Quantitative Aptitude
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CA – CPT
QUANTITATIVE APTITUDE
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
RATIO AND PROPORATION, INDICES, LOGARITHAMS
TABLE OF CONTENT
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.1
RATIO
1.2
PROPORTION
1.3
INDICES
1.4
SURDS
1.5
LOGARITHM
1.1
RATIO AND PROPORTION, INDICES, LOGARITHMS 1.1
CHAPTER - 1
RATIO
Ratio: A ratio is a comparison of the sizes of two or more quantities of the same kind (in same units) by division. Note: 1. Usually a ratio is expressed in the lowest ten, for e.g. 16:64=
16 16x1 1 = = =1:4. 64 16x4 4
2. Order of the terms in the ratio is important e.g. 2:5 5:2. 3. Ratio exists between quantities of the same kind e.g. ratio between age of one child and weight of another child cannot be found. 4. Quantities to be compared must be in the same units. E.g. Ratio between 15 minutes and 55 seconds 15:55. But 15x60 seconds 180 = =180:11 55 seconds 11
5. To compare ratios convert them into equivalent like fractions.
LIST OF FORMULAE 1. Ratio of two numbers a and b is a:b a is called antecedent (first term) and b is called consequent (second term). e.g. in the ratio 3:4, 3 is called antecedent and 4 is called consequent. 2. Inverse ratio: One ratio is the inverse of another if their product is 1. e.g. (i) Inverse ratio of a : b is b : a. (ii) Inverse ratio of 2:3 is 3:2. 3. Ratio of greater inequality: A ratio a : b is said to be of greater inequality if a b e.g. 5:4. 4. Ratio of less inequality: A ratio a : b is said to be of less inequality if a b e.g. 4:5. 5. Ratio of equality: A ratio a: b is said to be ratio of equality if a=b e.g. 4:4. 6. Compound ratio: The compound ratio of two ratios a:b and c:d is ac:bd. E.g. the compound ratio of 2:3 and 6:5 is 2x6: 3 x 5 = 4:5. 7. Duplicate ratio : A ratio compounded of It self is called duplicate ratio. e.g. (i) Duplicate ratio of a:b is a 2 : b2 . (ii) Duplicate ratio of 3:4 is 9:16. 8. Triplicate ratio: Triplicate ratio of a:b is FOUNTAINHEAD
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1.2
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS a3 : b3 . e.g. Triplicate ratio of 3:4 is 27:64.
9. Sub duplicate ratio: Sub duplicate ratio of a: b is 10. Sub triplicate ratio: Sub triplicate ratio of a: b is
a : b e.g. sub duplicate ratio of 4:25 is 2:5. 3
a : 3 b e.g. sub triplicate ratio of 27:8 is 3:2.
11. Commensurable quantities: If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable quantities. (i) e.g. ratio of 1.28:1.5 =
1.28 128 10 64 which is a ratio of integers 64 and 75. = x = 1.5 100 15 75
1:28, 1.5 are commensurable quantities. 12. Incommensurable quantities: If the ratio of two similar quantities cannot be exactly expressed as a ratio of two integers, the Quantities are said to be incommensurable
e.g. ratio
3 : 1 cannot be exactly expressed as a ratio of two integers. So
3 and 1 are in
commensurable quantities. 13. Continued ratio: Continued ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind e.g. The continued ratio of three similar quantities a, b, c is a:b:c.
14. a : b ma : mb, m 0 . 15. a : b 16. If
If
a b : ,m 0 . m m
a c e Sum of antecedent a c e . , then each ratio = b d f Sum of consequent b d f
a c e ma+nc+pe , then each ratio = b d f mb+nd+pf
Type–1: To find antecedent/consequent.
Multiple Choice Questions 1. The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is a. 16 b. 60 c. 22 d. 20 3 3x5 15 Solution: Ratio of two quantities = 3:4= = = . As antecedent is 15, consequent is 20. 4 4x5 20 2. The ratio of two quantities is 2:5. If the antecedent is 10, the consequent is …. a. 4 b. 25 FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.3
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
c. 20 d. 50 3. The ratio of two quantities is 7:4. If the consequent is 28, the antecedent is ….. a. 49 b. 28 c. 14 d. 24.5
Type–2: To perform operation on ratio. 1. The inverse ratio of 11 : 15 is 15:11 a. 15 : 11 b. 11 : 15 c. 121 : 225 d. none of these 2. The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is a. 5 b. 5 c. 7 d. none of these
Type–3: To find compound ratio of given ratios. 1. 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 2 9 5 8 720 Solution: Compound ratio of 2:3, 9:4, 5:6 and 8:10 is x x x = =1:1. 3 4 6 10 720 2. The compound ratio of 3:4, 10:21 and 14:17 is …. a. 35:11 b. 10:11 c. 5:11 d. 5:33 3. The compound ratio of 4:7, 3:5, 5:6 and 14:5 is a. 4:7 a. 5:7 b. 5:4 c. 4:5
Type–4: To find duplicate, triplicate, sub duplicate, sub triplicate ratio of given ratios: 1. The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and sub triplicate ratio of 125 : 512 is a. 4 : 512 b. 3 : 32 FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.4
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS c. 1 : 12 d. none of these 2
3
1 4 16 1 Solution: Duplicate ratio of 4:5 = . Triplicate ratio of 1:3 = . 25 5 3 27
Sub Duplicate ratio of 81:256 =
81 9 . Sub triplicate ratio of 125:512 = 256 16
3
125 5 512 8
16 1 9 5 1 x x x = 25 27 16 8 120 The duplicate ratio of 3 : 4 is a. 3 : 2 b. 4 : 3 c. 9 : 16 d. none of these The sub duplicate ratio of 25 : 36 is a. 6 : 5 b. 36 : 25 c. 50 : 72 d. 5 : 6 The triplicate ratio of 2 : 3 is a. 8 : 27 b. 6 : 9 c. 3 : 2 d. none of these The sub triplicate ratio of 8 : 27 is a. 27 : 8 b. 24 : 81 c. 2 : 3 d. none of these The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4 is a. 1:4 b. 1:3 c. 3:1 d. none of these The ratio compounded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is a. 2:7 b. 7:2 c. 2:21 d. none of these The ratio compounded of 4:5 and sub duplicate ratio of a:q is 8:15. Then value of a is …. (Dec.) a. 2 b. 3 c. 4 d. 9
Required compound ratio = 2.
3.
4.
5.
6.
7.
8.
FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.5
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS Type–5: Harder sums on duplicate, sub duplicate ratios: 1. If p : q is the sub duplicate ratio of p–x2 : q–x2 then x2 is a. __p____ p+q b. __q____ p+q c. __pq____ p+q d. none of these Solution: p:q is sub duplicate ratio of p x : q x . 2
p 2 q p 2 x2 pq 2 q 2 x2
pq( p q) ( p q)( p q)
x2
x2
2
p 2 q pq 2 p 2 x2 q 2 x2
p = q
p x2 q x2
p x2 p2 = q x2 q2
pq( p q) x2 ( p 2 q 2 )
pq pq
2. If 2s : 3t is the duplicate ratio of 2s – p : 3t – p then a. p2 = 6st b. p = 6st c. 2p = 3st d. none of these
Type–6: From given value of ratio, evaluate 1. 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 x y x y Solution: First method : x : y 3: 4 Let k k , k 3 4 3 4 3 3 3 2 2 2 2 36k 48k 84k 12 x y xy (3k ) (4k ) 3k (4k ) Now 3 = = 3 3 3 3 3 27k 64k 91k 3 13 x y (3k ) (4k )
x 3k , y 4k
Short cut : x : y 3: 4, then taking x = 3 and y = 4 in x 2 y xy 2 (3)2 4) (3)(4)2 36 48 84 12 x3 y 3 33 43 27 64 91 13
2. If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is a. 54 : 25 b. 8 : 25 c. 17 : 24 d. 18 : 25 3. If x : y 3: 4, the value of 3x 5 y : 5x 3 y is ….. a. 0 27 b. 29 FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.6
RATIO AND PROPORTION, INDICES, LOGARITHMS c. 4.
5.
6.
7.
CHAPTER - 1
29 27
d. 1 If x : y 3: 4, the value of is …. 3 y x : 2 x y 1 a. 2 9 b. 10 9 c. 11 13 d. 10 If A: B=2:5, then (10A+3B) 5A+|2B) = ….. a. 7:4 b. 7:3 c. 6:5 d. 7:9 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 2ax 3 yx If a : b 3: 4 and x : y 5 : 2, the value of ..... 4ay bx a. 3:2 b. 3:13 c. 27:34 d. 24:11
Type–7: Simple problem sums on ratio. 1. An alloy is to contain copper and zinc in the ratio of 9:4. The zinc required to melt with 24 kg of copper is ….. kg 2 a. 10 3 1 b. 10 3 2 c. 9 3 d. 9 Weight of copper 9 Solution: Weight of zinc 4 24 9 96 32 2 10 kg. 96 = 9x x x 4 9 3 3 2. Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally entire cost of the car, then the share of each of the remaining persons increased by …. (March 2007). FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.7
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
1 9 1 b. 8 1 c. 7 7 d. 8
a.
Solution: Let the rent be 8 x 7 = Rs.56.
Rent share for each of 8 people =
56 =Rs.7 8
56 Rent share increased by = 8–7 = Rs.1. =Rs.8 7 1 Rent share of each person increased = 7 3. Ratio of earning of A and B is 4:7. If the earnings of A increase by 50% and those of B decrease by 25%, the new ratio of their earning becomes 8:7. What is A’s earning? (August 2007). a. Rs.21,000 b. Rs.26,000 c. Rs.28,000 d. Data inadequate 150 Solution: A:B = 4:7 Let their earnings be 4x, 7x. A’s new earning = 4xx 6x . 100 75 21x 6 8 B’s new earning = 7x x Now ratio of new earnings = 6x x 100 4 21x 7 As data is inadequate, A’s earning cannot be found.
Rent share for each of 7 people =
Type–8: Problems on distribution. 1. Division of Rs. 324 between X and Y is in the ratio 11 : 7. X & Y would get Rupees a. (204, 120) b. (200, 124) c. (180, 144) d. none of these Solution: First method : Let x:y = 11:7 x = 11k and y = 7k, where k is common multiple… Now x+y=324. 324 18 From (1) x 11k 11x18 Rs.198 and 11k 7k 324 18k 324 k 18 y 7k 7x18 Rs.126 11 7 : (Dividing by 11+7=18). Now 18 18 11 7 x x324 11x18 Rs.198 and y x324=7x8=Rs.126 Verification: On verifying in 18 18 option (a), (b), (c), none is divisible by 11.
Shortcut: x : y 11: 7
2. Anand earns Rs. 80 in 7 hours and Promode Rs. 90 in 12 hours. The ratio of their earnings is a. 32 : 21 FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.8
RATIO AND PROPORTION, INDICES, LOGARITHMS
3.
