Ratio And ProportionIndicesLogarithms– CA Foundation, CPT notes, PDF
This article is about Ratio And ProportionIndicesLogarithms Business Mathematics and Logical Reasoning & Statistics for CA foundation CPT students. we also provide PDF file at the end.
What we will study in this chapter: Ratio and ProportionIndicesLogarithms
UNIT I : RATIO
LEARNING OBJECTIVE
After reading this unit a student will learn –
 How to compute and compare two ratios;
 Effect of increase or decrease of a quantity on the ratio;
 The concept and application of inverse ratio.
UNIT OVERVIEW
We use ratio in many ways in practical fields. For example, it is given that a certain sum of money is divided into three parts in the given ratio. If first part is given then we can find out total amount and the other two parts.
In the case when ratio of boys and girls in a school is given and the total number of student is also given, then if we know the number of boys in the school, we can find out the number of girls of that school by using ratios.
1.1 RATIO
A ratio is a comparison of the sizes of two or more quantities of the same kind by division.
If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. Thus, the ratio of a to b = a/b or a : b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent.
For example, in the ratio 5 : 6, 5 & 6 are called terms of the ratio. 5 is called first term and 6 is called second term.
1.1.2 Remarks
 Both terms of a ratio can be multiplied or divided by the same (nonzero) number. Usually, a ratio is expressed in lowest terms (or simplest form).
Illustration I:
12 : 16 = 12/16 = (3 x 4)/(4 x 4) = 3/4 = 3 : 4
 The order of the terms in a ratio is important.
Illustration II:
3 : 4 is not same as 4 : 3
 The ratio exists only between quantities of the same kind.
Illustration III:
 There is no ratio between a number of students in a class and the salary of a teacher.
 There is no ratio between the weight of one child and the age of another child.
 Quantities to be compared (by division) must be in the same units.
Illustration IV:
(i)  Ratio between 150 gm and 2 kg  =  Ratio between 150 gm and 2000 gm 
=  150/2000 = 3/40 = 3 : 40  
(ii)  Ratio between 25 minutes and 45 seconds  =  Ratio between (25 x 60) sec. and 45 sec. 
=  1500/45 = 100/3 = 100 : 3 
Illustration V:
 Ratio between 3 kg and 5 kg = 3/5
 To compare two ratios, convert them into equivalent like fractions.
Illustration VI: To find which ratio is greater __________
2\frac { 1 }{ 3 } :3\frac { 1 }{ 3 } ;3.6:4.8
Solution : 7/3 : 10/3 = 7 : 10 = 7/10
3.6 : 4.8 = 3.6/4.8 = 36/48 = 3/4
L.C.M of 10 and 4 is 20.
So, 7/10 = (7 x 2)/(10 x 2) = 14/20
And 3/4 = (3 x 5)/(4 x 5) = 15/20
As 15>14 so, 15/20 > 14/20 i.e. 3/4 > 7/10
Hence, 3.6 : 4.8 is greater ratio.
 If a quantity increases or decreases in the ratio a : b then new quantity = b of the original quantity/a
The fraction by which the original quantity is multiplied to get a new quantity is called the factor multiplying ratio.
Illustration VII: Rounaq weighs 56.7 kg. if he reduces his weight in the ratio 7 : 6, find his new weight.
Solution:  Original weight of Rounaq = 56.7 kg He reduces his weight in the ratio 7 : 6 His new weight = (6 x 56.7)/7 = 6 x 8.1 = 48.6 kg 
Applications:
Example 1: Simplify the ratio 1/3 : 1/8 : 1/6
Solution: L.C.M. of 3, 8 and 6 is 24.
1/3 : 1/8 : 1/6 = 1 x 24/3 : 1 x 24/8 : 1 x 24/6
= 8 : 3 : 4
Example 2: The ratio of the number of boys to the number of girls in a school of 720 students is 3 : 5. If 18 new girls are admitted in the school, find how many new boys may be admitted so that the ratio of the number of boys to the number of girls may change to 2 : 3.
Solution: The ratio of the number of boys to the number of girls = 3 : 5
Sum of the ratios So, the number of boys in the school And the number of girls in the school  = = =  3 + 5 (3 x 720)/8 (5 x 720)/8  = = =  8 270 450 
Let the number of new boys admitted be , then the number of boys become (270 + ).
After admitting 18 new girls, the number of girls become 450 + 18 = 468
According to given description of the problem, (270 + )/468 = 2/3
Or, 3 (270 + ) = 2 468
