NCERT Solutions for class 7 Maths Chapter 8: Comparing QuantitiesExercise 8.11. Find the ratio of: (a) ₹ 5 to 50 paise Answer: 10: 1 Explanation: To find the ratio, both the units should be same. 1₹ = 100 paise 5₹ = 5 × 100 = 500 paise Ratio = 500/50 Ratio = 10/1 Ratio = 10: 1 (b) 15 kg to 210 g Answer: 500: 7 Explanation: To find the ratio, both the units should be same. 1 kg = 1000 g 15 kg = 15000 g Ratio = 15000/210 Ratio = (15 × 2 × 500)/ (15 × 2 × 7) Cancelling out the common terms, we get: Ratio = 500/7 Ratio = 500: 7 (c) 9 m to 27 cm Answer: 100: 3 Explanation: To find the ratio, both the units should be same. 1 m = 100 cm 9 m = 900 cm Ratio = 900 cm/ 27 cm Ratio = (9 × 100) (9 × 3) Ratio = 100/3 Ratio = 100: 3 (d) 30 days to 36 hours Answer: 20: 1 Explanation: To find the ratio, both the units should be same. 1 day = 24 hours 30 days = 24 × 30 = 720 hours Ratio = 720 hours/36 hours 720 = 36 × 20 Ratio = (36 × 20)/ 36 Ratio = 20/1 = 20: 1 2. In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students? Answer: 12 computers Explanation: Number of computers for 6 students = 3 Number of computers for 1 student = 3/6 = 1/2 Number of computers for 24 students = 1/2 × 24 = 24/2 = 12 Thus, 12 computers will be needed for 24 students. 3. Population of Rajasthan = 570 lakhs and population of UP = 1660 lakhs. Area of Rajasthan = 3 lakh km2 and area of UP = 2 lakh km2. (i) How many people are there per km2 in both these States? Answer: Rajasthan: 190 people; UP: 830 people Explanation: People per km2 = Population/ Total area in km2 People per km2 inRajasthan = Population of Rajasthan /Total area in km2 inRajasthan People per km2 inRajasthan = 570 lakhs/ 3 lakh People per km2 inRajasthan = 190 (570 = 190 × 3) People per km2 inUP = Population of UP /Total area in km2 inUP People per km2 inUP = 1660 lakhs/ 2 lakh People per km2 inUP = 830 (1660 = 830 × 2) (ii) Which State is less populated? Answer: Rajasthan Explanation: The area with less number of people per square kilometer is less populated. People per km2 inRajasthan = 190 People per km2 inUP = 830 People per km2 inRajasthan < People per km2 inUP Hence, Rajasthan is less populated. Exercise 8.21. Convert the given fractional numbers to per cents. (a) 1/8 Answer: 12.5 % Explanation: To convert a fraction to the percentage, we need to multiply the given fraction by 100. 1/8 × 100 = 100/8 = 12.5 % Thus, fraction 1/8 is equivalent to 12.5%. (b) 5/4 Answer: 125 % Explanation: To convert a fraction to the percentage, we need to multiply the given fraction by 100. 5/4 × 100 = (5 × 100)/4 = 500/4 = 125 % Thus, fraction 5/4 is equivalent to 125%. (c) 3/40 Answer: 7.5 % Explanation: To convert a fraction to the percentage, we need to multiply the given fraction by 100. 3/40 × 100 = (3 × 100)/40 = 300/40 = 7.5 % Thus, fraction 3/40 is equivalent to 7.5%. (d) 2/7 Answer: 7.5 % Explanation: To convert a fraction to the percentage, we need to multiply the given fraction by 100. 