VIDYA SAGAR CAREER INSTITUTE LIMITED Mobile : 93514-68666 Phone : 0141-3215161, 3276512 Leader in CA & CS Education Mathematics : Chapter -1 Ratio, Proportion, Indices & Logarithms - 1.1 CHAPTER # 1 RATIO, PROPORTION, INDICES & LOGARITHMS ! A ratio is a comparison of the sizes of two or more same kind quantities. ! If a and b are two quantities of the same kind (in same units) then the fraction a/b is called ratio of a to b. It is written as a : b ! The quantities a and b are called the terms of ratio. a : b First Terms Second Terms (Antecedent) Consequent « a & b must be same kind. « The units of a & b must be same. « Ratio always represent in simplest term. « Importance of order. for Example - Find ratio between 150 gm & 2 kg. 150 gm 2 kg (Same kind quantities because both quantities belong to weight) 150 gm 2000gm (unit must be same, convert kg into gm) 150 : 2000 (Ratio always represent is simplest term) 15 : 200 3 : 40 So ratio of 150 gm to 2 kg is 3:40 we cannot write it 40 : 3 because the order of the term in a ratio is important. Types of Ratio 1. Inverse Ratio 2. Compound Ratio 3. Duplicate Ratio 4. Triplicate Ratio a b b a Inverse Ratio a b c d e f ace bdf C.R. , , a b a² b² D. R. a b a³ b³ T. R. A - RATIO VIDYA SAGAR CAREER INSTITUTE
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VIDYA SAGAR CAREER INSTITUTE LIMITEDMobile : 93514-68666 Phone : 0141-3215161, 3276512
1. A ratio a : b is said to be of greater inequality if a > b.2. A ratio a : b is said to be of less inequality if a < b.3. If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be
commensurable.4. If the ratio of two similar quantities can be expressed as a ratio of two non-integers, the quantities are said to be
incommensurable.5. Continued ratio is the relation between the magnitudes of three or more quantities.
Solved Example :
1. Find Ratio between 25 minute and 45 Seconds
25 min : 45 second (Same kind quantities)(25 x 60) sec. : 45 sec. (same unit)100 : 3 (simplify term)
12. The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is :(a) 16 (b) 60(c) 22 (d) 20
13. The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 25, the antecedent is :(a) 35 (b) √5(c) 7 (d) None of these
14. The ratio of two quantities is 5 : 6. If the consequent is 180, then the antecedent is :(a) 510 (b) 150(c) 30 (d) None of these
15. The inverse ratio of 11 : 15 is :(a) 15 : 11 (b) √11 : √15(c) 121 : 225 (d) None of these
16. The inverse ratio of the ratio √48 : √64 is : (a) √2 : 3 (b) √15 : √30(c) 2 :√3 (d) None of these
17. The duplicate ratio of 3 : 4 is : (a) √3 : 2 (b) 4 : 3(c) 9 : 16 (d) None of these
18. The sub duplicate ratio of 25 : 36 is :(a) 6 : 5 (b) 36 : 25(c) 50 : 72 (d) 5 : 6
19. The triplicate ratio of 2 : 3 is ______ : (a) 8 : 27 (b) 6 : 9(c) 3 : 2 (d) None of these