Or, 810 + 3 = 936 or, 3 = 126 or, = 42.
Hence the number of new boys admitted = 42.
1.1.3 Inverse Ratio
One ratio is the inverse of another if their product is 1. Thus a : b is the inverse of b : a and viceversa.
 A ratio a : b is said to be of greater inequality if a>b and of less inequality if a<b.
 The ratio compounded of the ratios a : b and c : d is ac : bd.
For example compound ratio of 2 : 3, 5 : 7 and 4 : 9 is 40 : 189.
 A ratio compounded of itself is called its duplicate ratio.
Thus is the duplicate ratio of a : b. similarly, the triplicate ratio of a : b is .
For example, duplicate ratio of 2 : 3 is 4 : 9. Triplicate ratio of 2 : 3 is 8 : 27.
 The subduplicate ratio of a : b is and the subtriplicate ratio of a : b is
For example subduplicate ratio of 4 : 9 is = 2 : 3
And subtriplicate ratio of 8 : 27 is = 2 : 3.
 If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be incommensurable. cannot be expressed as the ratio of two integers and therefore, and are incommensurable quantities.
 Continued Ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a : b : c.
Applications:
Illustration I: The continued ratio of ₹ 200, ₹ 400 and ₹ 600 is ₹ 200 : ₹ 400 : ₹ 600 = 1 : 2 : 3.
Example 1: The monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenditures are in the ratio 7 : 9. If each saves ₹ 50 per month, find their monthly incomes.
Solution: Let the monthly incomes of two persons be ₹ 4 and ₹ 5 so that the ratio is
₹ 4 : ₹ 5 = 4 : 5. If each saves ₹ 50 per month, then the expenditures of two persons are ₹(4 – 50) and ₹ (5 – 50).
or
Or, , or,
Hence, the monthly incomes of the two persons are ₹ 4 x 100 and ₹ 5 x 100 i.e. ₹ 400 and ₹ 500.
Example 2: The ratio of the prices of two houses was 16 : 23. Two years later when the price of the first has increased by 10% and that of the second by ₹ 447, the ratio of the prices becomes 11 : 20. Find the original prices of the two houses.
Solution: Let the original prices of two houses be ₹ 16 and ₹ 23 respectively. Then by the given conditions,
Or, , or,
Or, , or, ;
Hence, the original prices of two houses are ₹ 16 x 52 and ₹ 23 x 53, i.e., ₹ 848 and ₹ 1,219.
Example 3: 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.
Solution: Let x be the original number of workers and ₹ y the (average) wages per workers. Then the total wages before changes = ₹ xy.
After reduction, the number of workers = (11x )/15
After increment, the (average) wages per workers = ₹ (25y )/22
The total wages after changes = (₹ = ₹
Thus, the total wages of workers get decreased from ₹ to ₹ 5 /6
Hence, the required ratio in which the total wages decrease is : = 6 : 5.
EXERCISE 1(A)
Choose the most appropriate option (a) (b) (c) or (d).
 The inverse ratio of 11 : 15 is
(a) 15 : 11  (b)  (c) 121 : 225  (d) None of these 
 The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is
(a) 16  (b) 60  (c) 22  (d) 20 
 The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is
(a) 5  (b)  (c) 7  (d) None of these 
 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 
 The duplicate ratio of 3 : 4 is
(a)  (b) 4 : 3  (c) 9 : 16  (d) None of these 
 The subduplicate 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 subtriplicate 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 duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and sub0triplicate ratio of 125 : 512 is
(a) 4 : 512  (b) 3 : 32  (c) 1 : 12  (d) None of these 
 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 
 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 
 The angles of a triangle are in ratio 2 : 7 : 11. The angles are
(a)  (b) 
(c)  (d) None of these 
 Division of ₹ 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 
 Anand earns ₹ 80 in 7 hours and Pramod ₹ 90 in 12 hours. The ratio of their earnings is
(a) 32 : 21  (b) 23 : 12  (c) 8 : 9  (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 
 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 
 If : = 3 : 4, the value of is
(a) 13 : 12  (b) 12 : 13  (c) 21 : 31  (d) None of these 
 If : is the subduplicate ratio of then is
(a)  (b)  (c)  (d) None of these 