4.
5.
6.
CHAPTER - 1
b. 23 : 12 c. 8 : 9 d. none of these The angles of a triangle are in ratio 2: 7: 11. The angles are a. (20°, 70°, 90°) b. (30°, 70°, 80°) c. (18°, 63°, 99°) d. none of these The ratio of 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 The ratio of two numbers is 9:5 and their sum is 224. The numbers are…. a. 100, 124 b. 150,74 c. 144,80 d. 200,24 Two numbers are in the ratio of 2:3 and the difference of their squares si 320. The numbers are…. a. 12,18 b. 16,24 c. 14,21 d. None of these
Type–9: To find what should be added or subtracted to numerator and denominator to make some ratio or vice versa. 1.
2.
3.
FOUNTAINHEAD
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 Solution: Let x be subtracted from numerator and denominator. 19 x 1 76 4 x 31 x 76 31 4x x 45 3x 31 x 4 19 15 4 1 Verification : x 15. 31 15 16 4 The number which when subtracted from each of the terms of the ratio 29:37 reducing it to 3:4i. a. 2 b. 3 c. 4 d. 5 The number which when added to each of the terms of the ratio 29:37 reducing it to 4:5 is… a. 2 CA - CPT
QUANTITATIVE APTITUDE
1.9
RATIO AND PROPORTION, INDICES, LOGARITHMS
4.
5.
6.
7.
CHAPTER - 1
b. 3 c. 4 d. 5 Two numbers are in the ratio 2: 3. If 4 be subtracted from each, they are in the ratio 3 : 5. The numbers are a. (16,24) b. (4,6) c. (2,3) d. none of these Two numbers are in the ratio of 3:5. If 12 is added to each, they are in the ratio 2:3. The numbers are a. 15,25 b. 18,30 c. 36,60 d. 24,40 Two numbers are in the ratio of 7:8. If 3 is adding to each of them, then their ratio becomes 8:9. The numbers are…. a. 14,16 b. 24,27 c. 21,24 d. 16,18 In 40 litres mixture of glycerine and water, the ratio of glycerine and water is 3:1. The quantity of water added in order to make this ratio 2:1 is…. Litres a. 15 b. 10 c. 8 d. 5 30 2 Solution: Glycerine: water = 3:1 = 30:10. Let xlitres of water be added. 30 10 x 1 = 20 + 2x 10 = 2x x =5.
8.
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 9. 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 10. The students of two classes are in the ratio 5:7, if 10 students left from each class, the remaining students are in the ratio of 4:6, then the number of students in each class is … a. 30,40 b. 25,24 c. 40,60 d. 50,70
FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.10
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS Type–10: Miscellaneous sums: 1.
2.
3.
4.
x y then ratio ….. x 2 y 2x y 1 a. -1 3 1 b. ,1 3 c. 3,1 d. -3,-1 Anand earns Rs. 80 in 7 hours and Promode Rs. 90 in 12 hours. The ratio of their earnings is a. 32 : 21 b. 23 : 12 c. 8 : 9 d. none of these 80 Rs./hr Anand's earning 80 12 32 Solution: = 7 = x = Promode's earning 90 Rs./hr 7 90 21 12 The ratio between the speeds of two trains is 7: 8. If the second train runs 400 Kms. in 5 hours, the speed of the first train is a. 10 Km/hr b. 50 Km/hr c. 70 Km/hr d. none of these x km Speed of first train hr 7 x x 7 x80=70 km/hr Solution: = 8 80 8 Speed of second train 400 km hr 5 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. (40, 50) b. (50, 40) c. (400, 500) d. none of these Solution: Earnings of A:B = 4:5 = 4 x : 5x … (1) (x common multiple) Expenses of A:B = 7:9 = 74:94 … (2) (y common multiple) If
Now earnings – expenses = savings For B, 5x 9 y 50 …
For A, 4 x 7 y 50
…
(2)
(3) Multiplying (2) by 9 and (3) by 7,
36 x 63 y 450 35 x 63 y 350
x
100
From (1), their incomes are 4x = 400 and 5x = 500. Short cut: On verification 400:500 are in the ratio of 4:5. Their expenses (400-50) : (500-50) = 350:450 = 7:9. FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.11
RATIO AND PROPORTION, INDICES, LOGARITHMS 5.
6.
Monthly incomes of A and B are in the ratio of 4:3 and their expenses are in the ratio of 3:2. If each of them saves Rs.600 every month, then their annual incomes are … a. Rs.1800, Rs.2400 b. Rs.2400, Rs.1800 c. Rs.1200, Rs.900 d. Rs.1600, Rs.1200 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 Solution: :Q = 11:12 = 99:108 (Multiplying by 9 : P:R= 9 : 8= 99 : 88 (Multiplying by 11) P: Q: R = 99 : 108:88
7.
CHAPTER - 1
Q:R = 108:88 = 27:22.
In a film shooting, A and B received money in a certain ratio and B and C also received the money in the same ratio. If A gets Rs.1, 60,000 and C gets Rs.2, 50,000. Find the amount received by B. a. Rs.2,000 b. Rs.2,50,000 c. Rs.1,00,000 d. RS.1,50,000 Solution: Here A:B = a : b a 2 : ab (multiplying by a) B:C = a : b ab : b2 (multiplying by b) A:B:C = a 2 : ab : b2 Here a 2 160000 and b2 250000 a = 400 and b = 500
8.
Find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in the number of workers in the ratio 15:11 and an increment in their wages in the ratio 22:25. a. 5:6 b. 6:5 c. 3:10 d. 10:3 Solution: Let * be original number of workers and Rs.y be their average wage. Therefore total original wage = Rs. Xy.
11 25 x New average wage = Rs. y 15 22 11 25 5 5 Total new wage = x x y xy Wage decreased in ratio = xy : xy 6 : 5 15 22 6 6 9. The incomes of A and B are in the ratio 3:2 and their expenditures in the ratio 5:3. If each saves Rs.1500, then B’s income is Rs….. (Nov. 2007) a. 6000 b. 4500 c. 3000 d. 7500 On verification we get 10. The recurring decimal 2.7777 …. Can be expressed as…. 24 a. 9 New number of workers =
FOUNTAINHEAD
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QUANTITATIVE APTITUDE
1.12
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS 22 9 26 c. 9 25 d. 9 Solution: Short Cut :
b.
Using calculator answer can be verified. First method : Let x = 2.7777 (1)
25 9 11. The product of 3 numbers is 750 and their ratio is 1:2:3. The sum of three numbers is …. 10x = 27.777
(2) Subtracting (1) from (2), 9x = 25
x=
a. 750 b. 150 c. 60 d. 30 Solution: Let x be common multiple. The numbers are x,2 x,3x Now product = 750. x.2 x.3x 750 6 x3 750 x3 125 x 5 .Their sum = x 2x 3x 6x 6x5 30
12. If a carton containing a dozer mirrors is dropped, which of the following cannot be the ratio of broken ________________________________. a.
1.2
PROPORTION
Proportion: An equality of two ratios is called a proportion. OR if
a c , then, a, b, c, d are said b d
to be in OR If
a c e ; then a, b, c, d , e, f .... are in proportion. b d f
Note: 1. If a, b, c, d are in proportion, they are called first, second, third and fourth proportional terms respectively. 2. If a, b, c, d are in proportion, then a, d are called extreme terms while b,c are called mean a b terms. i.e. b c Continuous Proportion: If a,b,b,c are in proportion, a,b,c are said to be in continuous proportion. OR If
a b c ..., then, a,b,c,d ,…. Are in continuous proportion. b c d
Note: 1. If a,b,c are in continuous proportion, a,b,c are called first, mean and third proportional terms respectively. FOUNTAINHEAD
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RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
2. If a,b,c are in continuous proportion a,c are called extremes while b is called mean proportional term.
LIST OF FORMULAE
If a,b,c,d are in proportion, then
1. 2. ad=bc (cross multiplication) Product of extremes = product of means 3.
b d (Invertendo) a c
4.
a b (Alternendo) c d
5.
ab cd (componendo) b d
6.
a b c d (Dividendo) b d
7.
ab cd (componendo – dividend) a b c d
8.
a c (convertendo) a b c d
9. If
a c e a c e ... ..., then each of these ratios (addendo) is equal to b d f ... b d f
10. If
a c e a c e ... ...., then each of these ratios (subtrahendo) is equal to b d f b d f ...
11. If a,b,c are in continuous proportion, then b2 ac . i.e. (mean proportional term)2 = product of extremes.
Type–1: To find any proportional term. 24 here we have to find second proportional term = x Now 4, x, 9,
27 are in proportion. 2
Product of means = Product of extremes xx9 4x
FOUNTAINHEAD
27 2
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9 x 54
x6
QUANTITATIVE APTITUDE
1.14
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Multiple Choice Questions 1.
2.
3.
4.
5.
6.
7.
The fourth proportional to 4, 6, 8 is a. 12 b. 32 c. 48 d. none of these The number which has the same ratio to 26 that 6 has to 13 is a. 11 b. 10 c. 21 d. none of these The fourth proportional to 2a, a2, c is a. ac/2 b. ac c. 2/ ac d. none of these The numbers 14, 16, 35, 42 are not in proportion. The fourth term for which they will be in proportion is a. 45 b. 40 c. 32 d. none of these 4, *, 9, 13½ are in proportion. Then * is a. 6 b. 8 c. 9 d. none of these Fourth proportional to x,2 x,( x 1) is …. June 2009. a. x 2 b. x 2 c. 2 x 2 d. 2 x 2 If four numbers 1/2, 1/3, 1/5, 1/x are proportional then x is a. 6/5 b. 5/6 c. 15/2 d. none of these
Type–2: To find mean or third proportional term if three terms are in continuous proportion. 1.
FOUNTAINHEAD
The third proportional to 12, 18 is a. 24 b. 27 c. 36 d. none of these Solution: Let 12, 18, x be in continuous proportion. (Mean proportional term)2 = Product of CA - CPT
QUANTITATIVE APTITUDE
1.15
RATIO AND PROPORTION, INDICES, LOGARITHMS 324 x = 27 x 12 The mean proportional between 1.4 gms and 5.6 gms is a. 28 gms b. 2.8 gms c. 3.2 gms d. none of these Solution: 25 Let 1.4 gms, x gms and 5.6 gms be in continuous proportion. (Mean proportional termZ)2 = Product of extremes. x2 = 1.4 x 5.6 x2 = 7.84 x= 2.8 gms
extremes.
2.
3.
4.
5.
6.
7.
8.
FOUNTAINHEAD
CHAPTER - 1
(18)2 = 12 x x.
