RATIO, PROPORTION AND UNITARY METHOD

8 RATIO, PROPORTION AND UNITARY METHOD 1. The males and females in a village are in the ratio 8 : 3. If the number of females is 405. How many males are there in the village? Ans. Let number of males be x x : 405 = 8 : 3 x 8 = 405 3 x = 8 405 3 x = 8 135 x = 1080 Hence, number of males in the village is1080. 2. An alloy contains copper and zinc in the ratio 7 : 3. A certain quantity of alloy contains 12.6 gm of copper. How much does this alloy weigh? Ans. Let the weight of zinc be x gm. 12.6 : x = 7 : 3 12.6 7 = 7x = 3 12.6 x 3 x = 3 12.6 7 x = 3 1.8 = 5.4 Weight of zinc = 5.4 gm Weight of alloy = (12.6 + 5.4) gm = 18 gm. 3. Divide Rs 1625 among Anu, Vinay and Priya in the ratio 5: 7 : 13. Ans. Sum of ratio = 5 + 7 + 13 = 25 5 Anu share = Rs 1625 = Rs 5 65 = Rs 325 25 7 Vinay s share = Rs 1625 = Rs 7 65 = Rs 455 25 Priya s share = Rs 13 1625 25 = Rs 13 65 = Rs 845 ICSE Math Class VII 1 Question Bank

4. Divide 35 Chocolates between Sunil and Kamal in the ratio 1 1 : 2 3. Ans. Ratio = 1 : 1 2 3 = 3 : 2 Sum of the terms of ratios = 3 + 2 = 5 Sunil s share = 3 35 3 7 21 5 = = chocolates Kamal s share = 2 35 2 7 14 5 = = chocolates 5. Divide 468 gm of rice into three heaps containing the quantities in the ratio 1 : 1 : 1. 6 8 12 Ans. Ratio = 1 : 1 : 1 6 8 12 = 4 : 3 : 2 Sum of the terms of ratios = 4 + 3 + 2 = 9 4 468 gm = 4 52 gm = 208 gm 9 First heap contains = ( ) 3 468 gm = 4 52 gm = 156 gm 9 Second heap contains = ( ) 2 Third heap contains = 468 gm = ( 2 52 ) gm = 104 9 6. The angles of triangle are in the ratio 5 : 6 : 7. Find each angle. Ans. We know that the sum of the angles of a triangle = 180 Sum of the terms of the ratio = 5 + 6 + 7 = 18 First angle = 5 180 5 10 50 18 = = Second angle = 6 180 6 10 60 18 = = Third angle = 7 180 7 10 70 18 = = 7. Divide Rs 760 among Ajay, Vijay and Chitra such that Ajay gets 5 of what Vijay gets 6 and ratio between the shares of Vijay and Chitra is 3 : 4. Ans. Let the amounts of Ajay, Vijay and Chitra be a, b and c respectively then a = 5 6 b ICSE Math Class VII 2 Question Bank

a 5 and b = a : b = 5 : 6 6 and b : c = 3 : 4 Then, a : b : c = 15 : 18 : 24 a : b : c = 5 : 6 : 8 Sum of the terms of ratio = 5 + 6 + 8 = 19 Ajay s share = 5 760 5 40 19 = = Rs 200 Vijay s share = 6 760 6 40 19 = = Rs 240 Chitra s share = 8 760 8 40 19 = = Rs 320 8. Divide Rs 715 among A, B and C in such a way that B gets three times as much as A gets and C gets half of what B gets. Ans. B = 3A B 3 A = 1 B : A = 3 : 1 A : B = 1 : 3 1 C 1 and C = B 2 B = C : B = 1:2 B : C = 2 : 1 2 A : B :C = 2 : 6 : 3 Sum of the terms of ratio = 2 + 6+ 3 = 11 A s share = 2 715 2 65 11 = = Rs 130 B s share = 16 715 6 65 11 = = Rs 390 C s share = 3 715 3 65 11 = = Rs 195 9. Out of daily income of Rs 60, a labourer spends Rs 45 on food and shelter and saves the rest. Find the ratio of his (i) spending to income (ii) saving to income (iii) saving to spending Ans. Daily income = Rs 60 Income spend on food and shelter = Rs 45 ICSE Math Class VII 3 Question Bank