 If 2s : 3t is the duplicate ratio of 2 – : 3 – then
(a)  (b)  (c)  (d) None of these 
 If = 2 : 3 and = 4 : 5, then the value of is
(a) 71 : 82  (b) 27 : 28  (c) 17 : 28  (d) None of these 
 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 
 Daily earnings of two persons are in the ratio 4 : 5 and their daily expenses are in the ratio 7 : 9. If each saves ₹ 50 per day, their daily earnings in ₹ are
(a) (40, 50)  (b) (50, 40)  (c) (400, 500)  (d) None of these 
 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 
SUMMARY
 A ratio is a comparison of the sizes of two or more quantities of the same kind by division.
 If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. it is written as a : b. Thus, the ratio of a to b = 1/b or a : b.
 The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent.
 The ratio compounded of the two ratios a : b and c : d is ac : bd.
 A ratio compounded of itself is called its duplicate ratio. is the duplicate ratio of a:b. Similarly, the triplicate ratio of a : b is
 For any ratio a : b, the inverse ratio is b : a
 The subduplicate ratio of a : b is a : b and the subtriplicate ratio of a : b is
 Continued Ratio is the relation (or compassion) between the magnitudes of three or more Quantities of the same king. The continued ratio of three similar quantities a, b, c is written as a : b : c.
UNIT II: PROPORTIONS
LEARNING OBJECTIVES
After reading this unit a student will learn –
 What is proportion?
 Properties of proportion and how to use them.
UNIT OVERVIEW
 PROPORTIONS
If the income of a man is increased in the given ratio and if the increase in his income is given then to find out his new income, Proportion problem is used.
Again if the ages of two men are in the given ratio and if the age of one man is given, we can find out the age of another man by Proportion.
An equality of two ratios is called a proportion. Four quantities a, b, c, d are said to be in proportion if a : b = c : d (also written as a : b :: c : d) i.e. a/b = c/d i.e. if ad = bc.
The quantities a, b, c, d are called terms of the proportion; a, b, c and f are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).
If a : b = c : d then d is called fourth proportional.
If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc
i.e. product of extremes = product of means.
This is called cross product rule.
Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion if a : b = b : c i.e. a/b = b/c i.e. = ac.
If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional.
Thus, if b is mean proportional between a and c, then = ac i.e. b = .
When three or more numbers are so related that the ratio of the first to the second, the ratio of the second to the third, third to the fourth etc. are all equal, the numbers are said to be in continued proportion. We write it as
………………….. when
and are in continued proportion. If a ratio is equal to the reciprocal of the other, then either of them is in inverse (or reciprocal) proportion of the other. For example 5/4 is in inverse proportion of 4/5 and viceversa.
Note: in a ratio a : b, both quantities must be of the same kind while in a proportion a : b = c : d, all four quantities need not be of the same type. The first two quantities should be of the same kind and last two quantities should be of the same kind.
Applications:
Illustration I:
₹ 6 : ₹ 8 = 12 toffees : 16 toffees are in a proportion.
Here 1^{st} two quantities are of same kind and last two are of same kind.
Example 1: The numbers 2.4, 3.2, 1.5, 2 are in proportion because these numbers satisfy the property the product of extremes = product of means.
Here 2.4 x 2 = 4.8 and 3.2 x 1.5 = 4.8
Example 2: Find the value of if 10/3 : :: 5/2 : 5/4.
Solution: 10/3 : = 5/2 : 5/4
Using cross product rule, x 5/2 = (10/3) x 5/4
Or, = (10/3) x (5/4) x (2/5) = 5/3
Example 3: Find the fourth proportional to 2/3, 3/7, 4.
Solution: If the fourth proportional be , then 2/3, 3/7, 4, are in proportion.
Using cross product rule, (2/3) x = (3 x 4)/7
Or, = (3 x 4 x 3)/(7 x 2) = 18/7.
Example 4: Find the third proportion to 2.4 kg, 9.6 kg.
Solution: Let the third proportion to 2.4 kg are in continued proportion since = ac