2/7 × 100 = (2 × 100)/7 = 200/7 = 28.57% Or 200/7 = 196/7 + 4/7 (28 × 7 = 196) 200/7 = 28 4/7% 2. Convert the given decimal fractions to per cents. (a) 0.65 Answer: 65% Explanation: 0.65 = 65/100 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 65/100 × 100 = 65% Thus, the decimal fraction 0.65 is equivalent to 65%. (b) 2.1 Answer: 210% Explanation: 2/1 = 21/10 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 21/10 × 100 = (21 × 100)/10 = 210 % Thus, the decimal fraction 2.1 is equivalent to 210%. (c) 0.02 Answer: 2 % Explanation: 0.02 = 2/100 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 2/100 × 100 = 2% Thus, the decimal fraction 0.02 is equivalent to 2%. (d) 12.35 Answer: 1235 % Explanation: 12/35 = 1235/100 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 1235/100 × 100 = (1235 × 100)/100 = 1235 % Thus, the decimal fraction 12.35 is equivalent to 1235%. 3. Estimate what part of the figures is coloured and hence find the per cent which is coloured. (i) Answer: 1/4, 25% Explanation: 1 part of the figure is coloured among the 4 parts. Fraction = Number of coloured parts/ Total number of parts Fraction = 1/4 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 1/4 × 100 = 100/4 = 25 % Thus, fraction 1/4 is equivalent to 25%. (ii) Answer: 3/5, 60% Explanation: 3 parts of the figure is coloured among the 5 parts. Fraction = Number of coloured parts/ Total number of parts Fraction = 3/5 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 3/5 × 100 = (3 × 100)/5 = 300/5 = 60 % Thus, fraction 3/5 is equivalent to 60%. (iii) Answer: 3/8, 37.5% Explanation: 3 parts of the figure is coloured among the 8 parts. Fraction = Number of coloured parts/ Total number of parts Fraction = 3/8 To convert a fraction to the percentage, we need to multiply the given fraction by 100. 3/8 × 100 = (3 × 100)/8 = 300/8 = 37.5 % Thus, fraction 3/8 is equivalent to 37.5%. 4. Find: (a) 15% of 250 Answer: 37.5 Explanation: of means multiplication 15 % = 15/100 15/100 × 250 = (15 × 250)/100 = 3750/100 = 37.5 (b) 1% of 1 hour Answer: 36 seconds or 3/5 minute Explanation: of means multiplication 1 % = 1/100 1/100 × 1 hour 1 hour = 60 minutes 1 minute = 60 seconds 1 hour = 60 × 60 = 3600 seconds 1/100 × 3600 = (1 × 3600)/100 = 3600/100 = 36 seconds Or 1 % = 1/100 1/100 × 1 hour 1 hour = 60 minutes = 1/100 × 60 = (1 × 60)/100 = 6/10 = 3/5 minutes (c) 20% of ₹ 2500 Answer: ₹ 500 Explanation: of means multiplication 20% = 20/100 = 20/100 × 2500 = (20 × 2500)/100 = ₹ 500 (d) 75% of 1 kg Answer: 750 g or 0.75 kg Explanation: of means multiplication 75 % = 75/100 1 kg = 1000 g 75/100 × 1000 = (75 × 1000)/ 100 = 750 g Or 75/100 × 1 = (75 × 1)/ 100 = 0.75 kg 5. Find the whole quantity if (a) 5% of it is 600. Answer: 12000 Explanation: of means multiplication Let the whole quantity be A. 5 % of A = 600 5% = 5/100 5/100 × A = 600 A = (600 × 100)/5 A = 12000 Thus, the whole quantity is equivalent to 12000. (b) 12% of it is ₹ 1080. Answer: ₹9000 Explanation: of means multiplication Let the whole quantity be A. 12 % of A = 1080 12% = 12/100 12/100 × A = 1080 A = (1080 × 100)/12 A = 9000 Thus, the whole quantity is equivalent to ₹9000. (c) 40% of it is 500 km. Answer: 1250 km Explanation: of means multiplication Let the whole quantity be A. 40 % of A = 500 40% = 40/100 40/100 × A = 500 A = (500 × 100)/40 A = 1250 km Thus, the whole quantity is equivalent to 1250 km. (d) 70% of it is 14 minutes. Answer: 20 minutes Explanation: of means multiplication Let the whole quantity be A. 70 % of A = 14 70% = 70/100 70/100 × A = 14 A = (14 × 100)/70 A = 20 minutes Thus, the whole quantity is equivalent to 20 minutes. (e) 8% of it is 40 litres. Answer: Explanation: of means multiplication Let the whole quantity be A. 40 % of A = 500 40% = 40/100 40/100 × A = 500 A = (500 × 100)/40 A = 1250 km Thus, the whole quantity is equivalent to 1250 km. 6. Convert given per cents to decimal fractions and also to fractions in simplest forms: (a) 25% Answer: 0.25, 1/4 Explanation: Simplest form refers to a form where the numerator and the denominator of the fraction have only 1 as the common factor. 