:: Ratio and Proportion

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1 :: Ratio and Proportion 1. In a college, the ratio of the number of boys to girls is 8 :. If there are 160 girls, the total number of students in the college is:. [Central Excise ] Answer: 416 Let the number of boys and girls be 8x and x. Total number of students = 13x = 13 x 32 = The ratio of two numbers is 3 : 4 and their sum is 420. The greater of the two numbers is :. [Railways 1991] Answer: 240 Required number = (420 * (4/7)) = If a+b : b+c : c+a = 6 : 7 : 8 and a + b + c = 14, then the value of c is :. [C.B.I. 1993] Answer: 14 (a/3) = (b/4) = (c/7) then a = 3k, b = 4k, c = 7k a+b+c/c = 3k+4k+7k/7k = 14k/7k = 2 View Answer Report Error Let us discuss 4. If 1% of x is the same as 20% of y, then x : y is :. [C.B.I. 1990] 3 : 4 17 : 16 4 : 3 16 : 17 Answer: 4 : 3 1% of x = 2O% of y 1x/100 = 20y/100 x/y = 4/3. A sum of Rs. 427 is to be divided among A, B and C such that 3 times A s share, 4 tunes B s share and 7 times C s share are all equal. The share of C is : (Assistant Grade 1994) Rs. 84. Rs. 196 Rs. 147 Rs. 240 Answer: Rs A = 4B = 7C = k,then A = k/3, B = k/4 and C= k/7. A : B : C = k/3 : k/4 : k/7 = 28:21 :12. Cs share = Rs. [427 x (12/61)] = Rs An alloy is to contain copper and zinc in the ratio 9 :4. The zinc required (in kg) to be melted with 24 kg of copper, is 7 10 and 2/3 9 and 2/3 10 and 1/3 9

2 Answer: 10 and 1/3 9:4: 24:x 9x = 4 * 24 x = (4*24)/9 = 32/3 Kg. hence `0 and 1/3 7. Rs. 730 were divided among A, B, C in such a way that if A gets Rs. 3, then B gets Rs. 4 and if B gets Rs. 3.0 then C gets Rs. 3. The share of B exceeds that of C by :. [Central Excise ] Rs. 30 Rs. 70 Rs. 40 Rs. 210 Answer: Rs. 40 A:B = 3:4 and B:C = 7/2:3 = (8/7)*(7/2)*(8/7)*3 = 4:(24/7) A : B : C = 3 :4: 24/7 = 21 : 28 : 24. Bs share = Rs. [730 *(28/73)]= Rs C s share = Rs. [730 * (24/73)] = Rs Difference of their shares = If 0.4 : 1.4 :: 1.4 : x, the value of x is :. [Teachers Exam 1] Answer: * x = 1.4 * 1.4 x = (1.4*1.4)/0.4 = What number should be subtracted from both the terms of the ratio 1 : 19 so as to make it as 3 : 4? (Assistant Grade 1994) Answer: 3 Let x be subtracted. Then, (1 - x) / (19 - x) = 3/4 4(1 - x) = 3(19 - x) x = If p : q = 3 : 4 and q : r= 8 : 9, then p : r is :. [Assistant Grade] 1: 3 2 : 3 3 : 2 1: 2 Answer: 2 : 3 p/r = (p/q) * (q/r) = (3/4) * (8/9) = 2/3 so p : q = 2:3 11. The salaries of A, B, and C are in the ratio of 1 : 2 : 3. The salary of B and C together is Rs By what percent is the salary of C more than that of A? (Bank P.O. 1992) 100% 300% 200% 600% Answer: 600% Let the salaries of A, B, C hex, 2x and 3x respectively. Then,2x + 3x = 6000 = x = As salary = Rs. 1200, Bs salary = Rs. 2400, and Cs salary Rs Excess of Cs salary over As=[(2400/1200)x100] = 200%. 12. An amount of money is to be distributed among F, Q and R in the ratio 3 : : 7. If Qs share is Rs. 100, what is

3 the difference between Ps and Rs shares?. [Bank P.O. 1994] Rs Rs Rs, 100 Rs Answer: Rs Let P = 3x, Q = x and R = 7x. Then, x = 100 x = 300. P=900,Q=100 and R = 21OO. Hence, (R - p) = ( ) = What number should be added to each of the numbers 8, 21, 13 and 31 so that the resulting numbers, in this order form a proportion?. [U.D.C. 1993] Answer: (8+x)/(21+x) = (13+x)/(31+x) Then, (8 + x)(31 + x) = (13 + x)(21 + x) or39x = 34x or x=2 or x =. 14. The ratio of income of A to that of B is : 4 and the expenditure of A to that of B is 3: 2. If at the end of the year, each saves Rs, 800, the income of A is :. [Assistant Grade] Rs Rs Rs Rs Answer: Rs Let the incone of A and B be x and 4x and. the expenditures of A and B be 3y and 2y. Then, x 3y = 800 and 4x 2y= 800. On solving we get : x = 400. As income = x = Rs A profit of Rs is to be distributed among A, B, C in the proportion 3 : : 7. What will be the difference between B s and C s shares?. [Bank P.O. 1993] Rs Rs Rs Rs Answer: Rs Bs share = Rs *(/1) = Rs C s share = Rs * (7/1) = Rs.14000, Difference in Bs and Cs shares = Rs Rs. 120 are divided among A, B, C such that A s share is Rs. 20 more than B s and Rs. 20 less than C s. What is B s share.. [Railways 1991] Rs. 10 Rs. 20 Rs. 1 Rs. 2 Answer: Rs. 20 Let C = x. Then A = (x 20) and B = (x 40). x + x x - 40 = 120 Or x=60. A:B:C = 40:20:60 = 2:1 :3. Bs share = Rs. 120*(1/6) = Rs If 7 : x = 17. : 22., then the value of x is:. [Assistant grade] 9 6