20. The sub triplicate ratio of 8 : 27 is _________ : (a) 27 : 8 (b) 24 : 81(c) 2 : 3 (d) None of these
21. The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is :(a) 1 : 4 (b) 1 : 3(c) 3 : 1 (d) None of these
22. The ratio compounded of 2 : 3, 9 : 4, 5 : 6 and 8 : 10 is :(a) 1 : 1 (b) 1 : 5(c) 3 : 8 (d) None of these
23. The ratio compound of two ratios 4:3 and 7:3 is(a) 12:21 (b) 28:9(c) 9:28 (d) None of these
24. The compounded ratio of duplicate ratio of 3:5 and the sub triplicate ratio of 1:27 and reciprocal of 1:5 is equal to(a) 3:5 (b) 1:2 (c) 3:25 (d) 9:25
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25. The duplicate ratio of 2/3 : 3/5 is : (a) 10 : 9 (b) 100 : 90(c) 100 : 81 (d) 1000 : 810
26. The ratio compounded of 4 : 9, 9 : 7, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 is :(a) 2 : 7 (b) 7 : 2(c) 2 : 21 (d) None of these
27. If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is(a) 18 : 25 (b) 8 : 25(c) 17 : 24 (d) None of these
28. If x : y = 3 : 4, the value of x²y + xy² : x³ + y³ is :(a) 13 : 12 (b) 12 : 13(c) 21 : 31 (d) None of these
29. If p : q = 2 : 3 and x : y = 4 : 5, then the value of 5px + 3qy : 10px + 4qy is :(a) 71 : 82 (b) 27 : 28(c) 17 : 28 (d) None of these
30. If 15(2x² - y²) = 7xy and x, y are positive, then find x : y is : (a) 5 : 6 (b) 6 : 5(c) 3 : 5 (d) 4 : 5
31. If the ratio of (5x - 3y) and (5y -3x) is 3 : 4, then the value of x:y is :(a) 27 : 29 (b) 29 : 27(c) 3 : 4 (d) 4 : 3
32. If a : b = 4 : 1 then + is :
(a) 5/2 (b) 4(c) 5 (d) None of these
33. If A : B = 3 : 2 and B : C = 3 : 5, then A : B : C is : (a) 9 : 6 : 10 (b) 6 : 9 : 10(c) 10 : 9 : 6 (d) None o these
34. If a : b = n : d and b : c = n : d then :1 1 2 2
(a) a : b : c = n d : n d : d d (b) a : b : c = n d : n d : d d1 1 2 2 1 2 1 2 2 1 2 1
(c) a : b : c = n n : d n : d d (d) None of these1 2 1 2 1 2
35. The ratio of X : Y is 7 : 9 and Y : Z is 6 : 7. The ratio of X : Z is :(a) 2 : 3 (b) 3 : 2(c) 1 : 2 (d) 2 : 1
36. If p : q = 3 : 2, r : q = 4 : 3, then r : q : p is :(a) 6 : 8 : 9(b) 8 : 6 : 9(c) 9 : 6 : 8(d) None of these
ab
ba
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37. If A : B = 3 : 4, B : C = 8 : 9 and C : D = 15 : 16 then A : D will be : (a) 3 : 16 (b) 1 : 2(c) 5 : 8 (d) 9 : 16
38. If 2A = 3B = 4C then A : B : C is :(a) 2 : 3 : 4 (b) 4 : 3 : 2(c) 6 : 4 : 3 (d) 3 : 4 : 6
39. If A : B = 2 : 3, B : C = 4 : 5, C : D = 3 : 7, then A : B : C : D is :(a) 24 : 36 : 45 : 105 (b) 12 : 36 : 45 : 105(c) 24 : 36 : 65 : 105 (d) 24 : 48 : 45 : 105
40. A, B, C, D are four numbers so that A : B = 2 : 3, B : C = 4 : 5. C : D = 5 : 8, then A : D is :(a) 2 : 3 (b) 3 : 2(c) 1 : 3 (d) 3 : 1
41. If (7p+3q) : (3p-2q) = 43:2 , then p:q is(a) 5:4 (b) 4:5(c) 7:2 (d) 2:7
42. If a:b = 4:5, find the value of (5a - 3b): (6a + 3b)(a) 7:39 (b) 15:39 (c) 1:3 (d) 5:39
43. If a² + b² = 45 & ab = 18, then + is
(a) (b)