The mean proportional between 25, 81 is a. 40 b. 50 c. 45 d. none of these The mean proportional between 12x2 and 27y2 is a. 18xy b. 81xy c. 8xy d. none of these The third proportional between (a 2 b2 ) and (a b) is ….. ab a. a b a b b. ab ( a b) 2 c. ab ( a b)3 d. a b Mean proportional term between 3 and 27 is ….
3 a. b. 3 c. 9 d. 81 Should be subtracted from each of 6,8,11 so that they are in continuous proportion. a. 1 b. 2 c. 3 d. 4 1 1 Mean proportional between a 2 2 and b 2 2 is …… b a a. ab 1 1 b. 1 ab 1 c. ab ab CA - CPT
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RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
1 ab The third proportional between a 2 b2 and a b is…. a. a3 b3 b. a3 b3 c. a–b ab d. a b
d. ab 9.
Type–3: To convert one form of proportion into other form using properties of proportion: 1.
2.
3.
4.
5.
If u/v = w/p, then (u–v)/(u+v) = (w–p)/(w+p). The process is called a. Invertendo b. Alternendo c. Addendo d. none of these u w u v w p Solution: is obtained by applying dividend – compound. v p u v w p If x/y = z/w, implies y/x = w/z, then the process is called a. Dividendo b. Componendo c. Alternendo d. none of these If p/q = r/s = p–r/q–s, the process is called a. Subtrahendo b. Addendo c. Invertendo d. none of these If a b = 4 5 then a. a + 4 b – 5 = a - 4 b + 5 b. a + 4 b + 5 = a - 4 b – 5 c. a - 4 b + 5= a + 4 b – 5 d. none of these If a/b = c/d, implies (a+b)/(a–b) = (c+d)/(c–d), the process is called a. Componendo b. Dividendo c. Componendo and Dividendo d. none of these
Type–4: To find A: B: C from given relation. 1.
FOUNTAINHEAD
If A = B/2 = C/5, then A : B : C is a. 3 : 5 : 2 b. 2 : 5 : 3 c. 1 : 2 : 5 d. none of these
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CHAPTER - 1
A B C = = ÞA:B:C=1:2:5 1 2 5 If 2a 3b 5c, then a:b:c = …
Solution: 2.
a. b. c. d.
15:10:6 2:3:5 5:3:2 15:20:24
Solution: 2a 3b 5c
a b c 15 10 6
2a 3b 5c ( 30 30 30
Dividing by L.C.M. 30 of 2,3,5)
a:b:c = 15:10:6
Type–5: To find a: b: c from a: b, b: c, etc. 1.
2.
3.
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 of these Solution: A:B = 3:2 = 9:6 (Multiplying by B:C = 3:5 : 6: 10 (Multiplying by A:B:C = 9:6:10. A B 3 3 Shortcut: = 3 x 3 : 3 x 2 : 2 x 5 = 9:6:10. , are B C 2 5 If x : y = 2 : 3, y : z = 4 : 3 then x : y : z is a. 2 : 3 : 4 b. 4 : 3 : 2 c. 3 : 2 : 4 d. none of these If x : y 3: 7 and y : z 5 :11 then x : y : z ...... a. 9:21:11 b. 15:7:77 c. 15:35:77 d. 3:35:11
Type–6: From given continued ratio, evaluate some ratios. 1.
FOUNTAINHEAD
If x/2 = y/3 = z/7, then the value of (2x–5y+4z)/2y is a. 6/23 b. 23/6 c. 3/2 d. 17/6 x y z x y z Solution: Let k k , k and k x 2k , y 3k , z 7k 2 3 7 2 3 7 2x 5 y 4z 2(2k ) 5(3k ) 4(7 k ) 2y 2(3k ) 4k-15k+28k 17k 17 6k 6k 6 CA - CPT
QUANTITATIVE APTITUDE
Now
1.18
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Shortcut : 2 x 5 y 4 z 2(2) 5(3) 4(7) 17 x y z As , take x 2, y 3 and z 7 2 3 7 2y 2(3) 6 2.
3.
4.
5.
If a/3 = b/4 = c/7, then a+b+c/c is a. 1 b. 3 c. 2 d. none of these If a = b = c then a+b+c is 4 5 9 c a. 4 b. 2 c. 7 d. none of these If = = then (b – c)x + (c – a)y + (a – b)z is a. 1 b. 0 c. 5 d. none of these If
x y z , then the value of 4 x 3 y 4 z .... 3 4 6
a. 6.
7.
x y z 3x 2 y z , then the value of p is …. 2 3 4 p a. 2 b. 3 c. 4 d. 16 a b c If and 6a 7b 8c 90 then a = ….. 3 4 5 a. 6 b. 9 c. 12 d. 15 If
Type–7: If x: y is given, then to evaluate some ratio. 1.
If p/q = r/s = 2.5/1.5, the value of ps:qr is a. 3/5 b. 1:1 c. 5/3 d. none of these p r 2.5 5 ps 5x3 Solution: Now ps : qr 1 q s 1.5 3 qr 3x5
2.
If x : y = z : w = 2.5 : 1.5, the value of (x+z)/(y+w) is a. 1
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.19
RATIO AND PROPORTION, INDICES, LOGARITHMS b. 3/5 c. 5/3 d. none of these x z 2.5 5 x z 5 5 10 5 Solution: Now y w 1.5 3 y w 33 6 3 3.
5/2 4 5 none of these
If (a b) : ab 4 :1, then a. b. c. d.
If
a b is ….. b a
7 4 5 None of these
Solution: 5.
xz 5 (Addendo) yw 3
If a : b = 4 : 1 then √ + √ is a. b. c. d.
4.
Each ratio =
CHAPTER - 1
ab 4 :1 ab
a b 4 ab ab
a2 b2 4 ab ab
a b 4 b a
p 2 2p q is …. , then the value of q 3 2p q
2 p 4 2 p q 4 3 1 (Compodendo dividend) Putting p 2 and q 3 is q 3 2 p q 4 3 7 2 p q 2(2) 3 1 2 p q 2(2) 3 7
Type–8: From some relation, find x: y. 1.
If (5x–3y)/(5y–3x) = 3/4, the value of x : y is a. b. c. d.
2:9 7:2 7:9 none of these
Solution:
5x 3 y 3 4(5x 3 y) 3(5 y 3x) 20 x 12 y 15 y 9 x 5 y 3x 4
20 x 9 x 15 y 12 y
FOUNTAINHEAD
CA - CPT
29 x 27 y
QUANTITATIVE APTITUDE
x 27 y 29
1.20
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type–9: Problem sums. 1.
The sum of the 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 Solution: 10 years ago, their ages were in the ratio 7:8:9. Let their ages were 7 x,8x,9 x . Their present ages are 7 x 10,8x 10,9 x 10 Their sum = 150
7 x 10 8x 10 9 x 10 150 24 x 120 x 5 Their present ages are 7 x 10 7(5) 10 45, 8x 10 8(5) 10 50 and 9 x 10 9(5) 10 55 years.
Verification: If present ages 45, 50, 55. Ages 10 years ago 35, 40, 45 which are in 35:40:45 = 7:8:9. 2.
3.
4.
5.
FOUNTAINHEAD
Division of Rs. 750 into 3 parts in the ratio 4 : 5 : 6 is a. (200, 250, 300) b. (250, 250, 250) c. (350, 250, 150) d. 8 : 12 : 9 Two numbers are in the ratio 3 : 4; if 6 be added to each terms of the ratio, then the new ratio will be 4 : 5, then the numbers are a. 14, 20 b. 17, 19 c. 18 and 24 d. none of these 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. 72600, 48200, 121000 b. 48400, 121000, 72600 c. 72000, 49000, 121000 d. 48000, 121400, 72600 3 2 5 Solution: A:B:C = 126000:84000:210000 = 3:2:5 ( Dividing by 42000) = : : 10 10 10 3 2 A’s Profit = 242000 x 72600 Rs. B’s Profit = 242000 x = 48400 Rs. 10 10 5 C’s Profit = 242000 x = 121000 Rs. 10 In what ratio should tea worth Rs.10 per kg be mixed with tea worth Rs.14 per kg, so that the average price of the mixture may be Rs.11 per kg. a. 2:1 b. 3:1 c. 3:2 d. 4:3 Solution: Let x Kg of Rs.10/kg be mixed with y Kg of Rs.14/kg Weight of mixture = ( x y) kg. Cost of x kg + cost of y kg = cost of mixture 10x + 14y = 11 (x+y) 10x + 14y = 11x + CA - CPT
QUANTITATIVE APTITUDE
1.21
RATIO AND PROPORTION, INDICES, LOGARITHMS 11y 3y x
CHAPTER - 1
x 3 3:1 y 1
Shortcut method Method of alligation Price of first 10
14 price of second 11
price of mixture
(14–11) : (11–10) = 3:1 5.
Rs.407 are to be divided among A, B and C so that their shares are in the ratio
1 1 1 : : . 4 5 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
Type–10: Miscellaneous sums. 1.
a b 4 b c 0 2 a. 3 4 b. 9 16 c. 81 d. None
2.
b is mean proportional term between a and c.
a : c......
a b c 4 12 ...
4 3 3 b. 4 1 c. 36 d. 36 1 1 1 x y If : : = 2:3:4 then = ….. x y z yz 10 a. 9 10 b. 7 9 c. 10
a.
3.
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.22
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
7 10 4 x 5 y4 x If 4 161, then .... 4 x 5y y 3 a. 2 2 b. 3 2 c. 3 3 d. 4 5 x 4 y 15 x y If 5, then .... 5x 4 y 3 x y a. –11 b. 11 29 c. 19 29 d. 19 3 x 3x 14 If , then x = ……… 3x 2 1 13 a. 2 1 b. 2 c. 3 1 d. 3 If x y : y z : z x 6 : 7 :8 and x y z 14 then the value of x is ….
d. 4.
5.
6.
7.
a. b. c. d. 8.
9.
FOUNTAINHEAD
6 7 8 10 x2 1 x 1 ..... If 2, then the value of 2 x 1 x 1 3 a. 5 15 b. 17 4 c. 5 5 d. 4 x y If x:y=3:4, x : y 3: 4, then : is …. y x a. 3:4 b. 1:1 CA - CPT
QUANTITATIVE APTITUDE
1.23
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
c. 9:16 d. 16:9 10. If x : y 1: 2, y : z 3: 4, z : w 5: 6 then x : y : z : w is .... a. b. c. d.
15:30:40:48 15:30:48:40 30:15:40:48 30:15:48:40
Type–11: Based on variation 1.
If x varies inversely as square of y and given that y=2 for x=1, then the value of x for y=6 will be… a. 3 b. 9 1 c. 3 1 d. 9 1 k Solution: x 2 x 2 (k is a non zero constant) Now y 2 for x 1 y y k 4 4 1 k Now x = 2 x = 2 Now y = k=4 x= 1 2 36 9 y y 2
2.
x 2 y 3 when x 4, y 3 when x 2, y .... 2 a. 3 3 b. 2 4 c. 3 3 d. 4 A B,B C then AB ....
