 40 mL of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 mL of yellow to the mixture. What is the new ratio of blue and yellow in the paint?

SOLUTION Total groups = 3 + 5 = 8. Size of one group = 40 ÷ 8 = 5 mL. Blue = 3 × 5 = 15 mL;   yellow = 5 × 5 = 25 mL. After adding 20 mL of yellow: yellow = 25 + 20 = 45 mL, blue is still 15 mL. New ratio blue : yellow = 15 : 45 = 1 : 3 (dividing both by HCF 15).

4. To make soft idlis, you need to mix rice and urad dal in the ratio of 2 : 1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?

SOLUTION Total groups = 2 + 1 = 3. Size of one group = 6 ÷ 3 = 2 cups. Rice = 2 × 2 = 4 cups;   urad dal = 1 × 2 = 2 cups. (Check: 4 + 2 = 6 cups.)

5. I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of 3 : 5. I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?

SOLUTION Take the orange bucket as 1 whole. Red = 3/8 of a bucket and yellow = 5/8 of a bucket. Adding one full bucket of yellow: yellow = 5/8 + 1 = 13/8; red stays 3/8. New ratio red : yellow = (3/8) : (13/8) = 3 : 13.

Figure it Out (Page 176)

1. Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.

SOLUTION Ratio = 600 : 900. HCF(600, 900) = 300. 600÷300 : 900÷300 = 2 : 3.

2. Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?

SOLUTION Capacity of one bus = 162 ÷ 3 = 54 people. Buses for 204 people = 204 ÷ 54 = 3.78…, so we cannot use a fraction of a bus → we need 4 buses. Seats in 4 buses = 4 × 54 = 216. Since 216 − 204 = 12, not all buses will be full — 12 seats will be vacant.

3. The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?

SOLUTION Compare the population density (people per sq. km) of each city. Delhi: 30,000,000 ÷ 1,484 ≈ 20,216 people/sq km. Mumbai: 20,000,000 ÷ 550 ≈ 36,364 people/sq km. Mumbai has more people in each square kilometre, so Mumbai is more crowded.

4. A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?

SOLUTION Neck : rest of body = 4 : 6, so neck is 4 out of every (4 + 6) = 10 parts of the total height. That is, neck = (4/10) × height. Neck height = (4/10) × your height. For example, if your height is 150 cm, your neck would be (4/10) × 150 = 60 cm. (Use your own measured height.) (For the crane itself this gives neck = (4/10) × 155 = 62 cm.)

5. Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If 2½ palas of saffron costs 3/7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?”

SOLUTION Set up the proportion (palas) : (niskas) → 2.5 : (3/7) :: x : 9. By cross multiplication: (3/7) × x = 2.5 × 9 = 22.5. x = 22.5 × (7/3) = 22.5 × 7 ÷ 3 = 157.5 ÷ 3 = 52.5 palas. So 52.5 palas of saffron can be bought for 9 niskas.

6. Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is 1 : 2?

SOLUTION Let this happen after x years. Then Harmain is (1 + x) and her brother is (5 + x). (1 + x) : (5 + x) = 1 : 2 ⇒ 2(1 + x) = 1(5 + x) ⇒ 2 + 2x = 5 + x ⇒ x = 3. So after 3 years their ages are 4 and 8 (ratio 4 : 8 = 1 : 2). Harmain will be 4 years old.

7. The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?

SOLUTION For equal volumes, mass of gold : mass of water = 37 : 2. Let 1 L of gold weigh x kg, while 1 L of water = 1 kg. 37 : 2 :: x : 1 ⇒ 2x = 37 × 1 ⇒ x = 37 ÷ 2 = 18.5 kg.

8. It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter).

SOLUTION 10 tonnes = 10,000 kg, and 1 acre = 43,560 sq ft. So manure : area = 10,000 : 43,560 (kg per sq ft). Area of plot = 200 × 500 = 100,000 sq ft. Manure x: x : 100,000 :: 10,000 : 43,560 ⇒ x = (100,000 × 10,000) ÷ 43,560 ≈ 22,956.8 kg (about 22.96 tonnes).

9. A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?

SOLUTION 10 litres = 10,000 mL, and one mug = 500 mL, so the bucket = 10,000 ÷ 500 = 20 mugs. Time = 20 × 15 = 300 seconds = 5 minutes.

10. One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?

SOLUTION 1 acre = 43,560 sq ft costs ₹15,00,000. Cost is proportional to area: 43,560 : 15,00,000 :: 2,400 : x. x = (15,00,000 × 2,400) ÷ 43,560 = 360,00,00,000 ÷ 43,560 ≈ ₹82,645 (approx.).

11. A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?

SOLUTION Oxen take 6 hours per acre, so for 20 acres: 20 × 6 = 120 hours with a pair of oxen. The tractor is 4 times faster, so it needs one-fourth of the time: per acre = 6 ÷ 4 = 1.5 hours. For 20 acres with the tractor: 20 × 1.5 = 30 hours (equivalently 120 ÷ 4 = 30 hours).

12. The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?

SOLUTION Split 7.74 g in the ratio 3 : 1. Total groups = 4, so one group = 7.74 ÷ 4 = 1.935 g. Copper = 3 × 1.935 = 5.805 g;   nickel = 1 × 1.935 = 1.935 g. Cost of copper = ₹906 per kg = ₹906 per 1000 g, so for 5.805 g: (906 ÷ 1000) × 5.805 ≈ ₹5.26. Cost of nickel = ₹1,341 per 1000 g, so for 1.935 g: (1341 ÷ 1000) × 1.935 ≈ ₹2.59. Total cost of metals in one coin ≈ 5.26 + 2.59 = ₹7.85 (approx.).