(c) (d) None of these
44. If = - then the value of is :
(a) 1 (b) - 1/7(c) 1/7 (d) 7
45 If = , then the value of = ?
(a) (b) 7
(c) 7 : 2 (d) 15
46. If A : B = 2 : 5 then (10 A + 3 B) : (5A + 2 B) is equal to :(a) 7 : 4 (b) 7 : 3(c) 6 : 5 (d) 7 : 9
47. An amount of Rs. 950 is distributed among A, B & C in the ratio of 5:11:3, what is the difference between the share of B and A.(a) 300 (b) 340(c) 500 (d) None of these
1
a
1
b
13
23
12
p
q
2
32p + q
2p - q
x
2y
3
2
2x + y
x - 2y1
7
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48. The ages of two persons are in the ratio 5 : 7. Eighteen years ago their ages were in the ratio of 8 : 13, their present ages (in years) are :(a) 50, 70 (b) 70, 50(c) 40, 56 (d) none
49. If A, B and C started a business by investing Rs. 1,26,000, Rs. 84,000 and Rs. 2,10,000. If at the end of the year profit is Rs. 2,42,000 then the share of each is :(a) 72,600, 48,400, 1,21,000 (b) 48,400, 1,21,000, 72,600(c) 72,000, 49,000, 1,21,000 (d) 48,000, 1,21,400, 72,600
50. Find the two numbers which are in the ratio 7 : 15 and whose difference is 72 : (a) 35, 75 (b) 49, 105(c) 63, 135 (d) 56, 128
51. The number which when subtracted from each of the terms of the ratio 19 : 31 reducing it to 1 : 4 is ___ : (a) 15 (b) 5(c) 1 (d) None of these
52. What must be added to each term of the ratio 49 : 68, so that it becomes 3: 4? (a) 3 (b) 5(c) 8 (d) 9
53. The ratio of two numbers is 4:1. If 5 is added to both the numbers, the ratio of the new numbers obtained becomes 3:1. Then the numbers are(a) 30,20 (b) 40, 10(c) 20, 10 (d) None of these
54. What must be subtracted from each term of the ratio 27:43 to make it equal to 7:15?(a) 13 (b) 15(c) 17 (d) None of these
55. The ratio of the number of boys and girls in a school is 2:5. If there are 280 students in the school, find the number of girls in the school.(a) 200 (b) 250(c) 150 (d) None of these
56. The ratio of the sum and the difference of two numbers is 7:1. Find the ratio of those two numbers.(a) 5:3 (b) 4:3(c) 4:5 (d) None of these
57. Rs. 407 are to be divided among A, B and C so that their shares are in the ratio 1/4 : 1/5 : 1/6. The respective shares of A, B, C are:(a) Rs. 165, Rs. 132, Rs. 110 (b) Rs. 165, Rs. 110, Rs. 132(c) Rs. 132, Rs. 110, Rs. 165 (d) Rs. 110, Rs. 132, Rs. 165
58. The students of two classes were in the ratio 5 : 7, if 10 students left from each class, the remaining students are in the ration of 4 : 6 then now the number of students in each class is:(a) 30, 40 (b) 25, 24.(c) 40, 60. (d) 50, 70
59. Anand earns Rs. 80 in 7 hours and Promod Rs. 90 in 12 hours. The ratio of their earnings is :(a) 32 : 21 (b) 23 : 12(c) 8 : 9 (d) None of these
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60. The ratio between the speeds of two trains is 7 : 8. If he second train runs 400 Kms. in 5 hours, the speed of the first train is : (a) 10 Km/h (b) 50 Km/hr(c) 70 Km/hr (d) None of these
61. A scooter covers a distance of 200 km in 2 hour 40 minutes, while a car covers the same distance in 2 hours. The ratio of their speeds is :(a) 2 : 3 (b) 5 : 4(c) 4 : 5 (d) 3 : 4