Savings = Rs(60 45) = Rs 15 (i) Required ratio = Rs 45 : Rs 60 = 45 60 = 15 20 = 3 4 = 3 : 4 (ii) Required ratio = Rs 15 : Rs 60 = 15 60 = 5 20 = 1 1: 4 4 = (iii) Required ratio Rs 15 : Rs 45 = 15 45 = 5 15 = 1 1: 3 3 = 10. 5 grams of an alloy contains by weight of nickel to copper. Ans. Total weight of an alloy = 5 gm Weight of copper = 3 3 4 grams copper and the rest is nickel. Find the ratio 3 3 gm = 15 4 4 gm Weight of nickel = 5 1 gm 15 4 gm = 20 15 4 gm = 5 4 gm Ratio of weight of nickel to weight of copper = 5 4 gm :15 4 gm = 5 15 = 5 4 4 4 4 = 1 1: 3 15 3 = 11. Rajesh gets Rs 120 per month as pocket money. He spends one third of it on ice creams, one half on comic books and remaining on wafers. Find the ratio of the money spent on each item. Ans. Total pocket money = Rs 120 Amount spends on ice creams = 1 3 of Rs 120 = Rs 1 120 = Rs 40 3 Amount spends on comics = 1 2 of Rs 120 = Rs 1 120 = Rs 60 2 Amount spends on wafers = Rs 120 (Rs 40 + Rs 60) = Rs 120 Rs 100 = Rs 20 Required ratio = 40 : 60 : 20 = 2 : 3 : 1 12. If a : b = 5 : 6 and b : c = 2.8 : 3.5, find a : c. Ans. a :b = 5 : 6 a 5 b = b 2.8 and b : c = 2.8 : 3.5 6 c = 3.5 ICSE Math Class VII 4 Question Bank

13. If p : q = Ans. p : q = a a b 5 2.8 a : c = c = b c = 6 3.5 = 4 = 2 = 6 3 2 : 3 1 1 1 :1 3 2 and q : r = 1 : 1, 2 3 1 1 1 :1 3 2 find p : r. p : q = 4 : 3 4 p 3 4 2 = = 3 2 q 3 3 3 2 p 8 q = 1 1 and q : r = : 9 2 3 1 q 1 3 2 q 3 = = r 1 2 1 r = 2 3 p p q p : r = = r q r = 8 3 = 4 = 4 : 3 9 2 3 p : r = 4 : 3 1 2 14. If l : m = 2 :1 2 3 and m : n = 1 1 1 : 3, find l : m : n. 4 2 Ans. l : m m : n 1 2 5 5 l : m = 2 :1 = : 2 3 2 3 l 5 5 5 3 3 m = 2 3 = 2 5 = 2 1 1 5 7 m : n = 1 : 3 = : 4 2 4 2 m 5 2 5 n = 4 7 = 14 Then l : m = 3 : 2 = 3 5 : 2 5 = 15 : 10 m : n = 5 : 14 = 5 2 : 14 2 = 10 : 28 Hence, l : m : n = 15 : 10 : 28 15. Two numbers are in the ratio 5 : 8 and their difference is 12. Find the number. Ans. Ratio in two numbers = 5 : 8 (given) ICSE Math Class VII 5 Question Bank

Let first number be 5x then second number be 8x According to questions 8x 5x = 12 3x = 12 x = 12 4 3 = First number = 5x = 5 4 = 20 and second number = 8x = 8 4 = 32 Hence, numbers are 20 and 32 16. Two numbers are in the ratio 9 : 13 and their sum is 176. Find the numbers. Ans. Ratio in two numbers = 9 : 13 (given) Let first number be 9x then second number = 13x But their sum = 176 9x + 13x = 176 22x = 176 = 8 First number = 9x = 9 8 = 72 and second number = 13x = 13 8 = 104 Hence, numbers are 72 an 104 17. There are 60 toffees in a jar. You are offered the option of getting two third or three fifth of the total free of cost. Which option would you choose? Ans. Total number of toffees = 60 Case I Toffees I will get = 2 3 of 60 = 2 60 2 20 40 3 = = toffees Case II Toffees I will get = 3 of 60 = 3 12 = 36 toffees 5 Since, 40 toffees are more than 36 toffees, Hence, I will choose the first option. 18. Heights of Anshul, Ankita and Dhruv are 1.04 m and 91 cm respectively. Divide 100 sweets among them in the ratio of their heights (1 m = 100 cm) Ans. Height of Anshul = 1.04 m = (1.04 100) cm = 104 cm Height of Ankita = 1.30 m (given) = (1.30 100) cm = 130 cm Height of Dhruv = 91 cm (given) Ratio of their heights = 104 cm : 130 cm : 91 cm = 104 : 130 : 91 = 8 : 10 : 7 Sum of the terms of the ratio = 8 + 10 + 7 = 25 Anshul s share = 8 25 100 ICSE Math Class VII 6 Question Bank