Common Mistakes to Avoid

Watch out for these

  • Thinking that equal differences keep things proportional — only equal factors (multiplication) do (image A vs B in the chapter).
  • Comparing two quantities in different units — convert first (e.g. 1 kg = 1000 g, 4 hours = 240 minutes) before forming the ratio.
  • Mixing up the order of the terms in cross multiplication for a : b :: c : d — the rule is ad = bc.
  • When sharing x in the ratio m : n, dividing by m or n instead of by (m + n) to get the group size.
  • Assuming every “more/less” situation is direct proportion — higher speed gives less time, so the Rule of Three does not apply there.
  • Forgetting to reduce the final ratio to its simplest form.

Practice MCQs & Assertion–Reason

1. The simplest form of the ratio 60 : 40 is:

(a) 6 : 4    (b) 2 : 3    (c) 3 : 2    (d) 5 : 3

2. Two ratios a : b and c : d are proportional if:

(a) a + d = b + c    (b) ad = bc    (c) ac = bd    (d) a − b = c − d

3. Which of these is proportional to 4 : 9?

(a) 8 : 27    (b) 9 : 4    (c) 12 : 27    (d) 13 : 18

4. If ₹4,500 is divided in the ratio 2 : 3, the smaller part is:

(a) ₹900    (b) ₹1,800    (c) ₹2,700    (d) ₹2,250

5. In the proportion 6 : 10 :: 18 : x, the value of x is:

(a) 24    (b) 28    (c) 30    (d) 36

6. To divide a quantity x in the ratio m : n, the first part is:

(a) x/(m+n)    (b) m × x/(m+n)    (c) m × x/n    (d) (m+n)/x

7. The ancient Indian name for Rule of Three problems is:

(a) Lilavati    (b) Trāirāśika    (c) Bijaganita    (d) Aryabhatiya

8. 1 acre is equal to:

(a) 10,000 sq ft    (b) 43,560 sq ft    (c) 2.471 sq ft    (d) 1,000 sq ft

9. Acid and water are mixed in the ratio 1 : 5. In 240 mL of solution, the amount of acid is:

(a) 40 mL    (b) 48 mL    (c) 200 mL    (d) 60 mL

10. As the speed of a vehicle increases, the time taken to cover a fixed distance:

(a) increases proportionally    (b) decreases    (c) stays the same    (d) becomes zero

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

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: The ratios 4 : 7 and 12 : 21 are proportional.

Reason: Two ratios a : b and c : d are proportional when ad = bc.

A-R 2. Assertion: Adding the same number to both terms of a ratio always keeps it proportional to the original.

Reason: A ratio stays proportional only when both terms are multiplied (or divided) by the same factor.

A-R 3. Assertion: When 12 counters are shared in the ratio 3 : 1, the two shares are 9 and 3.

Reason: The size of each group is the total divided by (m + n).

A-R 4. Assertion: A problem of speed and time for a fixed distance can be solved by the Rule of Three.

Reason: Time decreases as speed increases, so speed and time are not in direct proportion.

A-R 5. Assertion: The ratio 60 : 40 in its simplest form is 3 : 2.

Reason: A ratio is reduced to simplest form by dividing both terms by their HCF.

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

Quick Revision Summary

  • A ratio a : b compares two quantities; a and b are its terms.
  • Reduce a ratio to simplest form by dividing both terms by their HCF.
  • Two ratios are proportional (a : b :: c : d) when they are equal in simplest form, i.e. ad = bc.
  • Rule of Three: given three terms, the fourth = (b × c) ÷ a.
  • Sharing x in the ratio m : n → parts are m × x/(m+n) and n × x/(m+n); group size = x ÷ (m + n).
  • Equal multiplication keeps a ratio proportional; equal addition/subtraction usually does not.
  • Always convert to the same units before forming or comparing ratios.
  • Not every situation is direct proportion — e.g. speed vs time for a fixed distance is inverse.

How to score full marks in this chapter

State the test you are using (simplest form or cross multiplication) before you calculate, and always check the answer by substituting it back into the proportion. Convert units first, show the group size when sharing in a ratio, and reduce every final ratio to its simplest form. For “Rule of Three” word problems, ask yourself whether the quantities really grow together — if more of one means less of the other, do not use direct proportion.

Frequently Asked Questions

What is Class 8 Maths Ganita Prakash Chapter 7 about?

Chapter 7, titled “Proportional Reasoning-1” in the textbook, teaches ratios, simplest form, proportion and cross multiplication, the ancient Rule of Three (Trāirāśika), sharing a quantity in a given ratio, and useful unit conversions. Part II of the book continues the topic as “Proportional Reasoning-2.”

How do you check if two ratios are proportional?

Reduce both ratios to simplest form — if they are equal, they are proportional. The quick test is cross multiplication: a : b :: c : d is true exactly when a × d = b × c.

How many exercises are there in Ganita Prakash Chapter 7?

There are four “Figure it Out” sets (on pages 165, 170, 175 and 176) along with several worked Examples, Math Talk and Try This boxes. All the Figure it Out questions are solved on this page.

Are these Class 8 Maths Ganita Prakash Chapter 7 solutions free?

Yes. All solutions are free and follow the official NCERT Ganita Prakash (Part I) textbook for 2026–27, with every calculation cross-checked against the textbook answer key.

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