So, 2.4/9.6 = 9.6/ or, = (9.6 x 9.6)/2.4 = 38.4
Hence, the third proportional is 38.4 kg.
Example 5: Find the mean proportion between 1.25 and 1.8.
Solution: Mean proportion between 1.25 and 1.8 is = 1.5.
 Properties of Proportion
 If a : b = c : d, then ad = bc
Proof. ; ad = bc (By cross – multiplication)
 If a : b = c : d, then b : a = d : c (Invertendo)
Proof. or , or
Hence, b : a = d : c.
 If a : b = c : d, then a : c = b : d (Alternendo)
Proof. or, ad = bc
Dividing both sides by cd, we get
, or , i.e. a : c = b : d.
 If a : b = c : d, then a + b : b = c + d : d (Componendo)
Proof. , or, =
Or, , i.e. a + b : b = c + d : d.
 If a : b = c : d, then a – b : b = c – d : d (Dividendo)
Proof. ,
, i.e. a – b: b = c – d : d.
 If a : b = c : d, then a + b : a – b = c + d : c – d (Componendo and Dividendo)
Proof. , or , = , or ………………………. 1
Again , or …………………………………………….2
Dividing (1) by (2) we get
, i.e. a + b : a – b = c + d : c – d
 If a : b = c : d = e : f = ………………………………, then each of these ratios (Addendo) is equal ( a + c + e + ………….) : (b + d + f + ………….)
Proof. …………………… (say) k,
a = bk, c = dk, e = fk, ……………………….
Now a + c + e………….. = k (b + d + f) ………… or
Hence, (a + c + e + …………..) : (b + d + f + …………..) is equal to each rati
Example 1: If a : b = c : d = 2.5 : 1.5, what are the value of ad : bc and a + c : b + d?
Solution: We have
From (1) ad = bc, or, , i.e. ad + bc = 1 : 1
Again from (1) =
= , i.e. a + c : b + d = 5 : 3
Hence, the values of ad :bc and a + c : b + d are 1 : 1 and 5 : 3 respectively.
Example 2: If , then prove that
Solution: We have
or = 2
Example 3: A dealer mixes tea costing ₹ 6.92 per kg. with tea costing ₹ 7.77 per kg. and sells the mixture at ₹ 8.80 per kg. and earns a profit of 17½% on his sale price. In what proportion does he mix mix them?
Solution: Let us first find the cost price (C.P.) of the mixture. If S.P. is ₹ 100, profit is 17½%. Therefore C.P. = ₹ (100 – 17½ ) = ₹ 82 ½ = ₹
If S.P. is ₹ 8.80, C.P. is (165 x 8.80)/(2 x 100) = ₹ 7.26
C.P. of the mixture per kg = ₹ 7.26
2^{nd} difference = Profit by selling 1 kg of 2^{nd} kind @ ₹ 7.26
= ₹ 7.77 – ₹ 7.26 = 51 Paise
1^{st} difference = ₹ 7.26 – ₹ 6.92 = 34 Paise
We have to mix the two kinds in such a ratio that the amount of profit in the first case must balance the amount of loss in the second case.
Hence, the required ratio = (2^{nd} diff.) : (1^{st}. diff.) = 51 : 34 = 3 : 2.
EXERCISE 1(B)
Choose the most appropriate option (a) (b) (c) or (d).
 The fourth proportional to 4, 6, 8 is
(a) 12  (b) 32  (c) 48  (d) None of these 
 The third proportional to 12, 18 is
(a) 24  (b) 27  (c) 36  (d) None of these 
 The mean proportional between 25, 81 is
(a) 40  (b) 50  (c) 45  (d) None of these 
 The number which has the same ratio to 26 that 6 has to 13 is is
(a) 11  (b) 10  (c) 21  (d) None of these 
 The fourth proportional to 2a, , c is
(a) Ac/2  (b) Ac  (c) 2/ac  (d) None of these 
 If four numbers 1/2, 1/3, 1/5, 1/ are proportional then is
(a) 6/5  (b) 5/6  (c) 15/2  (d) None of these 
 The mean proportional between and is
(a) 18  (b) 81  (c) 8  (d) None of these 
(Hint: Let z be the mean proportional and z =
 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 
 If a/3 = b/4 = c/7, then a + b + c/c is
(a) 1  (b) 3  (c) 2  (d) None of these 
 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 
 If , the value of is
(a) 1  (b) 3/5  (c) 5/3  (d) None of these 
 If = 3/4, the value of is
(a) 2 : 9  (b) 7 : 2  (c) 7 : 9  (d) None of these 
 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 
 If , then the value of is
(a) 6/23  (b) 23/6  (c) 3/2  (d) 17/6 
 If then is
(a) 2 : 3 : 4  (b) 4 : 3 : 2  (c) 3 : 2 : 4  (d) None of these 
 Division of ₹ 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 
 The sum of the ages of 3 persons is 150 years. 10 years ago their wages 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 
 The numbers 14, 16, 35, 42 are not in the proportion. The fourth term for which they will be in proportion is
(a) 45  (b) 40  (c) 32  (d) None of these 