4 7.. Answer: 9 7*22. = x*17. x = 7 * 22./17. x = X, Y and Z share a sum of money in the ratio 7 : 8 : 16. If Z receives Rs. 27 more than X, then the total money shared was:. [Central Excise ] Rs. 48 Rs. 279 Rs. 93 Rs. 8 Answer: Rs. 93 Let X = 7x, Y = 8x & Z = 16x. Then, total money = 31x. Now, Z - X = 27 so 16x 7x = 27 that is why x = 3. Total money 31*x = Rs A dog takes 3 leaps for every leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of the speed of the dog to that of the hare is :. [U.D.C. 1993] 8: 8:7 9: 9:7 Answer: 9: Dog : Hare = (3*3) leaps of hare : leaps of hare = 9 :. 20. The least whole number which when subtracted from both the terms of the ratio 6 : 7 to give a ratio less than 16 : 21, is. [Central Excise ] Answer: 3. Let x is subtracted. Then, ((6 - x)/(7 - x)) < 16 / 21 21(6 x) < 16(7 x) x > 14 = x > 2.8. Least such number is The compounded ratio of (2 : 3), (6: 11) and (11 :2) is. [Assistant Grade] 1: 2 11: 24 2: 1 36: 121 Answer: 2: 1 required ratio = (2/3) * () * (6/11) * (11/2) = 2/1 22. A sum of money is to the divided among F, Q andr in the ratio of 2 : 3 :. If the total share of P andr together is Rs 400 more than that of Q, what is R s share in it?. [Bank P.O. 1991] Rs. 400 Rs. 600 Rs. 00 Rs. 70 Answer: Rs. 00 Let P = 2x, Q = 3x and R=x. Now P+R-Q = 400 2x+x-3x = 400 hence x =1OO R = x = A certain amount was divided between Salim and Rahim in the ratio of 4 : 3. If Rahim s share was Rs. 2400, the total amount was. [Bank P.O. 1988]

5 Rs. 600 Rs Rs Rs Answer: Rs. 600 Let S = 4x and R = 3x. Total amount = 7x. Then, 3x = 2400 so x= 800. Total amount = 7x = Rs If three numbers in the be:. [Teacher s Exam.] ratio 3 : 2: be such that the sum of their squares is 1862, the middle number will Answer: 14 Let the numbers be 3x, 2x and x. Then, 9x + 4x + 2x = x = 1862 x = 49 x = 7. middle number = 2x = Pencils, Pens and Exercise books in a shop are in the ratio of 10: 2 : 3. If there are 120 pencils, the number of exercise books in the shop is:. [Assistant Grade] Answer: 36 Let Pencils = 10x, Pens = 2x & Exercise books = 3x. Now, 10x = 120 hence x = 12. Number of exercise books = 3x = A and B together have Rs If amount does B have? of A's amount is equal to of B's amount, how much A. Rs. 460 B. Rs. 484 C. Rs. 0 D. Rs. 664 Answer: Option B 4 2 A = B A = x 4 B A = 3 B 2 A = 3 B 2 A : B = 3 : 2. B's share = Rs x 2 = Rs Two numbers are respectively 20% and 0% more than a third number. The ratio of the two numbers is:

6 A. 2 : B. 3 : C. 4 : D. 6 : 7 Let the third number be x. Then, first number = 120% of x = 120x = 6x 100 Second number = 10% of x = 10x = 3x x 3x Ratio of first two numbers = : = 12x : 1x = 4 : A sum of money is to be distributed among A, B, C, D in the proportion of : 2 : 4 : 3. If C gets Rs more than D, what is B's share? A. Rs. 00 B. Rs. 100 C. Rs D. None of these Let the shares of A, B, C and D be Rs. x, Rs. 2x, Rs. 4x and Rs. 3x respectively. Then, 4x - 3x = 1000 x = B's share = Rs. 2x = Rs. (2 x 1000) = Rs Seats for Mathematics, Physics and Biology in a school are in the ratio : 7 : 8. There is a proposal to increase these seats by 40%, 0% and 7% respectively. What will be the ratio of increased seats? A. 2 : 3 : 4 B. 6 : 7 : 8 C. 6 : 8 : 9 D. None of these Answer: Option A Originally, let the number of seats for Mathematics, Physics and Biology be x, 7xand 8x respectively. Number of increased seats are (140% of x), (10% of 7x) and (17% of 8x). 140 x x, 10 x 7x and 17 x 8x x, 21x and 14x. 2 The required ratio = 7x : 21x : 14x 2 14x : 21x : 28x