25% = 25/100 = 0.25 Let's factorise the numbers in the numerator and denominator. = (5 × 5)/ (2 × 2 × 5 × 5) = 1/4 (b) 150% Answer: 1.5, 3/2 Explanation: Simplest form refers to a form where the numerator and the denominator of the fraction have only 1 as the common factor. 150% = 150/100 = 1.5 Let's factorise the numbers in the numerator and denominator. = (5 × 5 × 2 × 3)/ (2 × 2 × 5 × 5) = 3/2 (c) 20% Answer: 0.2, 1/5 Explanation: Simplest form refers to a form where the numerator and the denominator of the fraction have only 1 as the common factor. 20% = 20/100 = 0.2 Let's factorise the numbers in the numerator and denominator. = (5 × 2 × 2)/ (2 × 2 × 5 × 5) = 1/5 (d) 5% Answer: 0.05, 1/20 Explanation: Simplest form refers to a form where the numerator and the denominator of the fraction have only 1 as the common factor. 5% = 5/100 = 0.05 Let's factorise the numbers in the numerator and denominator. = (5)/ (2 × 2 × 5 × 5) = 1/20 7. In a city, 30% are females, 40% are males and remaining are children. What per cent are children? Answer: 30% Explanation: Percentage of females = 30 Percentage of males = 40 Remaining percentage = 100 - percentage of females - percentage of males = 100 - 30 - 40 = 30 Thus, the remaining percentage of children is 30%. 8. Out of 15,000 voters in a constituency, 60% voted. Find the percentage of voters who did not vote. Can you now find how many actually did not vote? Answer: 40%, 6000 Explanation: Percentage of voters who vote = 60 Percentage of voters who did not vote = 100 - percentage of voters who vote = 100 - 60 = 40% Number of voters who did not vote = 40% of 15000 = 40/100 × 15000 = (40 × 15000)/ 100 = 6000 9. Meeta saves ₹4000 from her salary. If this is 10% of her salary, what is her salary? Answer: ₹40,000 Explanation: Let the Meeta's total salary be A. 10% of A = ₹4000 10/100 × A = 4000 A = (4000 × 100)/10 A = 40,000 Thus, the total salary of Meeta is ₹40,000. 10. A local cricket team played 20 matches in one season. It won 25% of them. How many matches did they win? Answer: 5 matches Explanation: Number of matches own by the cricket team = 25 % of the total played matches = 25 % of 20 = 25/100 × 20 = (25 × 20)/100 = 500/100 = 5 Thus, the cricket team won 5 numbers of matches. Exercise 8.31. Tell what is the profit or loss in the following transactions. Also find profit per cent or loss per cent in each case (a) Gardening shears bought for ₹ 250 and sold for ₹ 325. Answer: Profit = ₹ 75; Profit % = 30 Explanation: Cost price of the gardening shears (CP) = ₹ 250 Selling price of the gardening shears (SP) = ₹ 325 If SP > CP, there is a profit. Profit = SP - CP Profit = ₹ 325 - ₹ 250 Profit = ₹ 75 Profit % = Profit/CP × 100 Profit % = 75/250 × 100 Profit % = 30 Thus, the profit is ₹ 75 and the profit percentage is 30%. (b) A refrigerator bought for ₹ 12,000 and sold at ₹ 13,500. Answer: Profit = ₹ 1500; Profit % = 30 Explanation: Cost price of the refrigerator (CP) = ₹ 12,000 Selling price of the refrigerator (SP) = ₹ 13,500 If SP > CP, there