Ankita s share = 10 25 of 100 = 10 100 25 = 1000 40 25 = sweets Dhruv s share = 7 7 700 of 100 = 100 = = 28 sweets 25 25 25 19. A rectangular field is 100 m by 80 m. Find the ratio of (i) length to its breadth (ii) breadth to its perimeter. Ans. Length of field (l) = 100 m (given) Breadth (b) = 80 m (given) Perimeter of field = 2 (l + b) = 2 (100 + 80) m = 2 180 = 360 m (i) Ratio of length to its breadth = 100 : 80 = 5 : 4 (ii) Ratio of breadth to its perimeter = 80 : 360 = 2: 9 20. The sum of three numbers, whose ratios are is 1639. Find the numbers. Ans. Sum of three numbers = 1639 Ratio between them = 1 1 1 3 : 4 : 6 3 5 8 1 1 1 3 : 4 : 6 3 5 8 10 21 49 400 : 504 : 735 = : : = (L C M of 3, 5, 8, = 120) 3 5 8 120 = 400: 504 : 735 Sum of the terms of ratio s = 400 + 504 + 735 = 1639 Ist number = 400 1639 of 1639 = 400 II nd number = 504 of 1639 = 504 1639 and III rd number = 735 of 1639 = 735 1639 21. The first, second and fourth terms of a proportion are 36,81 and 63 respectively. Find the third term. Ans. Let third term be x. Then 36,81, 63 are in proportion Product of extremes = product of means 36 63 = 81 x 36 63 x = = 4 7 = 28 81 Hence, the third term is 28. ICSE Math Class VII 7 Question Bank

22. Find the fourth proportional to : (i) 0.6, 1.5, 3 (ii) 1, 2,6 (iii) 3hrs 12 min, 24 min, 1 m 68 cm 3 5 Ans. (i) Let the fourth proportional be x then 0.6 : 1.5 : : 3: x Product of extremes = Product of means 0.6 x = 3 1.5 3 1.5 4.5 x = = = 7.5 0.6 0.6 The fourth proportional is 7.5 (ii) Let the fourth proportional be x then 1 2 : :: 6 : x 3 5 Product of extremes = Product of means 1 2 x = 6 3 5 x = 2 6 3 = 36 5 1 5 x = 7.2 The fourth proportional is 7.2. (iii) Let the fourth proportional be x then 3 hrs 12 min : 24 min :: 1 m 68 cm : x 3hrs12 min 1m 68cm = 24 min x (3 60 + 12) (100 + 68)cm 192 168 min = = cm 24 min x 24 x 168 24 x = cm = 21 cm 192 23. Find the third proportional to : (i) 1 2, 9 3 (ii) 1 kg 250 gm, 500 gm (i) Let the third proportional be x then 1 2 2 : : : : 9 3 3 x Product of extremes = Product of means ICSE Math Class VII 8 Question Bank

1 2 2 x = x = 4 9 9 3 3 9 = 4 The third proportional is 4 (ii) Let the third proportional be x, then 1 kg 250 gm : 500 gm : : 500 gm : x 1250 gm : 500 gm : : 500 gm : x Product of extremes = Product of means 1250 gm x = 500 gm 500 gm x = 500gm 500gm 1250 gm The third proportional is 200 gm. 24. Find the means proportion between : (i) 1.8 and 0.2 (ii) 1 2 and 1 32 = 200 gm (iii) 2 3 and 8 32 Ans. (i) Mean proportion between 1.8 and 0.2 = 1.8 02 = 0.36 = 0.6 (ii) Mean proportion between 1 2 and 1 32 = 1 1 1 1 = = 2 32 64 8 (iii) Mean proportion between 2 3 and 8 27 = 2 8 16 4 = = 3 27 81 9 25. A distance of 68 km is represented on a map by 1.7 cm. What distance is represented by 8.5 cm on the same map? Ans. Let the distance represented by 8.5 cm on map be x km then 68 : x : : 1.7 : 8.5 Product of extremes = Product of Means 68 8.5 = x 1.7 x 1.7 = 68 8.5 68 8.5 x = = 68 5 x = 340 1.7 The distance 8.5 cm on map represents 340 km 26. A map showing the area of certain towns has a scale showing 1 : 500000. What area will be represented on this map of a town of area 200000 m 2? Ans. Let x be the required area then 1 : 500000 : : x : 200000 Product of means = Product of extremes 500000 x = 1 200000 ICSE Math Class VII 9 Question Bank