 If , implies , 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) Subtahendo  (b) Addendo  (c) Invertendo  (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 
 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 
 12, 16, *, 20 are in proportion. Then * is
(a) 25  (b) 14  (c) 15  (d) None of these 
 4, *, 9, 13 ½ are in proportion. Then * is
(a) 6  (b) 8  (c) 9  (d) None of these 
 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 
 If then is
(a) 4  (b) 2  (c) 7  (d) None of these 
 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 then
(a)  (b) 
(c)  (d) None of these 
 If then is
(a) 5/2  (b) 4  (c) 5  (d) None of these 
 If then
is
(e) 5/2  (f) 4  (g) 5  (h) None of these 
SUMMARY
 p : q = r : s => q : p = s : r (Invertendo)
(p/q = r/s) => (q/p = s/r)
 a : b = c : d => a : c = b : d (Alternendo)
(a/b = c/d) => (a/c = b/d)
 a : b = c : d => a + b : b = c + d : d (Componendo)
(a/b = c/d) => (a + b)/b = (c + d)/d
 a : b = c : d => a – b : b = c – d : d (Dividendo)
(a/b = c/d) => (a – b)/b = (c – d)/d
 a : b = c : d => a + b : a – b = c + d : c – d (Componendo&Dividendo)
(a + b)/(a – b) = (c + d)/(c – d)
 a : b = c : d = a + c : b + d (Addendo)
(a/b = c/d = a + c/b + d)
 a : b = c : d = a – c : b – d (Subtrahendo)
(a/b = c/d = a – c/b – d)
 If a : b = c : d = e : f = ………. Then each of these ratios = (a – c – e – …….) : (b – d – f – …….)
 The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).
 If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc i.e. product of extremes = product of means. This is called cross product rule.
 Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion.
 If a : b = b : c i.e. a/b = b/c i.e. = ac
 If a, b, c are in continuous proportion, then the middle term b is called the man proportional between a and c, a is the first proportional and c is the third proportional.
 Thus, if b is mean proportional between a and c, then = ac, i.e. b = ac.
UNIT III: INDICES
After reading this unit a student will learn –
A meaning of indices and their applications.
Laws of indices which facilitates their easy applications.
UNIT OVERVIEW
1.3 INDICES
We are aware of certain operations of addition and multiplication and now we take up certain higher order operations with powers and roots under the respective heads of indices.
We know that the result of a repeated addition can be held by multiplication e.g
Now
It may be noticed that in the first case 4 is multiplied 5 times and in the second case ‘a’ is multiplied 5 times. In all such cases, a factor which multiplies is called the “base” and the number of times it is multiplied is called the “power” or the “index”. Therefore, “4” and “a” are the bases and “5” is the index for both. Any base raised to the power zero is defined to be 1; i.e. We also
define
If n is a positive integer, and ‘a’ is a real number, i.e. and (where N is the set of positive integers and R is the set of real numbers), ‘a’ is used to denote the continued product of n factors each equal to ‘a’ as shown below:
factors.
Here is the power of “a“ whose base is “a“ and the index or power is “n“.
For example, it is base and 4 is index or power.
Law 1
when m and n are positive integers; by the above definition,
to m factors, and to n factors.
factors.
Now, we extend this logic to negative integers and fractions. First, let us consider this for negative integer, that is m will be replaced by n.
Law 2
when m and n are positive integers and
By definition to m factors.
Therefore,
factors.
Now, we take a numerical value for a and check the validity of this
Law 3
when m and n are positive integers
By definition to n factors.
Following above,
(We will keep m as it is and replace n by p/q, where p and q are positive integers)
Now the qth power of is
If we take the qth root of the above we obtain
Now with the help of a numerical value for a let us verify this law.
Law 4
when n can take all of the values
For example
First, we look at n when it is a positive integer. Then by the definition, we have
When n is a positive fraction, we will replace n by p/q.
EXERCISE 1(C)
Choose the most appropriate option (a) (b) (c) or (d).
1. is expressed as
 