3.
a.
4.
FOUNTAINHEAD
4
c
c b. c. C d. C2 1 x 2 and y z 2 then x …. y a. Z 1 b. z c. z4 1 d. 4 z
CA - CPT
QUANTITATIVE APTITUDE
1.24
RATIO AND PROPORTION, INDICES, LOGARITHMS 1.3
CHAPTER - 1
INDICES LIST OF FORMULAE
1. If a R and n N, then a x a x a x … n times = an where a is called base and n is called power or index e.g. 2 x 2 x 2 x 2 x 2 = 25 where 2 is base and 5 is index. 2. a 0 1 where a 0, a R e.g. 2o = 1 an
3. 4.
n
1 1 1 , n N , a 0, a R x 5 5 , 2 x n a x x e.g. 1
a = a n , where a is a positive real number. e.g. 2
5.
n
2
am a
m
n
1
2
2, 3 4 4
1
3
Where a is a positive real number. e.g. 3
2
a a
2
3
Laws of indices (from 6 to 10): 6. am .an amn e.g. x2 .x3 x23 x5 7.
am x10 n , where e.g. a x103 x 7 m n 3 n x a =1, where m n e.g.
x4 1 x4 1
=a
nm
, where n m e.g.
x5 1 1 75 2 7 x x x 8.
(a m )n a mn e.g. 3
2 3 1 x 23 4 3 4 2 x x x
m m m 9. (ab) a b e.g.
(2 x)5 25 x5 32 x5 . m
4
a mbm 34 81 a 3 10. e.g. 4 4 x x b x FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.25
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
11. If a x a y , then x y e.g. 2x 24 , then x = 4. 12. If a n bn and a b , then n=0 e.g. if 2n = 3n, then n=0. 13. If x a y a , then it is not necessary that x y e.g. (2)2 = (–2)2 14. If x y , then x y n
1 n
Type–1: Based on simple expression:
Multiple Choice Questions 1.
4x–1/4 is expressed as a. –4x 1/4 b. x–1 c. 4/x1/4 d. none of these 1 4 4x 4 1 x4 Solution:
2.
The value of ( a. b. c. d.
)
where p, q, x, y ≠ 0 is equal to
0 2/3 1 none of these 0
2 p 2q3 Solution: 1 3xy
3.
4.
FOUNTAINHEAD
Which is True ? a. 20 > (1/2)0 b. 20 < (1/2)0 c. 20 = (1/2)0 d. none of these 31 .... x 2 x2 a. 3 1 b. 3x 2 c. 3x2 3 d. 2 x
CA - CPT
QUANTITATIVE APTITUDE
1.26
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type–2: Evaluate 1.
The value of 2(256)–1/8 is a. 1 b. 2 c. 1/2 d. none of these Solution: 2(256)
2.
1 8
2(28 )
Solution: 2 .4 2 (2 )
4.
5.
6.
7.
1 8
8 x
2x2
1 8
2x21
2 1 2
2½ .4¾ is equal to a. a fraction b. a positive integer c. a negative integer d. none of these 1
3.
3
2
4
1
2
2
3
4
1 2
2 x2
2x
3 4
1
2 2.2
3
2
2
1 3 2 2
22 4 . Which is an integer.
The value of 81/3 is a. 3 2 b. 4 c. 2 d. none of these The value of 2 × (32) 1/5 is a. 2 b. 10 c. 4 d. none of these The value of 4/(32)1/5 is a. 8 b. 2 c. 4 d. none of these The value of (8/27)1/3 is a. 2/3 b. 3/2 c. 2/9 d. none of these If 2x x 3 y x5z 360, then what are the values of x,y,z? a. 3,2,1 b. 1,2,3 c. 2,3,1 d. 1,3,2 2 360 2 180
Solution:
2 90 3 45 3 15
360 23 x32 x51 2 x x3 y x5 z x 3, y 2, z 1
5 5 1
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.27
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS Type– 3: Simple sums on simplification 1.
2.
The simplified value of 16x–3y2 × 8–1x3y–2 is a. 2xy b. xy/2 c. 2 d. none of these y 2 1 x3 Solution: 16 x 3 y 2 x81 x3 y 2 16 x 3 x x 2 2 x 8 y
x m3n x x 4 man is ….. x 6 m 6 n
Simplification of a. b. c. d.
xm xm xn xn
x m3n4 m9 n x5m6 n 6 m6 n x5m6 n6 m6 n x m x 6 m 6 n x {(33)2 × (42)3 × (53)2} / {(32)3 × (43)2 × (52)3} is a. 3/4 b. 4/5 c. 4/7 d. 1 (33 )2 x(42 )3x(53 )2 36 x46 x56 Solution: 2 3 3 2 2 3 = 6 6 6 =1 (3 ) x(4 ) x(5 ) 3 x4 x5 Solution:
3.
4.
has simplified value equal to a. b. c. d.
xy2 x2y 9xy2 none of these 1
5.
1
1
4 4 81x 4 4 3 4 x x 4 3x Solution: 8 2 3xy 2 1 8 y y (y ) 4 The value of ya–b × yb–c × yc–a × y–a–b is a. ya+b b. y c. 1 d. 1/ya+b
Solution: y ab x ybc xy ca xy ab 6.
FOUNTAINHEAD
= y abbcca x
1 y
a b
= y0x
1 y
a b
1 y a b
xa–b × xb–c × xc–a is equal to e. X f. 1 g. 0 h. none of these
CA - CPT
QUANTITATIVE APTITUDE
1.28
RATIO AND PROPORTION, INDICES, LOGARITHMS 7.
(
Show that e. f. g. h. 8.
)
)
(
)
√
x(
)
(
)
√
reduces to
1 0 –1 None
Simplification of a. b. c. d.
x(
√
CHAPTER - 1
x m2 n x3m8n is ….. x 5 m 6 n
xm xn xm xn
Type– 4: Harder sums on simplification 1.
The value of ( ) a. b. c. d.
2.
( )
( )
( )
1 0 2 none of these
Show that ( ) a. b. c. d.
( )
0 –1 3 1
x1 Solution: Solution: b x
a b
xb x c x
= x( ab)( ab) x x(bc )(bc ) x x(ca )(ca ) 3.
is given by
On simplification (
)x+y x (
b c
xc x a x
ca
= ( xab )ab x(xbc )bc x(x ca )ca
= a 2 b2 b2 c 2 c 2 a 2 )y+z
(
=x
= x0=1
)x+z m reduces to
a. 3 b. – 3 c. –1 3 1 d. 3 4.
Show that ( )b+c+a x( )c+a+b ( )a+b+c is given by a. b. c. d.
5.
1 0 –1 None
Show that ( ) ⁄
x( ) ⁄
x( ) ⁄
reduces to
a. -1 b. 0 c. 1 FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.29
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS d. None 6.
1
Solution: = x =x–
7.
)
x(
Show that (
)
)
reduces to
) x ( )(
(
x(
) x( )(
)
(
x(
)
is reduces to
) (
)
On simplification [
(
]a+b x [
]b+c x [
] reduces to
]a+b x [
]b+c x [
]reduces to
)
(
a. b. c. d.
)
)
)
11. On simplification [
(
)
(
)
5 2
7
3 9 2 x9 is equal to… 9 3 3 a. 1 3 b.
c. 3 FOUNTAINHEAD
x(
1 0 -1 none of these
a. b. c. d.
12.
= x0 = 1
=x
)
1 1 1 (a b)(c a) (b c)(a b) (c a)(b c)
1 3 –1 None
a. 1 b. c. d. 10.
b c c a a b (a b)(b c)(c a)
The value of ( ) ( a. b. c. d.
9.
1
1 1 1 ( a b )( a c ) ( bc )( ba ) ( c a )( c b )
Show that ( a. b. c. d.
8.
1
1 ac 1 ba 1 cb Show that x ab x x bc x x ca is given by a. 0 b. 1 c. – 1 d. None
3
CA - CPT
QUANTITATIVE APTITUDE
1.30
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS 3
d.
9 3 5
5
7
3 2 9 2 Solution: 9 3 3 x 9 1
= 3
7 7 5 2 4 4
7 25
1
=
x3
7
31/2 2 32 2 2 = 2 x3 1/2 3 3x3
=
4
147 5 4
x 3
3
1/2 .
3/4
13. [{(2) (4) . (8) a. A fraction b. an integer c. 1 d. none of these
5/6
. (16)
7/8
=
35/4 37 x x32 5 7/2 7/4 3 3 x3
34 1 34
. (32)9/10}4]3/25 is
5 7 9 3 4 1 5 7 9 3 1 Solution: (2) 2 .(4) 4 .(8) 6 .(16) 8 .(32)10 = 2 2.(22 ) 4 .(23 ) 6 .(24 ) 8 .(25 ) 10
= 2 2.2 2.2 2.2 2.2 1
3
5
7
9
4
2
3
25
1 3 5 7 9 4 = 2 2 2 2 2 2
3
25
=2
25 3 x4x 2 25
=26 Which is an integer
14. The True option is a. x2/3 = 3 x2 b. x2/3 = x3 c. x2/3 > 3 x2 d. x2/3 < 3 x2 15. The value of (8/27)–1/3 × (32/243)–1/5 is a. 9/4 b. 4/9 c. 2/3 d. none of these 16. Simplified value of (125)2/3 × 25 × 3 53 × 51/2 is a. 5 b. 1/5 c. 1 d. none of these 17. The value of {(x+y)2/3 (x–y)3/2/ x+y × (x–y)3}6 is a. (x+y)2 b. (x–y) c. x+y d. none of these 2
6
3
x6 x6 3 3 ( x y) ( x y) 2 ( x y) 3 ( x y) 2 Solution: 1 3 3 x y ( x y ) ( x y) 2 x 6 ( x y) 2 x 6 2
18. The value of (
) ⁄ x(
)
=
( x y)4 x y ( x y)
⁄ is
a. 0 b. 252 c. 250 FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.31
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS d. 248 19. The value of a. b. c. d.
x
x
x
is
1 -1 0 None
Type– 5: Based on k method. 1.
If x1/p = y1/q = z1/r and xyz = 1, then the value of p+q+r is a. 1 b. 0 c. 1/2 d. none of these Solution: Let x
1
p
y
1
q
z
1
r
k x
1
p
k, y
1
q
k and z
1
r
k
p 2 r p q r p q r k0 p q r 0 x k , y k and z k Now xyz 1 k .k .k 1 k
2.
If ax = b, by = c, cz = a, then xyz is a. 1 b. 2 c. 3 d. none of these Solution: a x b, b y c, c z a
x y (a ) c
xy a c
z xy z xyz 1 Now c a (a ) a a a xyz 1
3.