7 2 : 3 : 4.. In a mixture 60 litres, the ratio of milk and water 2 : 1. If the this ratio is to be 1 : 2, then the quanity of water to be further added is: A. 20 litres B. 30 litres C. 40 litres D. 60 litres Answer: Option D Quantity of milk = 60 x 2 3 litres = 40 litres. Quantity of water in it = (60-40) litres = 20 litres. New ratio = 1 : 2 Let quantity of water to be added further be x litres. Then, milk : water = 40 Now, = x x x = 80 x = 60. Quantity of water to be added = 60 litres. 6. The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio? A. 8 : 9 B. 17 : 18 C. 21 : 22 D. Cannot be determined Originally, let the number of boys and girls in the college be 7x and 8x respectively. Their increased number is (120% of 7x) and (110% of 8x). 120 x 7x and 110 x 8x x and 44x The required ratio = 42x : 44x = 21 : Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 7. What is Sumit's salary? A. Rs. 17,000 B. Rs. 20,000 C. Rs. 2,00 D. Rs. 38,000

8 Answer: Option D Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively. Then, 2x = 40 3x (2x ) = 40(3x ) 6x = 68,000 3x = 34,000 Sumit's present salary = (3x ) = Rs.( ) = Rs. 38, If 0.7 : x :: : 8, then x is equal to: A B C. 1.2 D Answer: Option B (x x ) = (0.7 x 8) x = 6 = The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to the third is : 8, then the second number is: A. 20 B. 30 C. 48 D. 8 Answer: Option B Let the three parts be A, B, C. Then, A : B = 2 : 3 and B : C = : 8 = x 3 : 8 x 3 = A : B : C = 2 : 3 : 24 = 10 : 1 : 24 B = 98 x 1 = : If Rs. 782 be divided into three parts, proportional to : :, then the first part is: A. Rs. 182 B. Rs. 190 C. Rs. 196 D. Rs. 204 Answer: Option D

9 Given ratio = : : = 6 : 8 : 9. 1 st part = Rs. 782 x 6 = Rs The salaries A, B, C are in the ratio 2 : 3 :. If the increments of 1%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries? A. 3 : 3 : 10 B. 10 : 11 : 20 C. 23 : 33 : 60 D. Cannot be determined Let A = 2k, B = 3k and C = k. A's new salary = k of 2k = x 2k = B's new salary = k of 3k = x 3k = C's new salary = of k = x k = 6k kk 33k New ratio : : 6k = 23 : 33 : If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number? A. 2 : B. 3 : 7 C. : 3 D. 7 : 3 Let 40% of A = 2 B 3 Then, 40A = 2B A 2B = 3 A = 2 x B 3 2 = 3 A : B = : The fourth proportional to, 8, 1 is: A. 18 B. 24 C. 19 D. 20 Answer: Option B Let the fourth proportional to, 8, 1 be x.

10 Then, : 8 : 1 : x x = (8 x 1) x = (8 x 1) = Two number are in the ratio 3 :. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is: A. 27 B. 333 C. 49 D. Answer: Option B Let the numbers be 3x and x. Then, 3x - 9 = 12 x (3x - 9) = 12(x - 9) 9x = 99 x = 11. The smaller number = (3 x 11) = In a bag, there are coins of 2 p, 10 p and p in the ratio of 1 : 2 : 3. If there is Rs. 30 in all, how many p coins are there? A. 0 B. 100 C. 10 D. 200 Let the number of 2 p, 10 p and p coins be x, 2x, 3x respectively. Then, sum of their values = Rs. 2x + 10 x 2x + x 3x = Rs. 60x x = x = 30 x 100 = Hence, the number of p coins = (3 x 0) = Ratio: The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b.. In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.. Eg. The ratio : 9 represents with antecedent =, consequent = 9. 9

11 Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Eg. 4 : = 8 : 10 = 12 : 1. Also, 4 : 6 = 2 : Proportion: The equality of two ratios is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion. Here a and d are called extremes, while b and c are called mean terms. Product of means = Product of extremes. Thus, a : b :: c : d (b x c) = (a x d). 3. Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c. Third Proportional: a : b = c : d, then c is called the third proportion to a and b. Mean Proportional: Mean proportional between a and b is ab. 4. Comparison of Ratios: We say that (a : b) > (c : d) a c >. b d. Compounded Ratio: 6. The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf). 7. Duplicate Ratios: Duplicate ratio of (a : b) is (a 2 : b 2 ). Sub-duplicate ratio of (a : b) is (a : b). Triplicate ratio of (a : b) is (a 3 : b 3 ). Sub-triplicate ratio of (a : b) is (a 1/3 : b 1/3 ). If a = c, then a + b = c + d. b d a - b c - d [componendo and dividendo] 8. Variations: We say that x is directly proportional to y, if x = ky for some constant k and we write, x y. We say that x is inversely proportional to y, if xy = k for some constant k and we write, x 1. y