62. The angles of a triangle are in ratio 2 : 7 : 11. The angles are : (a) (20º, 70º, 90º) (b) (30º, 70º, 80º)(c) (18º, 63º, 99º) (d) None of these
63. Division of Rs. 324 between X and Y is in the ratio 11 : 7. X and Y would get Rupees : (a) (204, 120) (b) (200, 124)(c) (180, 144) (d) (198, 126)
64. An amount of money is to be distributed among P, Q and R in the ratio 3 : 5 : 7. If Q's share is Rs. 1500, the differences between P's share and R's shares : (a) Rs. 1200 (b) Rs. 1500(c) Rs. 1600 (d) Rs. 1900
65. A boy has five rupee coins, two rupee coins and one rupee coins in the ratio 3 : 4 : 5. If he has an amount of Rs. 224, then find the number of one rupee coins with the boy :(a) 20 (b) 30(c) 5 (d) 40
66. An amount of Rs. 4686 is divided among A, B, C and D. The ratio of the shares of A and B is 3 : 2, while the ratio of the shares of B & C and C & D are 3 : 4 and 4 : 5 respectively. Find share of B :(a) Rs. 952 (b) Rs. 1260(c) Rs. 852 (d) Rs. 1540
67. Three numbers are in the ratio 2 : y : 5. The third number is greater than the first one by 21. Find the numbers, if the sum of numbers is 70 : (a) 14, 21 and 35 (b) 7, 5 and 21(c) 7, 14 and 21 (d) 14, 15 and 35
68. Daily earnings of two persons are in the ratio 4 : 5 and their daily expenses are in the ratio 7 : 9. If each saves Rs. 50 per day, their daily incomes in Rs. are :(a) (400, 50) (b) (500, 450)(c) (400, 500) (d) None of these
69. The sum of ages of 3 persons is 150 years. 10 years ago their ages were in the ratio 7 : 8 : 9. Their present ages are : (a) (45, 50, 55) (b) (40, 60, 50)(c) (35, 45, 70) (d) None of these
70. Two numbers are in the ratio 4 : 5. If 2 is subtracted from each of them, then their ratio becomes 3 : 4. The numbers are :(a) 9, 15 (b) 8, 10(c) 20, 25 (d) 12, 15
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71. Two numbers are in ratio 7 : 16. If 10 is added to both the numbers, then the ratio will be 1 : 2. Find the numbers : (a) 35 and 80 (b) 70 and 160(c) 60 and 150 (d) 45 and 90
72. There are oranges and apples in the ratio 3 : 14. If 10 oranges are eaten, then the ratio will become 1 : 6. Find the number of apples : (a) 120 (b) 60(c) 210 (d) 160
73. If the length and breadth of a room measure 14 feet 3 inches and 11 feet 6 inches respectively, then the ratio of length to breadth is :(a) 14 : 11 (b) 56 : 22(c) 57 : 36 (d) 57 : 46
74. If 35% of A's income is equal to 25% of B's income, then the ratio of theirs income is : (a) 4 : 3 (b) 5 : 7(c) 7 : 5 (d) 4 : 7
75. If 15% of x is the same as 20% of y, then x : y is :(a) 3 : 4 (b) 4 : 3 (c) 17 : 16 (d) 16 : 17
76. The ratio o two numbers is 7 : 10 and their difference is 105. The numbers are :(a) (200, 305) (b) (185, 290)(c) (245, 350) (d) None of these
77. Two numbers are in the ratio 2 : 3 and the difference of their squares is 320. The numbers are :(a) 12, 18 (b) 16, 24(c) 14, 26 (d) None of these
78. Two whole numbers whose sum is 64, cannot be in the ratio : (a) 5 : 3 (b) 7 : 1(c) 3 : 4 (d) 9 : 7
79. The ratio of the number of boys and girls in a school is 2 : 5. If there are 350 students in the school, the number of girls in the school are :(a) 200 (b) 275(c) 225 (d) 250
