Important Questions Class 8 Maths Chapter 7 Proportional Reasoning-1
Important Questions Class 8 Maths Chapter 7 Proportional Reasoning-1
Proportional Reasoning-1 is Chapter 7 of the Class 8 Maths book Ganita Prakash. It is about ratio and proportion. Students compare quantities using ratios, check whether two ratios are proportional, and solve real-life problems on direct proportion.
This page gives 12 important questions from Class 8 Maths Chapter 7 Proportional Reasoning-1 with step-by-step answers prepared by Extramarks experts. The questions cover ratios, proportions, sharing in a ratio, mixtures and recipes, and similar rectangles. Solve each question on paper first and then check your working with the answer.
Q1. (i) Are the ratios 3 : 4 and 72 : 96 proportional?
(ii) What is the HCF of 72 and 96?
Answer: (i) HCF of 3 and 4 = 1
HCF of 72 and 96 = 24 (∵ 72 = 24 × 3 and 96 = 24 × 4)
∴ The ratios 3 : 4 and 72 : 96 are proportional, because both ratios in their simplest form are the same.
(ii) HCF of 72 and 96 is 24.
Q2. Divide ₹ 4,500 into two parts in the ratio 2 : 3.
Answer: Given ratio = 2 : 3
Amount to be divided = ₹ 4,500
= 2 × 900 = ₹ 1,800
= 3 × 900 = ₹ 2,700
∴ The two parts are ₹ 1,800 and ₹ 2,700.
Verification: 1,800 : 2,700 = 18 : 27 = 2 : 3 and 1,800 + 2,700 = 4,500.
Q3. Prashanti and Bhuvan started a food cart business near their school. Prashanti invested ₹ 75,000 and Bhuvan invested ₹ 25,000. At the end of the first month, they gained a profit of ₹ 4,000. They decided that they would share the profit in the same ratio as their investment. What is each person’s share of the profit?
Answer: Investment of Prashanti = ₹ 75,000
Investment of Bhuvan = ₹ 25,000
∴ Ratio of investment = 75000 : 25000 = 3 : 1
Total profit = ₹ 4,000
= ₹ 3,000
= ₹ 1,000
Verification: ₹ 3,000 + ₹ 1,000 = ₹ 4,000, the total profit.
Q4. Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width-to-height ratio in your notebook? 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?
Answer: For the given rectangle, width = 32 mm and height = 18 mm. So, the ratio is 32 : 18.
We shall draw smaller and bigger rectangles similar to the given rectangle by using different ‘factors of change’.
A new, similar rectangle is shown in the figure.
Let the ‘factor of change’ be 2.
∴ New width = 2 × 32 = 64 mm and new height = 2 × 18 = 36 mm
The rectangles drawn by classmates may all be different, but they are not wrong. They are all similar to the given rectangle because their width-to-height ratio is the same.
Q5. 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.
Answer: Using a scale, we measure the width and height of the given rectangles.
The ratios ‘Width : Height’ for rectangles A, B, C, D and E are 1 : 3, 3 : 2, 9 : 4, 7 : 2 and 3 : 1 respectively.
The ratios of A and E are 1 : 3 and 3 : 1.
∴ For A and E, one side is 3 times the other side, so only rectangles A and E are similar (E is A turned on its side).
Q6. Shop A mixes orange pulp and water in the ratio 5:3. Shop B mixes orange pulp and water in the ratio 3:2. A restaurant mixes 8 litres of juice from Shop A and 10 litres of juice from Shop B to make a new juice. Find the total amount of orange pulp and water in the new mixture, and write the final ratio of orange pulp to water.
Answer: Shop A: 8 litres in the ratio 5 : 3. Total parts = 8, so each part = 1 litre.
Pulp = 5 × 1 = 5 litres, Water = 3 × 1 = 3 litres
Shop B: 10 litres in the ratio 3 : 2. Total parts = 5, so each part = 2 litres.
Pulp = 3 × 2 = 6 litres, Water = 2 × 2 = 4 litres
Total orange pulp = 5 + 6 = 11 litres; Total water = 3 + 4 = 7 litres
Hence, the final mixture has 11 litres of orange pulp and 7 litres of water, and the ratio is 11 : 7.
Q7. There are 2000 students in a school. 800 students opted basketball, 850 students opted cricket and the remaining opted table tennis. If a student can opt only one game, find the ratio of –
(a) number of students who opted basketball to the number of students who opted table tennis.
(b) number of students who opted cricket to the number of students who opted basketball.