is a profit. Profit = SP - CP Profit = ₹ 13500 - ₹ 12000 Profit = ₹ 1500 Profit % = Profit/CP × 100 Profit % = 1500/12000 × 100 Profit % = 12.5 Thus, the profit is ₹ 1500 and the profit percentage is 12.5%. (c) A cupboard bought for ₹ 2,500 and sold at ₹3,000. Answer: Profit = ₹ 500; Profit % = 20 Explanation: Cost price of the cupboard (CP) = ₹ 2500 Selling price of the cupboard (SP) = ₹ 3000 If SP > CP, there is a profit. Profit = SP - CP Profit = ₹ 3000 - ₹ 2500 Profit = ₹ 500 Profit % = Profit/CP × 100 Profit % = 500/2500 × 100 Profit % = 20 Thus, the profit is ₹ 500 and the profit percentage is 20%. (d) A skirt bought for ₹250 and sold at ₹ 150. Answer: Loss = ₹ 100; Loss % = 20 Explanation: Cost price of the skirt (CP) = ₹ 250 Selling price of the skirt (SP) = ₹ 150 If SP < CP, there is a loss. Loss = CP - SP Loss = 250 - 100 Loss = ₹ 100 Loss % = Loss/CP × 100 Loss % = 100/250 × 100 Loss % = 40 Thus, the loss is ₹ 100 and the loss percentage is 40 %. 2. Convert each part of the ratio to percentage: (a) 3 : 1 Answer: 75%, 25% Explanation: Ratio percentage of the first number = Given ratio/ Total × 100 Ratio percentage = 3/ (3 + 1) × 100 Ratio percentage = 3/4 × 100 Ratio percentage = 75 % Ratio percentage of the second number = Given ratio/ Total × 100 Ratio percentage = 1/ (3 + 1) × 100 Ratio percentage = 1/4 × 100 Ratio percentage = 25 % (b) 2: 3: 5 Answer: 20%, 30%, 50% Explanation: Ratio percentage of the first number = Given ratio/ Total × 100 Ratio percentage = 2/ (2 + 3 + 5) × 100 Ratio percentage = 2/10 × 100 Ratio percentage = 20 % Ratio percentage of the second number = Given ratio/ Total × 100 Ratio percentage = 3/ (2 + 3 + 5) × 100 Ratio percentage = 3/10 × 100 Ratio percentage = 30 % Ratio percentage of the third number = Given ratio/ Total × 100 Ratio percentage = 5/ (2 + 3 + 5) × 100 Ratio percentage = 5/10 × 100 Ratio percentage = 50 % (c) 1:4 Answer: 20%, 80% Explanation: Ratio percentage of the first number = Given ratio/ Total × 100 Ratio percentage = 1/ (1 + 4) × 100 Ratio percentage = 1/5 × 100 Ratio percentage = 20 % Ratio percentage of the second number = Given ratio/ Total × 100 Ratio percentage = 4/ (1 + 4) × 100 Ratio percentage = 4/5 × 100 Ratio percentage = 80 % (d) 1: 2: 5 Answer: 12.5 %, 25%, 62.5 % Explanation: Ratio percentage of the first number = Given ratio/ Total × 100 Ratio percentage = 1/ (1 + 2 + 5) × 100 Ratio percentage = 1/8 × 100 Ratio percentage = 12.5 % Ratio percentage of the second number = Given ratio/ Total × 100 Ratio percentage = 2/ (1 + 2 + 5) × 100 Ratio percentage = 2/8 × 100 Ratio percentage = 25 % Ratio percentage of the third number = Given ratio/ Total × 100 Ratio percentage = 5/ (1 + 2 + 5) × 100 Ratio percentage = 5/8 × 100 Ratio percentage = 62.5 % 3. The population of a city decreased from 25,000 to 24,500. Find the percentage decrease. Answer: 2% Explanation: Difference = 25000 - 24500 = 500 Percentage decrease = Difference in the population/Original population × 100 Percentage decrease = 500/25000 × 100 Percentage decrease = 2% 4. Arun bought a car for ₹ 3,50,000. The next year, the price went up to ₹3,70,000. What was the Percentage of price increase? Answer: 5 5/7% or 5.55 % Explanation: Cost price of the car (CP) = ₹ 3,50,000 Selling price of the car (SP) = ₹3,70,000 SP > CP Hence, there is a profit. Increased amount = ₹3,70,000 - ₹ 3,50,000 