2. The value of is
 
3. The value of is
 
4. The value of is
 
5. The value of is
 
6. The value of is
 
7. is equal to
 
8. has simplified value equal to
 
9. is equal to
 
10. The value of where i s equal to
 
11. is
 
12. Which is True?
 
13. If and then the value of is
 
14. The value of
 
15. The True option is
 
16. The simplified value of is
 
17. The value of is
 
18. The value of is
 
19. Simplified value of is
 
20. is
 
21. is equal to
 
22. is equal to
 
23. If then the simplified form of
 
24. Using tick the correct of these when
 
25. On simplification is equal to
 
26. The value of
 
27. If , then is
 
28. If , then is
 
29. The value of
 
30. If , is

UNIT IV: LOGARITHM
LEARNING OBJECTIVES
After reading this unit a student will learn –
 After reading this unit, a student will get fundamental knowledge of logarithm and its
application for solving business problems.
UNIT OVERVIEW
1.4 LOGONITHMS:
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, i.e. to make it equal to the given number. If there are three quantities indicated by say a, x and n, they are related as follows:
If where
then x is said to be the logarithm of the number n to the base ‘a’ symbolically it can be expressed as follows:
i.e. the logarithm of n to the base ‘a’ is x. We give some illustrations below:
(i)
i.e. the logarithm of 16 to the base 2 is equal to 4
(ii)
i.e. the logarithm of 1000 to the base 10 is 3
(iii)
i.e. the logarithm of to the base 5 is 3
(iv)
i.e. the logarithm of 8 to the base 2 is 3
 The two equations and are only transformations of each other and should be remembered to change one form of the relation into the other.
 The logarithm of 1 to any base is zero. This is because any number raised to the power zero is one.
since
 The logarithm of any quantity to the same base is unity. This is because any quantity raised to the power 1 is that quantity only.
Since
ILLUSTRATIONS:
 If find the value of a.
We have
 Find the logarithm of 5832 to the base
Let us take
We may write
Hence, x = 6
Logarithms of numbers to the base 10 are known as a common logarithm.
1.4.1 Fundamental Laws of Logarithm
 The logarithm of the product of two numbers is equal to the sum of the logarithms of the
numbers to the same base, i.e.
Proof:
Let
Multiplying (I) and (II), we get
 The logarithm of the quotient of two numbers is equal to the difference of their
logarithms to the same base, i.e.
Proof:
Let l so that
Dividing (I) by (II) we get
Then by the definition of logarithm, we get
Similarly,
Illustration I:
 Logarithm of the number raised to the power is equal to the index of the power multiplied by the logarithm of the number to the same base i.e.
Proof:
Let so that
Raising the power n on both sides we get
(by definition)
i.e.
Illustration II: 1(a) Find the logarithm of 1728 to the base
Solution: We have and so, we may write
1(b) Solve
Solution: The given expression
1.4.2 Change of Base
If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation.
Proof:
Let
Then by definition,
Also
Therefore,
Putting m = a, we have
Example 1: Change the base of into the common logarithmic base.
Solution: Since
Example 2: Prove that
Solution: Change all the logarithms on L.H.S. to the base 10 by using the formula.
, we may write
L.H.S.
R.H.S
Logarithm Tables:
The logarithm of a number consists of two parts, the whole part or the integral part is called the characteristic and the decimal part is called the mantissa where the former can be known by mere inspection, the latter has to be obtained from the logarithm tables.
Characteristic:
The characteristic of the logarithm of any number greater than 1 is positive and is one less than the number of digits to the left of the decimal point in the given number. The characteristic of the logarithm of any number less than one (1) is negative and numerically one more than the number of zeros to the right of the decimal point. If there is no zero then obviously it will be 1. The following table will illustrate it.
Number  Characteristic  
3 7 4 6 2 3 6.21  1 3 0  [One less than the number of digits to the left of the decimal point ] 
Number  Characteristic  
.8 .07 .00507 .000670  1 2 3 4  [One more than the number of zeros on the right immediately after the decimal point.] 
Zero on positive characteristic when the number under consideration is greater than unity:
Since
All numbers lying between 1 and 10 i.e. numbers with 1 digit in the integral part have their logarithms lying between 0 and 1. Therefore, their integral parts are zero only.