Answer: Number of students for table tennis = 2000 – (800 + 850) = 2000 – 1650 = 350
(a) Basketball : Table tennis = 800 : 350 = 16 : 7
(b) Cricket : Basketball = 850 : 800 = 17 : 16
Q8. For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted?
Answer: For 120 students, the rice required is 15 kg. Let x kg of rice be required for 80 students.
∴ 120 : 15 :: 80 : x
⇒ x = 10
∴ 10 kg of rice is required.
Alternative Method:
⇒ 3x = 30 ⇒ x = 10
∴ 10 kg of rice is required.
Q9. 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?
Answer: Ratio of acid and water = 1 : 5
Quantity of solution = 240 mL
= 40 mL
= 200 mL
∴ The solution contains 40 mL of acid and 200 mL of water.
Verification:
Q10. Kesang wanted to make lemonade for a celebration. She made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to the drink. Her father expected more people to join the celebration. So he asked her to make 18 more glasses of lemonade. To make the lemonade with the same sweetness, how many spoons of sugar should she add? How can we find the factor of change in the ratio?
Answer: Number of glasses of lemonade = 6, number of spoons of sugar = 10
In the second case, the number of glasses of lemonade = 18. Let the number of spoons of sugar = x.
Since the sweetness is the same, the ratios 3 : 5 and 18 : x are in proportion.
⇒ 3x = 5 × 18 = 90
∴ Number of spoons of sugar = 30.
Number of glasses of lemonade in the first case = 6 and in the second case = 18.
Q11. A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹ 200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹ 800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive? Why?
Answer: We have 1 kg = 1000 g.
In Himachal Pradesh, 200 g of tea costs ₹ 200. Ratio of weight to price = 200 : 200 = 1 : 1
In Meghalaya, 1000 g of tea costs ₹ 800. Ratio of weight to price = 1000 : 800 = 5 : 4
So, the ratios are not proportional.
In Himachal Pradesh, 200 g costs ₹ 200, so 1000 g costs ₹ 1000.
Since ₹ 1000 > ₹ 800, the tea from Himachal Pradesh is more expensive.
Q12. 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?
Answer: Ratio of copper and nickel in the ₹ 10 coin = 3 : 1
Mass of one ₹ 10 coin = 7.74 grams
= 5.805 grams
= 1.935 grams
Cost of 1000 grams of copper = ₹ 906
Cost of 1000 grams of nickel = ₹ 1341
Cost of copper in the coin = 5.805 × 906 ÷ 1000 ≈ ₹ 5.26
Cost of nickel in the coin = 1.935 × 1341 ÷ 1000 ≈ ₹ 2.59
Total cost of metals in a ₹ 10 coin ≈ ₹ 5.26 + ₹ 2.59 = ₹ 7.85
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Q.1 Show that 189 is not a perfect cube.
Marks:1
Ans
Prime factors of 189 = 3 — 3 — 3 — 7
Making triplets, one triplet is formed 3 — 3 — 3, leaving one more factor 7.
So, 189 cannot be expressed as a product of triplets.
Hence, 189 is not a perfect cube.
Q.2 Find the cube root of the following by prime factorisation method.
(i) 8000
(ii) 13824
Marks:2
Ans
Q.3 Find the cube root of 32768 through estimation.
Marks:1
Ans
The given number is 32768.
Form groups of three starting from the rightmost digit of 32768.
In this case it is 768 and 32 has only 2 digits.
Taking 768, 8 in unit digit, the required cube root is 2.
The other group i.e. 32, as cube of 3 is 27 and cube of 4 is 64, 32 lies between 27 and 64.
The smallest among 3 and 4 is 3.
The ones place of 3 is itself.
Take 3 as tens place of the cube root of 32768.
Thus, cube root of 32768 is 32.
Q.4 Find the smallest number by which 704 must be divided to obtain a perfect cube.
Marks:2
Ans
CBSE Class 8 Maths Important Questions
FAQs (Frequently Asked Questions)
Proportional reasoning compares quantities that change by the same multiplicative factor. It is used in ratios, similar shapes, scaling, unit rates, recipes, mixtures, and map-like comparisons.
A ratio compares two quantities in a fixed order. For example, (15 : 35) can mean 15 mL coffee decoction for 35 mL milk.
Two ratios are proportional if their simplest forms are the same. They can also be checked using cross multiplication.
Rule of Three finds an unknown fourth quantity from three known proportional quantities. It uses the formula (d = (b × c) / a).
Add the ratio terms first. Divide the whole by that sum, then multiply each ratio term by the value of one part.