If 2x = 3y = 6-z, a. b. c. d.
is
1 0 2 none of these
Solution: Let 2x 3y 6 z k 1
1
1
1
2x k ,3y k and 6 z k 2 k x ,3 k
k Now 2 x 3 = 6 k .k k 1 1 1 1 1 1 0 x y z x y z x
y
2
4.
xf .4x 5 y 20z , then z is equal to a. xy x y b. xy 1 c. xy xy d. x y
5.
If 2a = 3b= (12)c then
FOUNTAINHEAD
CA - CPT
1 1 x y
k
1
y
and 6=k
1
2
1 z
reduces to
QUANTITATIVE APTITUDE
1.32
RATIO AND PROPORTION, INDICES, LOGARITHMS
6.
7.
8.
CHAPTER - 1
a. 1 b. 0 c. 2 d. None If 3a=5b=(75)c then the value of ab-c(2a+b) reduces to a. 1 b. 0 c. 3 d. 5 If 2a=3b=(12)c then the value of ab-c(a+2b) reduces to a. 0 b. 1 c. 2 d. 3 If ap=bq=cr=ds and ab = cd then the value of
reduces to
a. ⁄ b. ⁄ c. 0 d. 1 9.
If 2a=4b=8c and abc=288 then the value
is given by
a. b. c. d. Solution: 2a 4b 8c a 22b 23c let a 2b 3c k a k ,2b k and 3c k k k k k Now abc 288 k . . 288 k 3 1728 123 k 12 b and c 2 3 2 3 k 12 k 12 1 1 1 1 1 1 a k 12, b 6 and c 4 Now + + 2 2 3 3 2a 4b 8c 2x12 4x6 8x4 4 4 3 11 96 96 10. If(5.678)x=(0.5678)y=10z then a. b. c. d. None x Solution: 5.678 (0.56780) y 10z (5.678) x 10z and (0.5678) y 10z z x
z y
x
z z
5.678 10 z z z z 10 10 x y 1 5.678 = 10 and 0.5678 = 10 0.5678 x y 10 y 1 1 1 1 1 1 0 ( dividing by z). z x y x y z
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.33
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type– 6: Based on expansion / factorization: 1.
2.
1 1 1 1 1 1 1 18 8 8 8 4 4 2 Value of a a a a a a a a 2 is 1 a. a a 1 a b. a 1 2 c. a a 2 1 2 d. a a 2 1 1 1 1 1 1 1 18 8 8 8 4 4 2 2 a a a a a a a a Solution: 2 1 1 1 1 1 1 1 1 1 1 1 14 8 8 4 4 4 2 4 4 2 2 a a a a a a a a a a a a 2 2 2 2 2 1 1 1 1 1 1 1 1 12 12 12 1 2 2 2 4 4 2 2 a a a a a a a a a a a a a a
x
2n
n1
y 2 x2 y 2 n
n1
...
a. x 2 y 2 b. x 2 y 2 c. x 2 y 2 d. x 2 y 2 3.
4.
1
n n 1n n1 x ... n a. x b. x n1 c. x n1 d. None of these b aab ba x x x b On simplification a a b x ba x
a. b. c. d.
reduces to
1 –1 0 None
aab bba x Solution: a b x ab ba
FOUNTAINHEAD
a b
CA - CPT
a b
aab abb x a b x ab ab
QUANTITATIVE APTITUDE
a b
aabb x ab x ab
a b
x x
a b
1ab 1
1.34
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
3 3 3 Type– 7: Based on if a b c 0, then a b c 3abc
1.
If x = 3 + 3 - , then
– 9x is
a. 15 b. 10 c. 12 d. none of these
2.
3.
4.
3
If a. b. c. d.
⁄
⁄
+
1 3
1 3
prove that
1 3
3
-12x is given by
12 13 15 17
5.
If x= ⁄ + a. 25 b. 26 c. 27 d. 30
⁄
prove that 5x3 -15x is given by
6.
If a ⁄ + a. x + x-1 b. x - x-1 c. 2x d. 0 If a= √ a. 3 b. 0 c. 2 d. 1
⁄
then a3-3a is
7.
FOUNTAINHEAD
3
13 13 13 13 ( x ) 3 3 3 x x 3 3 x 3 3 0 3 3 Solution: 1 x3 3 31 3x 3x3 9 x 9 1 10 x3 3x 3 3 Using (a–b)3 = a3–b3–3ab(a–b) tick the correct of these when x = p1/3 – p–1/3 a. x3+3x = p + 1/p b. x3 + 3x = p – 1/p c. x3 + 3x = p + 1 d. none of these If ax ⁄ + bx ⁄ + c =0 then the value of a3 x2 +b3 x+c3 is given by a. 3abcx b. –3abcx c. 3abc d. –3abc 1 3
CA - CPT
-√
then the value of a3 +3a-2 is
QUANTITATIVE APTITUDE
1.35
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS Type– 8: Miscellaneous examples: 1.
2
–1 –1/2
[1–{1–(1–x )–1} ]
a. b. c. d.
is equal to
X 1/x 1 none of these
Solution: 1 1 1 x
1 1 2 2 1
1 1 1 1 2 1 x
1
1
1 x2 2 x2 1 x2 2 1 1 x2 x 2 x2 2.
1 2
1
x
2x
1 2
1 2
1 x 2 11 1 2 1 x
1 2
x
On simplification, 1/(1+am–n+am–p) + 1/(1+an–m+an–p) + 1/(1+ap–m+ap–n) is equal to
a. b. c. d.
0 a 1 1/a
Solution:
1 1 z
1
c
z
a c
1 1 z
1 c
a
bc
z
b a
1 1 z
c a
z cb
1
z z z z z z b 1 1 z a z a z b z b z c z c z a z b z c a z z b z c z b z c z a z c z a z b 1
b
a b
a
z a z b z c = a 1 z z b z c
3. 4.
If xyz = 1 then the value of a. b. c. d.
is
1 y 1 x 1 1 1 1 1 1 1 1 1 1 x y y (1 y z ) x(1 z x 1 ) 1 x y 1 y z 1 z x
1 y 1 x 1 xyz 1 1 1 1 1 x y y 1 y z x xz 1 1 y 1 x 1 y 1 x 1 1 y x 1 x y 1 y 1 1 x x y 1 1 yz 1 x y 1 . . - would reduce to zero if a + b + c is given by a. 1 b. – 1 c. 0 d. None Solution: x
FOUNTAINHEAD
+
1 0 2 None
Solution:
5.
+
CA - CPT
a2b1c1
.x
b2c1a1
.x
c2a1b1
a2 bc
b2 ca
x 0 x . x .x 3
QUANTITATIVE APTITUDE
c2 ab
x
3
x
a b2 c2 bc ca ab
x3
1.36
RATIO AND PROPORTION, INDICES, LOGARITHMS 6. a. b. c. d.
CHAPTER - 1
a 2 b2 c 2 3 a3 b3 c3 3abc a b c 0 (or a=b=c) bc ca ab would reduce to one if a + b + c is given by 1 0 –1 None
1 1 1 c a c a x x 1 x x 1 x x b 1 1 xc xb 1 xc xb b x x c 1 x c ( xc x a 1) xb ( x a x b 1) xb x c 1 1 x ca x c x ab 1 xb b c This would reduce to 1 only if all denominators are x x 1 for that we require c a b and a b c i.e. a b c 0
Solution:
7.
b
On simplification reduces to a. – 1 b. 0 c. 1 d. 10 2 x3 x3 x y x5x y 3 x6 y 1 2 x3 x32 x y x5x y 3 x2 y 1 x3 y 1 2 x3 x32 x y x5x y 3 x2 y 1 x3 y 1 Solution: 6 x1 x10 y3 x15x (2x3) x1x10(2x5) y3x(3x5) x 2 x1 x3x1 x2 y3 x5 y3 x3x x5x
2 x3.2 y1 32 x y x3 y 1 5x y 3 x x1 x x y3 x 2 x1.2 y3 3 x3 5 .5 x 3 y 1 2 x y y 1 x y 3 2 3 5 x1 y3 x x1 x x y3 x = 1 x 1 x 1 = 1 2 3 5
=
8.
If a. b. c. d.
9.
(
)
then x–y is given by
–1 1 0 None
Show that
a. 1 b. – 1 c. 4 d. 0 10. x 1 x 9 4 3.3 3x x 3 3 3 a. 1 b. – 1 c. 3 d. 0
(
)
( )
( )
√
is given by …..
2 x 14 1 x 3 . 3 Solution: x 33 x.3.3 2 FOUNTAINHEAD
CA - CPT
is given by
2 x 12 12x 3 .3 x 33 x.3.3 2
QUANTITATIVE APTITUDE
1 1 x 2 x 2 2 2
3
3x
3
1
x 2
5x
32
1
x 5 1 2
1
3
1.37
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS =(
11. The value of z is given by the following if z a. 2 b.
)
c. d. Solution: Z
z
Z z
z
1
Z
z. z 2
1 1 ( Z .z ) Z z 2 1 2 z
z
z
Z
z
3 2
3 Z2
z
3 2
1 z2 3 3 3 3 9 Z z z2 z 2 2 z 2 4 2 3 2
12. If 2x 2x1 4 then x x is … a. 2 b. 1 c. 64 d. 27 Solution: 2x 2x1 4 2 x 2x 8 23 x 3
2x 1 1 4 2 x 1 4 2 x x 4 2 2 2
Now x x = 3
= 27
13. Ifab=ba then the value of ( ) a. b. c. d.
reduces to
A B 0 None a b
a Solution: a b 14. If m = a. - 1 b. 0 c. 1 d. None
a 1 b
a b
=a x
and (
1 a b
a
a 1 b
b )
a b
a a a a 1 1 1 a b a b b b b a = = a a 0 ( a b a b ) a
the value of xy is given by
y x Solution: (m y n x ) z b2 b x b y b2 z
____________________________________________ 2 xyz 2 xyz 1
15. If a. b. c. d.
then the value of 1 –1 0 None
16. If to a. 0 b. 1 c. – 1 FOUNTAINHEAD
reduces to
CA - CPT
then the value of
QUANTITATIVE APTITUDE
reduces
1.38
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS d. None ⁄
17. If a. b. c. d.
1.4
⁄
⁄
and
⁄
then the value of 3 (
) is
67 65 64 62
SURDS
1. Surd is typical real number. 2. If
n
a is an irrational number, where n N, n 1 and a is a rational number, is called a surd. e.g.
2,
3
4. b is an irrational number, then a ±
3. Quadratic surd: If a,b are rational numbers, b>0 and is called quadratic surd e.g. 2 +
3,4–2
4. Conjugate surd: Quadratic surd a +
b
3.
b and a- b are called conjugate surds of each other.