All numbers lying between 10 and 100 have two digits in their integral parts. Their logarithms lie between 1 and 2. Therefore, numbers with two digits have integral parts with 1 as characteristic.
In general, the logarithm of a number containing n digits only in its integral parts is (n – 1) + a decimal. For example, the characteristics of log 75, log 79326, log 1.76 are 1, 4 and 0 respectively.
Negative characteristics
Since
All numbers lying between 1 and 0.1 have logarithms lying between 0 and 1, i.e. greater than 1 and less than 0. Since the decimal part is always written positive, the characteristic is 1.
All numbers lying between 0.1 and 0.01 have their logarithms lying between 1 and 2 as characteristic of their logarithms.
In general, the logarithm of a number having n zeros just after the decimal point is –
(n + 1) + a decimal.
Hence, we deduce that the characteristic of the logarithm of a number less than unity is one more than the number of zeros just after the decimal point and is negative.
Mantissa
The mantissa is the fractional part of the logarithm of a given number.
Number  Mantissa  Logarithm 
Thus with the same figures, there will be a difference in the characteristic only. It should be remembered, that the mantissa is always a positive quantity. The other way to indicate this is
Negative mantissa must be converted into a positive mantissa before reference to a logarithm
table. For example
It may be noted that is different from is a negative number whereas,
in is negative while is positive.
Illustration I: Add
Antilogarithms
If x is the logarithm of a given number n with a given base then n is called the antilogarithm (antilog) of x to that base.
This can be expressed as follows:
If then n = antilog x
For example, if log 61720 = 4.7904 then 61720 = antilog 4.7904
Number Logarithm
206 2.3139
20.6 1.3139
2.06 0.3139
.206 1.3139
.0206 2.3139
Example 1: Find the value of log 5 if log 2 is equal to .3010.
Solution: log5 = log 10/2= log 10 – log 2
= 1 – .3010
= .6990
Example 2: Find the number whose logarithm is 2.4678.
Solution: From the antilog table, for mantissa .467, the number = 2931
for mean difference 8, the number = 5
for mantissa .4678, the number = 2936
The characteristic is 2, therefore, the number must have 3 digits in an integral part.
Hence, Antilog 2.4678 = 293.6
Example 3: Find the number whose logarithm is –2.4678.
Solution: 2.4678 = – 3 + 3 – 2.4678 = – 3 + .5322 =
For mantissa .532, the number = 3404
For mean difference 2, the number = 2
for mantissa .5322, the number = 3406
The characteristic is 3, therefore, the number is less than one and there must be two zeros
just after the decimal point.
Thus, Antilog (2.4678) = 0.003406
The relation between Indices and Logarithm
Example 1: Find the logarithm of 64 to the base
Solution:
Example 2: If prove that
Solution:
Therefore
Example 3: If then prove that
1+abc = 2bc
Solution:
SUMMARY
Ex. log (2 x 3) = log 2 + log 3
Ex. log (3/2) = log3 – log2
Ex. log
Ex. etc.
Ex. = 1
Ex.
Ex.
 (Inverse logarithm Property)
 The two equations ax = n and x =logan are only transformations of each other and should be remembered to change one form of relation into the other.
Since
Notes :
 If base is understood, base is taken as 10
 This log 10 = 1, log 1 = 0
 Logarithm using base 10 is called the Common logarithm and logarithm using base e is called Natural logarithm {e = 2.33 (approx..) called exponential number}.
EXERCISE 1(D)
Choose the most appropriate option. (a) (b) (c) or (d).
 Log 6+ log 5 is expressed as
(a) log 11  (b) log 30  (c) log 5/6  (d) none of these 
 is equal to
(a) 2  (b) 8  (c)  (d) none of these 
 log 32/4 is equal to
(a) log 32/log 4  (b) log 32 – log 4  (c) log 5/6  (d) none of these 
 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 
 The value of log 0.0001 to the base 0.1 is
(a) 4  (b) 4  (c) 1/4  (d) none of these 
 If 2 log x = 4 log 3, the is equal to
(a) 3  (b) 9  (c) 2  (d) none of these 
 64 is equal to
(a) 12  (b) 6  (c) 1  (d) none of these 
 1728 is equal to
(a)  (b) 2  (c) 6  (d) none of these 
 log (1/81) to the base 9 is equal to
(a) 2  (b) ½  (c) 2  (d) none of these 
 log 0.0625 to the base 2 is equal to
(a) 4  (b) 5  (c) 1  (d) none of these 
 Given = 0.3010 and = 0.4771 the value of log 6 is
(a) 0.9030  (b) 0.9542  (c) 0.7781  (d) none of these 
 The value of 16
(a) 0  (b) 2  (c) 1  (d) none of these 
 The value of log to the base 9 is