80. P, Q and R are three cities. The ratio of average temperature between P and Q is 11 : 12 and that between P and R is 9 : 8. The ratio between the average temperature of Q and R is :(a) 22 : 27 (b) 27 : 22(c) 32 : 33 (d) None of these
81. The ages of two persons are in the ratio 5 : 7 eighteen years ago their ages were in the ratio of 8 : 13, their present ages (in years) are :(a) 50, 70 (b) 70, 50(c) 40, 56 (d) None of these
82. Sides of two squares are in the ratio of 3 : 4. Their perimeters are in the ratio of : (a) 5 : 6 (b) 7 : 8(c) 3 : 4 (d) 6 : 7
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83. A bag contains Rs. 187 in the form of 1 Rupees, 50 Paise, and 10 Paise coins in the ratio of 3:4:5. Find the number of each type of coins?(a) 30,40, 50 (b) 102, 136, 170(c) 60,80, 100 (d) None of these
84. A bag contains one rupee, 50 paise and 25 paise coins in the ratio 10:14:18. If the total amount in the bag is Rs. 430, find the number of coins of each kind.(a) 200, 280, 360 (b) 280, 200, 360 (c) 360, 280. 200 (d) None of these
85. The ratio of the incomes of two persons is 9:7 and the ratio of their expenditures is 4:3. If each of them saves Rs. 200 per month, then their monthly incomes.(a) Rs. 1,800, Rs. 1,400 (b) Rs. 1,600, Rs. 1,200 (c) Rs. 1,400, Rs. 1,200 (d) None of these
86. The income of A and B are in the ratio 3 : 2 and their expenditure in the ratio 5 : 3. If each saves Rs. 1,500, then B's income is:(a) Rs. 6,000 (b) Rs. 4,500(c) Rs. 3,000 (d) Rs. 7,500
87. An employer reduces the number of employees in the ratio of 9:8 and increases their wages in the ratio of 14:15. In what ratio is the wages bill decreased?(a) 20:22 (b) 20:33(c) 21:20 (d) None of these
88. Find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in he number of workers in the ratio 15 : 11 and an increment in their wages in the ratio 22 : 25 : (a) decreased 6 : 5 (b) increased 6 : 5(c) decreased 5 : 6 (d) None of these
89. The compounded ratio of 4:9 and sub-duplicate ratio of 9:16 is(a) 1:4 (b) 1:3(c) 3:1 (d) None of these
90. The ratio Compounded of 4:5 and sub-duplicate of “a”:9 is 8:15. Then value of “a” is:(a) 2 (b) 3(c) 4 (d) 5
Answer Key
1 c 2 b 3 a 4 c 5 d 6 c 7 a 8 c 9 b 10 b 11 c 12 d 13 a
14 b 15 a 16 c 17 c 18 d 19 a 20 c 21 d 22 a 23 b 24 a 25 c 26 c
27 a 28 b 29 c 30 a 31 a 32 a 33 a 34 c 35 a 36 b 37 c 38 c 39 a
40 c 41 b 42 d 43 c 44 c 45 b 46 a 47 a 48 a 49 a 50 c 51 a 52 c
53 b 54 a 55 a 56 b 57 a 58 c 59 a 60 c 61 d 62 c 63 d 64 a 65 d
66 c 67 a 68 c 69 a 70 b 71 a 72 c 73 d 74 b 75 b 76 c 77 b 78 c
79 d 80 b 81 a 82 c 83 b 84 a 85 a 86 a 87 c 88 a 89 b 90 c
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a
b=
c
d( (b
a=
d
c
a
c=
b
d
a + b
b=
c + d
d
B - PROPORTION
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1/3 - 1/365. Using (a - b)³ = a³ - b³ - 3ab(a - b) tick the correct of these when x = p - p : (a) x³ + 3x = p + 1/p (b) x³ + 3x = p - 1/p(c) x³ + 3x = p + 1 (d) None of these
1/3 -1/3 366. If x = 3 + 3 , then 3x - 9x is : (a) 15 (b) 10(c) 12 (d) None of these
1/3 - 1/3 367. If x = 4 + 4 prove that 4x - 12 x is given by :(a) 12 (b) 13(c) 15 (d) 17
68. is equal to %
(a) 1 (b) √3
(c) 3√3 (d)
69.