Note : (i) Sum of two conjugate quadratic surd is a rational number, (ii) Product of two
conjugate quadratic surds is a rational number e.g. a b a b
= a 2 b is a rational
number. 2 5. a b c a b c e.g. 5 3 5 4x3 5 12 . Also 5 72 5 36x2 5 y 2.
6. If a,b,c,d are rational numbers and b,d>0 and
b,
d are irrational numbers such that
a b c d , then a c and b d . e.g. If a b 3 5 , then a = 3 and b = 5.
7. a Q+ and b Q+ and a 2 b is a quadratic surd. If a 2 4b is a perfect square rational number,
a a 2 4b a a 2 4b then a 2 b . 2 2 8. Rationalising factor: The number multiplying by which to a surd gives a rational number is called rationalizing factor. Note : Rationalising factor of a b is a b . e.g.
(1) Rationalising factor of 2 3 is 2 – (2) Rationalising factor of
e.g.
FOUNTAINHEAD
Rationalising factor of
CA - CPT
b is
5 is
QUANTITATIVE APTITUDE
3.
b.
5.
1.39
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS Type– 1: Based on rationalization:
Multiple Choice Questions 1.
The rationalizing factor of 3
2 - 2 3 is ….
a.
2 + 3 b. 2 - 3 c. 3 2 - 2 3 d. 18 + 12 Solution: Ratioalising factor of 3 2.
Reciprocal of 3-2 a. 2
2 1 b. 3 2 c. 3-2 d. 3 + 2
2 -2
3 =3
2 + 2 3 = 9x2+ 4x3 = 18 12
2 is ….
-3
2 2 2
Solution: Reciprocal of 3-2 3 2 2 3 2 2 9 8 If + + √
2
1 3 2 2
=
=
1 3 2 2 x 3 2 2 3 2 2
= 3.
=
then the value of P is
a. b. c. d. 4.
7/11 3/11 -1/11 -2/11 1 1 1 1 ... ... 2 1 3 2 4 3 n n 1 a. n-1 b. n+1 n +1 c. n -1 d.
Type– 2: Square root of a quadratic surd. 1.
FOUNTAINHEAD
⁄ If then the value of ⁄ is a. b. c. 2 d. -2 Solution: = 2 + 2 2 + 1 (Dividing 2 into 2 parts such that = ( 2 )2 + 2 1 1 1 1 1 1 2 2 2 2 = ( 2 + 1) a = 2 + 1 Now a +a =a 2 + 1 = 2+1+ 2+1 a2
CA - CPT
QUANTITATIVE APTITUDE
2 + (1)2
1.40
RATIO AND PROPORTION, INDICES, LOGARITHMS 2 1
= 2.
3.
2 1 2 1 2 1 2 2 2 1
If then the value of ⁄ a. 2 b. 2 c. 2 d. -2 The square root of x + √ is given by a.
[√
√
]
b.
[√
√
]
c. [√ d. [√
√ √
⁄
] ]
1 2 x 2 ( x y)( x y) 2 1 1 = ( x y) 2 ( x y)( x y) ( x y) = 2 x y x y 2 1 x y x y x x2 y 2 2 The square root of 3 + is
Solution: x x 2 y 2
=
4.
a. √ ⁄ b.
CHAPTER - 1
2
√ ⁄
(√ ⁄
√ ⁄ )
c. Both the above d. None 5.
11 2 30 .....
a. b. c. d. 6.
6 5 6 5
15 2 15 2
17 4 15 ....
a. 15 1 b. 15 1 c. 2 3 5 d. 2 3 5
Type– 3: Based on evaluation from given value of a 1.
If
then the value of
is
a. 21 b. 1 c. 12 d. None FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.41
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS 3 2 3 2 3 2 3 2 6 2 x 5 2 6 3 2 3 2 3 2 3 2 a2 10a 1 Now 2a4 21a3 12a2 a1
Solution: a
a 5 2 6
____________________________________ = 2a (a 2 10a) a(a 2 10a) 2(a 2 10a) 19a 1 = 2a 2 (1) a(1) 2(1) 19a 1 2
= 2a 2 a 2 19a 1 = 2(a 2 10 a) 1 2(1) 1 1
2.
If a = 3a. 10 b. 14 c. 0 d. 15
3.
If a = ( a. b. c. d.
then the value of
is
)then the value of
is
0 1 5 –1
Solution: a
=
1 1 2 2 5 21 2(5 21) (5 21) x 2 a 5 21 5 21 5 21 25 21
2(5 21) 5 21 1 5 21 5 21 10 5 4 2 a 2 2 2 2 2 2 ( ) 1 1 1 a 2 a 2 a 2 2 a 2a. 2 a a a ( 2 )2 2 = 52 – 2 = 23
=
3
4.
1 3 1 1 1 3 3 3 ( ) a a a 3 a 3a a a a a a 3 3 5 3(5) = 125 – 15 = 110 Now a a3 5a2 5a2 a a 1 = 110 – 5 (23) + 5 = 0 3 1 If x , then x 2 10 x 9 ...... 5 2 6 a. 0 b. – 512 c. 512 3
3
3
1000
5.
If x 4 5, then the value of 2 x3 13x2 2 x 33 is …. a. 0 b. – 21 c. 21 54
Type– 4: To evaluate from given values of a and b. 1.
If a =
b=
then the value of
is
a. 10 b. 100 c. 98 d. 99 FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.42
RATIO AND PROPORTION, INDICES, LOGARITHMS 3 2 3 2 3 2 3 2 ab ,b 3 2 3 2 3 2 3 2
Solution: a
=
3 2
3
CHAPTER - 1
3 2 2 3 2 2
2
=
3 2 6 23 2 6 2 10 3 2
1 1 b2 a 2 98 98 a b a b 2ab 10 2(1) 100 2 98 Now 2 2 2 2 a b ab 1 2
2.
2
2
If a =
2
b=
then the value of a + b is
b=
then the value of
a. 10 b. 100 c. 98 d. 99 3.
If a = a. b. c. d.
is
10 100 98 99
Type– 5: Based on compendo-dividendo 1.
If a = a. b. c. d.
then the value of
is given by
1 –1 2 –2
Solution: a
4 6 2 3x2 2 a 2 3 a2 2 2 3 2 3 2 3 2 3 2 2 2 3 a2 2 2 3 2 3
=
3 3 2 3 2
a2 2 a2 3 3 3 2 3 2 3 3 3 2 3 2 3 a2 2 a2 3 3 2 2 3 3 2 3 2
a 2 3
2 2 2 3
a2 3 2 2 2 3 a2 3 2 2 2 3
3 2 3 2 3
3 3 2 3 2 3 2 3 2 2 2 3 2 3 2 3 2 3 2
Type– 6: Miscellaneous sums 1.
If x=
and y =
then
is
a. 5 b. c. d. 4
Solution: Add 45 FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.43
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
1 3 1 4 3 3 3 3 15 1 x Now x 5 2 3 x 3 4 3 4 3 1 3 3 3 4 3 2 3 3 3 4 3 3 3 4 3 3 = 3 3 2 3 3 4 3 3 3 8 3 3 3 3x5 3 5x3 5 = x x 3 3 2 3 6 18 18 6 x 3
2.
If a = √
then the value of [ (
)] is
a. 14 b. 7 c. 2 d. 1 74 3 74 3 74 3 Solution: a x 74 3 74 3 74 3 a7 4 3
a 7 4 3 2
7 4 3 49 48
2
74 3
2
a2 14a 49 48 a(a 14) 1 a a 14 1 1 2
3.
If x = √ a. b. c. d.
√
2
the value of X is given by
-2 1 2 0
Solution: x 2 2 2...x
4.
FOUNTAINHEAD
x 2 x x2 2 x x2 x 2 0 x 2 x 1 0
x 2 (impossible) or x 1 The square root of is given by a. b. CA - CPT
QUANTITATIVE APTITUDE
1.44
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS c. d. Solution: Let
3
9 3 11 2 a 3 v 2
= a 3 3 a 3 b 2 3 a 3 b 2 b 2
9 3 11 2 a 3 v 2
3
3
2
2
2
= 3 3a3 9 2a 2b 6 3ab2 2 2b3 =
3 3a3 6ab2 2 9a 2b 2b3
3a3 6ab2 9 and 9a 2b 2b3 11 on observation a 1, b 1
5.
3
2 9 3 11 2 3 2 3 1 3
The cube root of 16 2 11 5 is ….
a. 5 5 1 b. 5 5 1 c. 5 1 d. 5 1
6.
If
4
2 5 2 5
2 5 2 5
49 20 6 3 2, then 49 20 6 ....
a. 5 2 6 b. 5 2 6 3 2 c. d. 5 6
1.5
LOGARITHM
Logarithm: The logarithm of a number to a given base is the index or the power to which the base must be raised to produce the number. e.g. If a x n hen log a n x, a is base x is index (i) (ii)
25 = 32, then log232 = 5 43
1 1 , then log4 3 64 64
Note: In log a n x, n is any positive real number, a is positive real number other than 1 and x is a real number. FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.45
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
LIST OF FORMULAE 1. a x n log a n x e.g. (i) 33 = 27, then log3 27=3 2. loga 1 = 0, a 1, a is positive real number e.g.
(i) log5 1 = 0
3. loga a = 1 e.g. (i) log5 5 = 1
(ii) log7 7 = 1
4. loga a x x e.g.(i) log223 = 3
(ii) log3 34 = 4
5. alog a n n e.g. (i) 3log3 5
log (ii) e e 6
5
(ii) log5625 = 4, then 54 = 625. (ii) log10 1 = 0
6
6. Logarithm of product : e.g. (i) loga (2x3(=loga 2+loga 3 (ii) log a 5 log a x log a (5x) m 7. Logarithm of quotient : log a log a m log a n n 3 e.g. (i) log10 log10 3 log10 5 5
x (ii) log10 x log10 4 log10 4
n 8. Logarithm of power: log a m n log a m
5 e.g.(i) log a x 5log a x
4 (ii) 4log a 3 log a 3
9. e.g. (i) log 3 2
log10 2 log10 3
(ii)
log10 4 log 5 4 log10 5
(ii)
1 log 5 4 log 4 5
10. 11.
e.g. (i) log3 2
1 log 2 3
12. logb ax logc bx log a c 1 13. If log a x log a y, then x y 14. alog b blog a
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.46
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Characteristic: is integral part of a logarithm of a number. Rules to find characteristic: 1. If a number is greater than or equal to 1, characteristic = Number of digits before decimal pointe.g. (i) Characteristic of 7=1-1=0 (ii) Characteristic of 15.7=2-1=1 (iii) Characteristic of 23400.56 = 5 – 1 = 4. If a number is less than 1 characteristic = Number of zeroes just after decimal point + 1. (Note: Here negative sign is represent as bar) e.g. (i) Characteristic of 0.5 0 1 1 (number of zeroes just after decimal point =0). (ii) Characteristic of 0.07 1 1 2 (iii) Characteristic of 0.000305 3 1 4 (iv) it is a fractional part of logarithm of a number. It is found using logarithm table. Number Characteristic Mantissa Logarithm of a number 786.6
2
0.8957
2.8957
2304
3
0.3624
3.3624
5=5.00
0
0.6990
0.6990
0.007
3
0.8451
3 .8451
Antilogarithm: If x is the logarithm of a given number n with base a, then n is called antilogarithm of x to the base a. i.e. If x loga n, then antiloga x n. e.g. log10 2 0.3010, then antilog10 (0.3010) 2. Rules to find antilog: 1. If characteristic is greater than or equal to zero, number of digits before decimal point = characteristic + 1. 2. If characteristic is less than 0, number of zeroes just after decimal point = 1 characteristic / - e.g. Number Characteristic Mantissa Antilogarithm
2.2345 3.4096
FOUNTAINHEAD
CA - CPT
2 3
QUANTITATIVE APTITUDE
0.2345 0.4096
171.6 0.002568
1.47
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type– 1: Based on law of product
Multiple Choice Questions 1.