x = 1 200000 = 2 x = 0.4 500000 5 Required area = 0.4 m 2. 27. 8 kg of tomatoes cost Rs 54. What is the cost of 14 kg of tomatoes? Ans. Cost of 8 kg of tomatoes = Rs 54 Cost of 1 kg of tomatoes = Rs 54 8 Cost of 14 kg of tomatoes = Rs 54 14 8 27 7 189 = = Rs 2 2 = Rs 94.50 28. 12 apples weigh 1 kg 50 gm. How many apples will weigh 4 kg 725 gm? Ans. 1 kg 50 gm = (1000 + 50) gm = 1050 gm and 4 kg 725 gm = (4000 + 725) gm = 4725 gm 1050 gm is the weight of = 12 apples 12 1 gm is the weight of = 1050 apples 12 4725 gm is the weight of = 4725 1050 apples = 2 4725 175 apples = 54 apples. 29. A scooter consumes 2.5 litres of petrol in covering a distance of 85 km. What distance will it cover in 8 litres of petrol? Ans. Distance covered in 2.5 litres of petrol = 85 km 1 litre of petrol = 85 2.5 85 8 litres of petrol = 8 2.5 km 680 10 1360 = km km = 272 km 25 5 30. A journey of 124 km costs Rs 465. How much will a journey of 240 km cost? Ans. For a journey of 124 km, the cost is Rs 465 For a journey of 1 km the cost will be = Rs 465 124 And for a journey of 240 km, the cost will be 465 465 60 = Rs 240 = Rs = Rs15 60 = Rs 900 124 31 ICSE Math Class VII 10 Question Bank

31. A typist takes 1 hour 45 minutes to type 24 pages. How long will he takes to type 64 pages? 45 3 7 Ans. For typing 24 pages, time taken = 1 hour 45 min = 1 = 1 = hours 60 4 4 7 For typing 1 pages, time taken = hrs 4 24 7 64 7 2 14 An for typing 64 pages, time taken = = = hrs 4 24 3 3 = 2 4 3 hrs = 4 hrs 2 60 3 minutes = 4 hrs 40 minutes 32. If 3 5 metre of a cloth cost Rs 36, what will be the cost of 1 4 Ans. Cost of 3 5 m of cloth = Rs 36 Cost of 1 m of cloth = Rs 36 5 3 And cost of 1 4 m of cloth = Rs 36 5 1 = Rs 15 3 4 metre of this cloth? 33. 30 workers can dig a trench in 6 days. How many workers will required to dig it in 5 days? Ans. In 6 days, a trench is dig by = 30 workers In 1 day, it will be dug by = 30 6 workers In 5 days = 30 6 workers = 36 workers 5 34. In a fort 360 men hen had provisions for 21 days. If 60 more men join them, how long will the provisions last? For 360 men provisions will lasted for = 21 days For 1 man the provisions will last for = (21 360) days For (360 + 60) i.e. 420 men, the provision will last for 21 360 = 420 days = 360 18 20 = days 35. In an army camp, 300 soldiers had provisions for 13 days. If 40 of them are transferred to the other camp how long will the provisions last? ICSE Math Class VII 11 Question Bank

Ans. For 300 soldiers, the provisions is for = 13 days For 1 soldier, the provisions is for = 13 300 days And for (300 40) i.e. 260 soldiers the 13 300 provisions if for = 260 days = 300 = 15 days 20 36. Moving at the rate of 70 km/hr, a car completes a journey in 18 minutes. How long would it take to complete this journey if the speed is increased to 84 km/h? Ans. If the speed of car is 70 km/hr, car completes journey in = 18 min If the speed of car is 1 km/hr, car will complete the journey in = 18 70 min If the speed of car is 84 km/hr, car will complete the journey in 18 70 18 5 = min = min = 15 min. 84 6 37. 7 workers working 10 hours a day, can complete a work in 12 days. In how many days will 5 workers do it working 8 hours a day? Ans. 7 workers working 10 hours in a day can complete the work in 12 days 1 worker working 10 hours in a day will complete the work in = (12 7) days 1 worker working one hour in a day can complete the work in = (12 7 10) days 5 workers working 8 hours in a day will complete the work in = 12 7 10 days = 3 7 days = 21 days 5 8 38. A typist takes 80 minutes to type 24 pages. How long will he take to type 87 pages? Ans. For typing 24 pages time taken is = 80 minutes For typing 1 page time taken is = 80 24 minutes For typing 87 pages, time taken 80 87 = minutes = 290 minutes 24 39. 400 men have provisions for 23 weeks. They are joined by 60 men. How long will the provisions last? Ans. 400 men have provisions for = 23 weeks 1 man will have provisions for = (23 400) weeks ICSE Math Class VII 12 Question Bank