(a) 1/2 (b) 3/2(c) 2/3 (d) 1/3
1/8 -1/8 1/8 -1/8 1/4 -1/4 1/2 -1/270. Value of (a +a )(a - a )(a +a ) (a +a ) is:(a) a + 1/a (b) a - 1/a
2 2(c) a + 1/a² (d) a - 1/a²
1y - x1+ a
1x - y1+ a
√93
5/29
3√3
7/2
x 9
3
9√3
n n - 12 + 2n + 1 n2 - 2
Answer Key
1 b 2 a 3 b 4 c 5 a 6 b 7 a 8 c 9 c 10 c 11 b 12 a 13 a
14 b 15 d 16 b 17 c 18 d 19 c 20 d 21 a 22 c 23 a 24 c 25 d 26 b
27 d 28 a 29 c 30 c 31 b 32 c 33 a 34 b 35 a 36 d 37 b 38 d 39 b
40 b 41 a 42 b 43 b 44 b 45 b 46 b 47 c 48 a 49 a 50 a 51 b 52 b
53 b 54 a 55 a 56 a 57 a 58 a 59 a 60 c 61 a 62 c 63 c 64 c 65 b
66 b 67 d 68 a 69 b 70 b
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17. The logarithm of a number consists of two parts, the whole part or the integral part is called the_______and the decimal part is called the_________.(a) Characteristic, Number (b) Characteristic, Mantissa(c) Mantissa, Characteristic (d) Number, Mantissa
18. log (1 x 2 x 3) is equal to(a) log 1 + log 2 + log 3 (b) log 3(c) log 2 (d) None of these
19. The base of log 13 into the common logarithmic base is6
(a) (b)
(c) (d) None of these
20. The value of + is
(a) 0 (b) 1(c) -1 (d) None of these.
21. log log 27 is equal to_______3 3
(a) 0 (b) -1 (c) 3 (d) 1 614
5 22. The value of (log a x log b x log c) is equal tob c a
(a) 5 (b) 0(c) 1 (d) None of these
23. If log (x+1) = 2 then x is equal to______√3
(a) 1 (b) 3 (c) 2 (d) 0
24. The value of log log 81 is equal to4 3
(a) 1 (b) 0(c) 3 (d) None of these
log 106
log 136
log
log 1310
610
log10
log 610
13
1log (ab)a
1log (ab)b
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2-log 6361. The value of 3 is (a) 9/5 (b) 3 /2(c) 9/4 (d) None of these
62. If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, there log 30(a) 2.5717 (b) 2.4771(c) 1.4771 (d) None of these
If log (2a - 3b) = Log a - Log b, then a = :
(a) (b)
(c) (d)
10064. If log 2 = 0.3010, find the number of digits in 2(a) 36 (b) 31(c) 38 (d) None of these
64Number of digits in the numeral for 2 . [Given log 2 = 0.30103] :(a) 18 digits (b) 19 digits(c) 20 digits (d) 21 digits
The value is:
(a) 3 log 2 (b) 7 log 310 10
(c) 3 log z (d) None.e
67. Given that log 2 = x and log 3 = y, the value of log 60 is expressed as :10 10 10
(a) x - y + 1 (b) x + y + 1(c) x - y - 1 (d) None of these
68. Given that log 2 = x, log 3 = y, then log 1.2 is expressed in terms of x and y as :10 10 10
(a) x + 2y - 1 (b) x + y - 1(c) 2x + y - 1 (d) None of these
69. Given that log x = m + n and log y = m - n, the value of log 10x/y² is expressed in terms of m and n as :(a) 1 - m + 3n (b) m - 1 + 3n(c) m + 3n +1 (d) None of these
63.
65.
66.
0.7214 x 20.3769.8√³
23b2b-1
23b2b+1
2b2b+1
3b2b-1
log 83
log 16.log 109 4
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