2.
log 6 + log 5 is expressed as a. log 11 b. log 30 c. log 5/6 d. none of these Solution: log6 log5 log(6x5) log30 If log x + log y = log (x+y), y can be expressed as a. x–1 b. x c. x/x–1 d. none of these Solution: log x log y log( x y) log xy log( x y) xy x y
3.
4.
5.
6.
FOUNTAINHEAD
ux
log (1 × 2 × 3) is equal to a. log 1 + log 2 + log 3 b. log 3 c. log 2 d. none of these log(1+2+3) is exactly equal to a. log 1 + log 2 + log 3 b. log(1×2×3) c. Both the above d. None Given log2 = 0.3010 and log3 = 0.4771 the value of log 6 is a. 0.9030 b. 0.9542 c. 0.7781 d. none of these The value of a. b. c. d.
7.
xy y x y( x 1) x
is
0 1 -1 None
The value of a. 0 b. 10 c. – 1 d. None
CA - CPT
is
QUANTITATIVE APTITUDE
1.48
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type– 2: Law of Quotient 1.
2.
log 32/4 is equal to a. log 32/log 4 b. log 32 – log 4 c. 23 d. none of these 32 Solution: log log32 log 4 4 log(2a 3b) log a log b, thena ....
3b 2 a. 2b 1 3b b. 2b 1 c.
b2 2b 1
d.
3b 2 2b 1
y Type– 3: Based on log a x y x a
1.
log10000 x
1 x ..... 4
1 100 1 b. 10 1 c. 20 d. None of these
a.
1 1 1 1 4 4 4 Solution: log10000 x x 10000 (10 ) 101 10 4
x Type– 4: Based on log a a x
1.
2.
FOUNTAINHEAD
log28 is equal to a. 2 b. 8 c. 3 d. none of these solution: log 2 8 log 2 23 3 log (1/81) to the base 9 is equal to a. 2 b. ½ CA - CPT
QUANTITATIVE APTITUDE
1.49
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS c. –2 d. none of these 1 solution: log9 log9 o 2 2 81
3.
log 0.0625 to the base 2 is equal to
4.
a. 4 b. 5 c. 1 d. none of these The value of log 0.0001 to the base 0.1 is a. –4 b. 4 c. ¼ d. none of these
Type– 5: Based on 1.
log yn x m
m log y x n
log 2√3 1728 is equal to a. 2 3 b. 2 c. 6 d. none of these Solution: log 2 3 1728
1 log 1 123 (Here 2 3 4x3 12 12 2 and 1728=123 12 2
3 log12 12 3x2 6 1 2 log √2 64 is equal to a. 12 b. 6 c. 1 d. none of these The logarithm of 64 to the base 2√2 is a. 2 b. 2 c. ½ d. none of these
= 2.
3.
4.
5.
FOUNTAINHEAD
The value of log to the base 9 is a. – ½ b. ½ c. 1 d. none of these The logarithm of 21952 to the base of 2 a. Equal b. Not equal c. Have a difference of 2269 CA - CPT
QUANTITATIVE APTITUDE
and 19683 to the base of 3
are
1.50
RATIO AND PROPORTION, INDICES, LOGARITHMS 6.
d. None The sum of the series a. b. c. d. None
CHAPTER - 1
is given by
n Type– 6: Based on log a x n log a x
1.
2.
If 2 log x = 4 log 3, the x is equal to a. 3 b. 9 c. 2 d. none of these Solution: 2log x 4log3 4 log x log3 2 log144 .... a. 2log 4 2log 2 b. 4log 2 2log3 c. 3log 2 4log3 d. 3log 2 4log3 2 144
log x 2log3
log x log32
x9
2 72 2 36
log144
Solution: 2 18
3 9
log(24 x32 ) log 24 log 32
4log 2 2log 3
3 3 1
Type– 7: Simplification 1.
The value of log2 log2 log2 16 a. 0 b. 2 c. 1 d. none of these Solution: log 2 log 2 log 2 16 = log 2 log 2 log 2 24 = log 2 log 2 4
2.
FOUNTAINHEAD
log a a n n = log 2 log 2 22
= log 2 2 1
The value of log2 [log2 {log3 (log3273)}] is equal to a. 1 b. 2 c. 0 d. none of these CA - CPT
QUANTITATIVE APTITUDE
1.51
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type– 8: Expressing log in x,y,m,n 1.
Given that log102 = x, log103 = y, then log101.2 is expressed in terms of x and y as a. x + 2y – 1 b. x + y – 1 c. 2x + y – 1 d. none of these 22 x3 12 Solution: log10 1.2 log10 log10 = log10 22 log10 3 log10 10 10 10 = 2log10 2 log10 3 1
2.
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 10 x Solution: log 2 = log10 log x log y 2 = log106log x 2log y y = 1 (m n) 2(m n)
3.
= 2x y 1
= 1 m n 2m 2n
= 1 m 3n
Given that log102 = x and log103 = y, the value of log1060 is expressed as a. x – y + 1 b. x + y + 1 c. x – y – 1 d. none of these
Type– 9: Harder Simplification 1.
The simplified value of 2 log105 + log108 – ½ log104 is a. ½ b. 4 c. 2 d. none of these 1 1 Solution: 2log10 5 log10 8 log10 4 = log10 52 log10 8 log10 4 2 2 25x8 2 = log10 25 log10 8 log10 2 = log10 log10 100 log10 10 2 10
2.
The value of is
is
a. b. c. d. 3.
FOUNTAINHEAD
0 1 2 –1 16 25 81 7log 5log 3log ..... 15 24 80 a. 0 CA - CPT
QUANTITATIVE APTITUDE
1.52
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
b. 1 c. log2 d. log3
Type– 10: Simplification 1.
log [1 – {1 – (1 – x2)–1}–1]–1/2 can be written as a. log x2 b. log x c. log 1/x d. none of these
Solution: log 1 1 1x 1 x 2 1 1 log 1 2 1 x
1 log 2 x 2.
1 2
log
1 1 2 2 1
1 2
1 1 log 1 1 2 1 x
1 x log 1 2 x 2
1 2
1 2
x 1 x log x2 2
2
1 2
1 log x x 1
The simplified value of log √ a. b. c. d.
is
log 3 log 2 log ½ none of these 4
3
1 Solution: log 729 9 .27
4
log 36 3
6 3
1 3
log 4 26 3 32
log 4 362
1
4 3 3
3
log 4 36 3 32.34
log 4 34 log3
Type– 11: Based on logb a.logc b.log a c 1 1.
The value of (logba × logcb × logac)3 is equal to a. 3 b. 0 c. 1 d. none of these Solution: (logb ax logc bx log a c) ( ).
2. a. b. c. d. FOUNTAINHEAD
( ).
3
log a log b log c x x log b log c log a
3
= 13 = 1
( )is equal to
0 1 –1 None CA - CPT
QUANTITATIVE APTITUDE
1.53
RATIO AND PROPORTION, INDICES, LOGARITHMS 3.
4.
(
(
).
).
CHAPTER - 1
( ) is equal to
a. 0 b. 1 c. -1 d. None The value of the following expression a. t b. Abcdt c. (a + b +c + d + t) d. None
is given by
Type–12: Solving equations involving logarithm 1.
If log2x + log4x + log16x = 21/4, these x is equal to a. b. c. d.
8 4 16 none of these
21 4 1 1 21 4 2 1 21 21 1 1 log 2 x log x x log 2 x 1 log 2 x log 2 x 2 4 4 4 4 4 2 4 Solution: log 2 x log 4 x log16 x
7 21 log 2 x 4 4
log 2 x
21 4
21 4 x 4 7
log 2 x log 22 x log 24 x
log 2 x 3
2.
If log 2 x log 4 x 6, then the value of x is …..
3.
a. 16 b. 32 c. 64 d. 128 On solving the equation a. b.
⁄
[
(
x 23 8
)] = 2 we get the value of t as
⁄ ⁄
⁄ c. d. None 2
1 5 1 Solution: log 1 logt log 4 32 2 log5 log 22 2 log5 log 2 2 2 2 4 2 5
4
4.
FOUNTAINHEAD
5 1 5 625 5 1 logt t 4 t 2 16 2 2 4 [ ( )] =1 we get the value of t as On solving the equation a. 8 b. 18 c. 81 d. 6561 CA - CPT
QUANTITATIVE APTITUDE
1.54
RATIO AND PROPORTION, INDICES, LOGARITHMS 5.
CHAPTER - 1
If log 2 log3 log 2 x 1, then x .... a. 128 b. 256 c. 512
6.
7.
d. None of these Log ( ) +log a= 10 if the value of a is given by a. 0 b. 10 c. – 1 d. None On solving the equation logt + log(t-3) = 1 we get the value of t as a. 5 b. 2 c. 3 d. 0
Type–13: To find value of logarithm to any base from given base 10 value. 1.
The value of log825 given log 2 = 0.3010 is a. 1 b. 2 c. 1.5482 d. None of these Solution: log8 25 100 log10 4
log10 25 log10 8
2 2log10 2 3log10 2
Type–14:
log10 100 log10 4 = log10 8
2 2(0.3010) 3(0.3010)
log10 102 log10 22 = log10 23
2 0.6020 1.3980 1.5482 0.9030 0.9030
1 logb a log a b
1.
(
)
(
)
(
)
is equal to
a. 0 b. 1 c. 3 d. -1 Solution:
FOUNTAINHEAD
1 1 1 1 log a (bc) 1 log b (ca) 1 log c ( ab)
1 1 1 log a a log a (bc) log b b log b (ca) log a c log a ( ab)
CA - CPT
QUANTITATIVE APTITUDE
1.55
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
1 1 1 log a abc logb abc log a abc
log abc a logabc b logabc c
log abc abc 1
2.
