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Solutions

Overview

Step-by-step NCERT solutions for Proportional Reasoning-1 (Chapter 7, NCERT Class 8 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Proportional Reasoning-1 textbook chapter.

Solved

What these solutions cover

All 26 questions in Proportional Reasoning-1 are solved in the PDF. Here's what's inside, exercise by exercise:

Problem Solving with Proportional Reasoning

  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
  2. Give 3 ratios that are proportional to 4 : 9.
  3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ___ 12 : ___ 20 : ___ 27 : ___
  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 shown in the figure.)
  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?
  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.
    • (a) [wall pattern with grey and red coloured bricks]
    • (b) [wall pattern with grey and brown coloured bricks]
  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?

Trairasika — The Rule of Three

  1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a 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. [Diagram shows an L-shaped house: outer top wall = 15 ft, outer left wall =…

Sharing, but Not Equally!

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

Unit Conversions

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

Worked examples

3 questions from Proportional Reasoning-1, solved step by step.

Problem Solving with Proportional Reasoning

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

Two ratios a : b and c : d are proportional if ad = bc (cross multiplication).

(i) 4 : 7 :: 12 : 21 — Check: 4 × 21 = 84 and 7 × 12 = 84. Equal, so TRUE.

(ii) 8 : 3 :: 24 : 6 — Check: 8 × 6 = 48 and 3 × 24 = 72. Not equal, so FALSE.

(iii) 7 : 12 :: 12 : 7 — Check: 7 × 7 = 49 and 12 × 12 = 144. Not equal, so FALSE.

(iv) 21 : 6 :: 35 : 10 — Check: 21 × 10 = 210 and 6 × 35 = 210. Equal, so TRUE.

Alternatively, 21 : 6 in simplest form = 7 : 2, and 35 : 10 in simplest form = 7 : 2. Same, so TRUE.

(v) 12 : 18 :: 28 : 12 — Check: 12 × 12 = 144 and 18 × 28 = 504. Not equal, so FALSE.

(vi) 24 : 8 :: 9 : 3 — Check: 24 × 3 = 72 and 8 × 9 = 72. Equal, so TRUE.

Alternatively, both ratios simplify to 3 : 1. TRUE.

Answer: (i), (iv), and (vi) are true.

Trairasika — The Rule of Three

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

1 year = 52 weeks.

Distance in 1 year = 940 million km = 940,000,000 km.

Set up proportion: the ratio of time to distance must be proportional.

52 weeks : 940,000,000 km :: 1 week : x km

By cross multiplication:

52 × x = 940,000,000 × 1

x = 940,000,000 ÷ 52

x = 18,076,923 km (approximately, since 940,000,000 / 52 = 235,000,000 / 13 ≈ 18,076,923.08)

Answer: The Earth travels approximately 18,076,923 km in a week.

Sharing, but Not Equally!

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

Sum of ratio terms = 2 + 3 = 5.

Size of each group = 4500 ÷ 5 = ₹900.

First part = 2 × 900 = ₹1,800.

Second part = 3 × 900 = ₹2,700.

Check: 1800 + 2700 = 4500 ✓

Check ratio: 1800 : 2700 = 2 : 3 ✓

Answer: The two parts are ₹1,800 and ₹2,700.

All 26 questions in Proportional Reasoning-1 are solved step by step in the Solutions PDF above.

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More solutions in Ganita Prakash

01A Square and a Cube02Power Play03A Story of Numbers04Quadrilaterals05Number Play06We Distribute, Yet Things Multiply08Fractions in Disguise09The Baudhayana-Pythagoras Theorem10Proportional Reasoning-211Exploring Some Geometric Themes12Tales by Dots and Lines13Algebra Play14Area
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More NCERT Solutions for Class 8

Read the Proportional Reasoning-1 textbook chapter / PDF, or browse all NCERT Class 8 Mathematics solutions.

← PreviousCh 6: We Distribute, Yet Things MultiplyNext →Ch 8: Fractions in Disguise

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