Increased amount = ₹ 20,000 Percentage increase = Increase amount/CP × 100 Percentage increase = 20,000/3,50,000 × 100 Percentage increase = 5.55% Or Percentage increase = 40/7 % = 35/7 % + 5/7 % = 5% + 5/7% = 5 5/7% 5. I buy a T.V. for ₹ 10,000 and sell it at a profit of 20%. How much money do I get for it? Answer: ₹ 12,000 Explanation: Cost price of the TV (CP) = ₹ 10,000 Profit % = 20 20 % of 10,000 = 20/100 × 10000 Profit = ₹ 2000 Selling Price of TV = CP + Profit Selling Price of TV = ₹ 10,000 + ₹ 2000 Selling Price of TV = ₹ 12,000 6. Juhi sells a washing machine for ₹ 13,500. She loses 20% in the bargain. What was the price at which she bought it? Answer: Explanation: Selling price of the washing machine (SP) = ₹ 13,500 Loss % = 20 Cost price of the washing machine = [(100)/ (100 - Loss %) × SP = 100/80 × 13,500 = (100 × 13,500)/ 80 = ₹ 16875 Thus, the washing machine was bought at the price of ₹ 16875. 7. (i) Chalk contains calcium, carbon and oxygen in the ratio 10:3:12. Find the percentage of carbon in chalk. Answer: 12 % Explanation: Percentage of carbon = Ratio of carbon/ Total ratio × 100 Percentage of carbon = 3/ (10 + 3 + 12) × 100 Percentage of carbon = 3 / 25 × 100 Percentage of carbon = (3 × 100)/25 = 12 % Thus, the percentage of carbon in the chalk is 12 %. (ii) If in a stick of chalk, carbon is 3g, what is the weight of the chalk stick? Answer: 25 g Explanation: Let the weight of chalk stick be A. Percentage of carbon in the chalk = 12 % 12 % of A = 3 g 12/100 × A = 3 A = (3 × 100)/ 12 A = 25 g Thus, the weight of the chalk stick is 25 g. 8. Amina buys a book for ₹ 275 and sells it at a loss of 15%. How much does she sell it for? Answer: ₹ 233.75 Explanation: Cost price of the book = ₹ 275 Loss % = 15 % Loss = Loss % × CP Loss = 15 % × 275 Loss = 15/100 × 275 Loss = (15 × 275)/100 Loss = ₹ 41.25 Selling price of the book = Cost price - Loss Selling price of the book = ₹ 275 - ₹ 41.25 Selling price of the book = ₹ 233.75 Thus, Amina sold the book for ₹ 233.75. 9. Find the amount to be paid at the end of 3 years in each case: (a) Principal = ₹ 1,200 at 12% p.a. Answer: ₹ 1,632 Explanation: Amount = principal + Interest Interest = (P × R × T)/100 Principal = ₹ 1,200 Rate = 12 % Time = 3 years Interest = (1200 × 12 × 3)/ 100 Interest = ₹ 432 Amount = principal + Interest Amount = ₹ 1,200 + ₹ 432 Amount = ₹ 1,632 (b) Principal = ₹ 7,500 at 5% p.a. Answer: ₹ 8,625 Explanation: Amount = principal + Interest Interest = (P × R × T)/100 Principal = ₹ 7,500 Rate = 5 % Time = 3 years Interest = (7500 × 5 × 3)/ 100 Interest = ₹ 1125 Amount = principal + Interest Amount = ₹ 7,500 + ₹ 1125 Amount = ₹ 8,625 10. What rate gives ₹ 280 as interest on a sum of ₹ 56,000 in 2 years? Answer: 0.25 % Explanation: Interest = (P × R × T)/100 Principal = ₹ 56,000 Rate =? Time = 2 years Interest = ₹ 280 R = (Interest × 100)/ (P × T) R = (280 × 100)/ (56,000 × 2) R = 1/4 % R = 0.25 % Thus, 0.25 % is the rate that gives ₹ 280 as interest on a sum of ₹ 56,000 in 2 years. 11. If Meena gives an interest of ₹45 for one year at 9% rate p.a. What is the sum she has borrowed? Answer: ₹ 500 Explanation: Interest = (P × R × T)/100 Principal =? Rate = 9% Time = 1 year Interest = ₹ 45 P = (Interest × 100)/ (R × T) P = (45 × 100)/ (9 × 1) P =₹ 500 Thus, Meena has borrowed rupees 500 that gives an interest of ₹45 for one year at 9% rate p.a. |