= (400 + 60) men i.e.460 men will have provisions for = 23 400 460 weeks = 20 weeks 40. A bus consumes 25 litres of diesel in covering a distance of 90 kilometres. How much diesel is needed to cover 288 kilometres? Ans. Diesel needed to cover 90 kilometres = 25 litres Then diesel needed to cover 1 kilometre = 25 90 litres 25 Diesel needed to cover 288 kilometres = 288 90 litres = 7200 90 litres = 80 litres 41. If 15 men can pack 540 parcels per day, how many men are needed to pack 396 parcels per day? Ans. Men needed to pack 540 parcels = 15 15 Then men needed to pack 1 parcel = 540 men 15 Men needed to pack 396 parcel = 396 540 men = 5940 men = 11 men 540 42. 200 men have provisions for 30 days. If 50 men left, the same provisions would last for the remaining men in how many days? Ans. 200 men have provisions for = 30 days 1 man will have provisions for = 30 200 days (200 50) men i.e. 150 men will have provisions for 30 200 = days = 40 days 150 43. 8 men can finish a certain amount of provisions in 40 days. If 2 more men join with them, find in how many days the same amount of provisions be sufficient? Ans. 8 men can finish a provision in = 40 days 1 man will finish it in (40 8) days (8 + 2) men i.e = 10 men will finish it in 40 8 10 days = 32 days 44. If the interest on Rs 200 be Rs 25 in a certain time what will be the interest on Rs 750 for the same time? Ans. Interest on Rs 200 is = Rs 25 ICSE Math Class VII 13 Question Bank

25 Interest on Rs 1 will be = Rs 200 and interest on Rs 750 will be = Rs 25 750 200 = Rs 750 8 = Rs 93.75 45. If 3 dozen eggs cost Rs 30, find the cost of 3 scores of eggs. (1score =20) Ans. 3 dozen = (3 12) eggs = 36 eggs 3 scores of eggs = (3 20) = 60 eggs The cost of 36 eggs is = Rs 30 30 The cost of 1 egg will be = Rs = Rs 50 36 Cost of 60 eggs will be = 30 60 36 = Rs 50 46. If 9 lines of a print in a column of a book, contains 36 words. How many words will a column of 51 lines contain? Ans. In 9 lines of print, number of words = 36 In 1 line of print words number of = 36 9 In 51 lines of print, number of words = 36 51 204 9 = 47. A wrist watch loses 10 sec in every 8 hours, in how much time will it lose 15 sec.? Ans. 10 sec is lost in = 8 hours 1 sec will be lost in = 8 10 hours 8 15 sec. will be lost in = 15 hours = 12 hours 10 48. In 2 days and 20 hours, a watch gains 20 sec : find how much time will the watch take to gain 35 sec? Ans. 2 days 20 hours = (2 24 + 20) = (48 + 20) = 68 hours Now 20 sec. are gained in = 68 hours 1 sec, will be gained in = 68 20 hours 68 35 sec will be gained in = 35 20 hours = 119 hours = (119 24) days = 4 days 23 hours ICSE Math Class VII 14 Question Bank

49. 50 men mow 32 hectares of land in 3 days. How many days will 15 men take to mow it? Ans. Land is same in both the cases. Now 50 men can mow land of 32 hectares in = 3 days 1 man will mow it in = 3 50 days 15 men will mow it in = 3 50 15 days = 10 days 50. The wages of 10 workers for a six days week is Rs 1200. What are the one day wages of 4 workers? Ans. 10 workers earn in 6 days = Rs 1200 1200 Amount is 1 worker will earn in 6 days = Rs 10 1200 Amount is 1 worker will earn in 1 day = Rs = Rs 20 10 6 4 workers will earn in 1 days = Rs (20 4) = Rs 80 ICSE Math Class VII 15 Question Bank