The value of a. b. c. d.
4.
)
(
)
(
)
(
)
(
5.
)
is equal to
0 1 2 –1
If a. b. c. d.
is
0 1 –1 None
3. a. b. c. d.
(
then the value if z is given by abc a+b+c a(b + c) (a + b)c ⁄ ( )
⁄ ( )
⁄ ( )
is equal to
a. 0 b. 1 c. 3
6.
d. - 1 has a value of a. b. c. d.
a b (a + b) None
Type–15: Based on alog b blog a 1.
The value of (
⁄
)
.(
⁄
)
.(
⁄
)
is
a. 0 b. 1 c. – 1 d. None
(bc)
b
c
a
b
b
c
c
a
a
log c
log a
log b
log c
log c
log a
log a
log b
log b
= a log
FOUNTAINHEAD
c a
.(ca) a
log b
.blog
CA - CPT
b c
.(ab) a
log b
.clog
b c
=b c
log a
.c
.c
c a x a b
.a b a x c b
.a
= a log .blog .clog
QUANTITATIVE APTITUDE
b c x c a
.b
c b
a c
= a log .blog .clog
b a
1.56
RATIO AND PROPORTION, INDICES, LOGARITHMS c = b
log a
a . c
log b
b . a
log c
c log a log b blog c = log a x log b x log c b c a
2.
3.
CHAPTER - 1
alog b blog a )
=1(
has a value of a. 1 b. 0 c. – 1 d. None The value of is a. 0 b. 1 c. – 1 d. None
⁄
.
⁄
.
⁄
Type–16: Based on ratio 1.
the value of
If a. b. c. d.
.
.
is given by
0 1 –1 None
Solution: Let
log a log b log c k yz zx x y
log a log b log c k, k and k yz zx x y
log a k ( y z),log b k ( z x) and log c k ( x y)
log a y z .b z x .c x y log a y z log b z x log c x y = ( y z)log a ( z x)log b ( x y)log c = ( y z).k ( y z ) ( z x).k ( z x) ( x y).k ( x y) = k y 2 z 2 z 2 x 2 x 2 y 2 0 log1.
a y z .b z x .c x y 1
2.
If a. b. c. d.
FOUNTAINHEAD
the value of abc is 0 1 –1 None
CA - CPT
QUANTITATIVE APTITUDE
1.57
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS Type–17: Solve by assuming k 1.
If
the value of
is
a. 0 b. 1 c.
–1 d. None b c
c a
a b
b c
b c
c a
c a
a b
(bc)log .(ca)log .(ab)log = blog .clog .clog .a log .a log .blog c
=a
a
log a log b
c b
log a
=
2.
b
.b
a
log c log b
a . c
log b
b
.c
c
log c log a
b . a
=a
log c
=
.b
b a x b
log c
.c
b c x a
log c
c log a log b blog c x x blog a c log b a log c
If a. b. c. d.
c a x b
log a
=a
a b
c
a
b
log b
log c
log a
.b
alog b blog a )
=1(
the value of
.c
is
0 1 –1 None
1 1 1 1 1 1 log a log b log c k log a k , log b k , log c k 2 3 5 2 3 5 log a 2k ,log b 3k ,log c 5k 4log a 8k ,log b log c 3k 5k 8k
Solution: Let
log a 4 8k ,log bc 8k log a 4 log bc a 4 bc a 4 bc 0
3.
If a = a.
then the value of
(abcd) is
b. c. 1+2+3+4 d. None
Type–18: To obtain relation between variances: 1.
If a. 2 b. 5 c. 7 d. 3
(loga+logb) then the value of
is
ab ab 1 ab log ab log a log b 2log log ab log 3 2 3 3 2
Solution: log
a 2 2ab b 2 ab a2 2ab b2 9ab a 2 b2 7ab 9
a 2 b2 7 ab ab FOUNTAINHEAD
CA - CPT
(
dividing by ab)
QUANTITATIVE APTITUDE
a b 7 b a 1.58
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS 2.
a b ab 1 (log a log b), then ..... b a 4 2
If log
a. b. c. d.
12 14 16 18
Type–19: From given relation obtain relation in log 1.
If a. 0 b. 1 c. -1 d. 3
then the value of
(
)
(
) is equal to
Solution: a3 b3 0 (a b)(a 2 ab b2 ) 0
a 2 ab b2 0
a b 0 a2 2ab b2 3ab
a b 3ab log(a b)2 log(3ab) 2log(a b) log3 log a log b 2
log(a b)
2.
1 1 log3 log a log b log(a b) log3 log a log b 0 2 2 then the value of is
If a. b. c. d.
0 1 –1 7
Type–20: 1.
If (
)
(
)
then the value of
is
a. 3 b. – 3 c. d.
Solution: 4.8 0.48 1000 x log 4.8 y log0.48 log1000 3 x
y
x log 4.8 3 and y log0.48 3 log 4.8
FOUNTAINHEAD
3 3 and log0.48= x y
3 3 3 3 4.8 1 1 1 log 4.8 log 0.48 log 3x 2 log10 1 x y x y 0.48 x y 3
CA - CPT
QUANTITATIVE APTITUDE
1.59
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Type–21: 1.
then the value of alog ( ⁄ ) is
If a. b. c. d.
3log x log x 6log x 5log x
Solution: x
2 a3
y
3a
2a
x
6 a
.y
5a
x 2 a3 y 2 a x3a 6 a x 5 a 1 3 a 9 1 x y y .x
a
x x x x9 x3 a log 3log x y y y
Type–22: Add 91 1.
If x = a. 0 b. 1 c. – 1 d. 2
then the value of xyz – x – y – z is
Solution: log a bc, y logb ca and z log c ab a x bc(1), b y ca...(2) and c z ab....(3) Now a xyz x yz =
(a x ) yz a x .b y .a z
(bc) yz (b y ) z (c z ) y (ca)2 (ab)3 c z .a z .a y .b y c z .b y = x y z from(1) = = from(2),(3) = x y z = x a .a .a a x .a y .a z a x .a y .a z a a .a .a
ab.ca a 2 .a x from(2),(3) = = = a 2 xyz x y z 2 x x a a 2.
If I = 1+ a. b. c. d.
then the value of
is
0 1 –1 3
Type–23: To find number of digits 1.
FOUNTAINHEAD
The number of digits in the numeral for 264 is …. a. 18 b. 19 c. 20 d. 21 CA - CPT
QUANTITATIVE APTITUDE
1.60
RATIO AND PROPORTION, INDICES, LOGARITHMS
CHAPTER - 1
Solution: Let x 264 log x 64log 2 log x 64(0.30103) 19.26592 Number of digits in 264 19 1 20
Type–24: To find number of zeroes just after decimal point: 1.
The number of zeroes between decimal point and first significant figure (in the numeral for (0.007)15 [Given log 7 = 0.8451] a. 31 b. 32 c. 33 d. 34 Solution: Let x (0.007)15 log x 15log0.007 = 15(3.8451) = 15(-3+0.8451) = -45 + 12.6765 = 33.665 Number of zeroes between decimal point and first significant figure = 33-1=32
Type–25: Miscellaneous Sums 1.
( ) For any three consecutive integers x y z the equation is a. True b. False c. Sometimes true d. cannot be determined in the cases of variables with cyclic order Solution: x, y, z are three consecutive integers. Suppose that x n 1, y n and z n 1
log(1 xz) 2log y log[1 (n 1)(n 1)] 2log n
= log[1 n2 1]log n2
= log n2 log n2 = 0 The statement is true. 2.
If
then the value of n is
a. b. c. d. Solution: x
en en 1 en e n (Invertendo) en e n x en e n
1 n 1 x en e n en e n 1 x 2en e2 n n n n n (Compondo) n 1 x 1 x e e e e 1 x 2e 1 1 x 1 x log e 2n n = log e 2 1 x 1 x
3.
FOUNTAINHEAD
log3 8 .... log9 16.log 4 10 CA - CPT
QUANTITATIVE APTITUDE
1.61
CHAPTER - 1
RATIO AND PROPORTION, INDICES, LOGARITHMS a. 3log10 2 b. 7log10 3 c. 3log e 2 d. None of these log 3 23 x log 4 3log 3 2x log 22 3x2log 2 Solution: = log 32 24 x log10 4 log 2x log10 2x log10 3 2 4.
If a log24 12; b log36 24 and c log 48 36, then 1 abc ..... a. b. c. d.
2ab 2bc 2ca 2b2
Solution: 1 abc 1 log 24 12.log36 24.log 48 36 =1+
1 =
5.
3log10 2
log12 log 48
log 48 log12 log48x12 = log 48 log48
=
=
log12 log24 log36 x x log24 log36 log48
log 242 log 48
2log24 log36 = 2log36 24xlog 48 36 = 2bc x log36 log48
log4 ( x2 x) log 4 ( x 1) 2 Find x a. b. c. d.
16 0 –1 None of these
x2 x x( x 1) 2 log 4 x 2 x 42 2 log 4 x 1 x 1
Solution: log 4
6.
Find the value of log 24 log(23 ) log(4)2 a. b. c. d.
x 16
x
X 10 1 none of these x
3 2 Solution: log10 25 log10 (2 ) log10 (4) = log10 5 log10 8 log10 16
x
x
7.
FOUNTAINHEAD
2 5x16 x = log10 log10 10 1 1 8 If log a b log a c 0, then…. a. b=c b. b=-c c. b=c=1 d. b and are reciprocals Solution: log a bc log a 1 bc 1
CA - CPT
QUANTITATIVE APTITUDE
b and c are reciprocals 1.62
RATIO AND PROPORTION, INDICES, LOGARITHMS 8.
CHAPTER - 1
2log x 2log x2 2log x3 ... 2log xn Will be ….. n(n 1)log x a. 2 b. n(n 1)log x
c. n2 log x d. None of these Solution: 2log.x 2log x2 2log x3 ... 2log xn = 2log x 4log x 6log x ... 2n log x = (2 4 6 ... 2n)log x = n(n 1)log x
9.
log x 3 11 log10 x Solve 10 2 2 3 a. 10-1 b. 102 c. 10 d. 103 Solution: 3log10 x 9 22 2log10 x 12 log10 x 12 22 9 1 x 101
10. If n m! where m is a positive integer > 2), then the value of 1 1 1 1 ... .... log 2 n log3 n log 4 n log m n a. 1 b. 0 c. – 1 d. 2 1 1 1 1 Solution: log n 2 log n 3 log n 4 ... log n m ... log 2 n log3 n log 4 n log n m = log(2x3x4x...xm) log m! log n n 1
FOUNTAINHEAD
CA - CPT
QUANTITATIVE APTITUDE
1.63
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