(a) – ½  (b) ½  (c) 1  (d) none of these 
 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 
 The value of [ { ( )}] is equal to
(a) 1  (b) 2  (c) 0  (d) none of these 
 , these is equal to
(a) 8  (b) 4  (c) 16  (d) none of these 
 Given that the value of is expressed as
(a) x – y + 1  (b) x + y + 1  (c) x – y – 1  (d) none of these 
 Given that , then is expressed in terms of and y as
(a) x + 2y – 1  (b) x + y – 1  (c) 2x + y – 1  (d) none of these 
 Given that log x = m + n and log y = m – n, the value of is expressed in terms of m and n as
(a) 1 – m + 3m  (b) m – 1 + 3n  (c) m + 3n + 1  (d) none of these 
 The simplified value of 2 is
(a) ½  (b) 4  (c) 2  (d) none of these 
 log can be written as
(a)  (b)  (c) log 1/  (d) none of these 
 The simplified value of is
(a) log 3  (b) log 2  (c) log ½  (d) none of these 
 The value of is equal to
(a) 3  (b) 0  (c) 1  (d) none of these 
 The logarithm of 64 to the base is
(a) 2  (b)  (c) ½  (d) none of these 
 The value of given log 2 = 0.3010 is
(a) 1  (b) 2  (c) 1.5482  (d) none of these 
ANSWERS
Exercise 1(A)
1.  (a)  2.  (d)  3.  (c)  4.  (a)  5.  (c)  6.  (d)  7.  (a)  8.  (c) 
9.  (a)  10.  (c)  11.  (d)  12.  (d)  13.  (a)  14.  (c)  15.  (d)  16.  (a) 
17.  (c)  18.  (b)  19.  (b)  20.  (c)  21.  (a)  22.  (c)  23.  (a)  24.  (c) 
25.  (c)  
Exercise 1(B)  
1.  (a)  2.  (b)  3.  (c)  4.  (d)  5.  (a)  6.  (c)  7.  (a)  8.  (c) 
9.  (c)  10.  (b)  11.  (c)  12.  (d)  13.  (a)  14.  (d)  15.  (d)  16.  (a) 
17.  (a)  18.  (b)  19.  (d)  20.  (a)  21.  (c)  22.  (d)  23.  (c)  24.  (a) 
25.  (b)  26.  (b)  27.  (c)  28.  (b)  29.  (a)  30.  (b)  
Exercise 1 (C)  
1.  (c)  2.  (c)  3.  (c)  4.  (b)  5.  (a)  6.  (a)  7.  (b)  8.  (d) 
9.  (b)  10.  (c)  11.  (d)  12.  (c)  13.  (b)  14.  (d)  15.  (a)  16.  (c) 
17.  (a)  18.  (c)  19.  (d)  20.  (b)  21.  (a)  22.  (c)  23.  (b)  24.  (b) 
25.  (c)  26.  (a)  27.  (b)  28.  (a)  29.  (a)  30.  (b)  
Exercise 1 (D)  
1.  (b)  2.  (c)  3.  (b)  4.  (a)  5.  (b)  6.  (b)  7.  (a)  8.  (c) 
9.  (c)  10.  (d)  11.  (c)  12.  (c)  13.  (a)  14.  (c)  15.  (c)  16.  (a) 
17.  (b)  18.  (c)  19.  (a)  20.  (c)  21.  (b)  22.  (a)  23.  (c)  24.  (d) 
25.  (c) 
ADDITIONAL QUESTION BANK
 The value of is
(a) 0  (b) 252  (c) 250  (d)248 
 The value of is
(a) 1  (b) 1  (c) 0  (d) None 
 On simplification reduces to
(a) 1  (b) 0  (c) 1  (d) 10 
 If then x – y is given by
(a) 1  (b) 1  (c) 0  (d) None 
 Show that is given by
(a) 1  (b) 1  (c) 3  (d) 0 
 Show that is given by
(a) 1  (b) 1  (c) 4  (d) 0 
 Show that is given by
(a) 0  (b) 1  (c) 3  (d) 1 
 Show that reduces to
(a) 1  (b) 0  (c) 1  (d) None 
 Show that reduces to
(a) 1  (b) 3  (c) 1  (d) None 
 Show that reduces to
(a) 1  (b) 3  (c) 0  (d) 2 
 Show that reduces to
(a) 1  (b) 0  (c) 1  (d) None 
 Show that is given by
(a) 1  (b) 1  (c) 0  (d) 3 
 Show that is given by
(a) 0  (b) 1  (c) 1  (d) None 
 Show that is given by
(a) 1  (b) 0  (c) 1  (d) None 
 Show that reduces to
(a) 1  (b)  (c)  (d) 
 would reduce to zero if a + b + c is given by
(a) 1  (b) 1  (c) 0  (d) None 
 The value of z Is given by the following if
(a) 2  (b) 3/2  (c) 3/2  (d) 9/4 
 would reduce to one if a + b + c is given by
(a) 1  (b) 0  (c) 1  (d) None 
 would reduce to
(a)  (b)  (c) 1  (d) 0 
 If then
(a)  (b)  (c)  (d) None 
 If prove that is given by
(a) 12  (b) 13  (c) 15  (d) 17 
 If prove that is given by
(a) 25  (b) 26  (c) 27  (d) 30 
 If then the value of is given by
(a)  (b)  (c)  (d) 
 If then the value of is given by
(a) 0  (b) 1  (c) 1  (d) None 
 If and the value of q(p + r)/pr is given by
(a) 1  (b) 1  (c) 2  (d) None 
 On simplification reduced to
(a) 1  (b) 1  (c) 0  (d) None 
 On simplification reduces to
(a)  (b)  (c)  (d) 
 On simplification reduces to
(a)  (b)  (c)  (d) 
 On simplification , reduces to
(a) 3  (b) 3  (c) 1/3  (d) 1/3 
 The value of is given by
(a) 1  (b) 0  (c) 1  (d) None 
 If then the value of is
(a) 1  (b) 0  (c) 2  (d) None 
 If then reduces to
(a) 1  (b) 0  (c) 2  (d) None 
 If then the value of reducing to
(a) 0  (b) 2  (c) 3  (d) 1 
 If , then the value of abc(2a+b) reduces to
(a) 1  (b) 0  (c) 3  (d) 5 
 , then the value of abc(a+2b) reduces to
(a) 0  (b) 1  (c) 2  (d) 3 
 If and abc=288 then the value of is given by
(a) 1/8  (b) 1/8  (c) 11/96  (d) 11/96 
 If and ab = cd then the value of reduces to
(a) 1/a  (b) 1/b  (c) 0  (d) 1 
 If , then the value of reduces to
(a) a  (b) b  (c) 0  (d) None 
 If and the value of is given by
(a) 1  (b) 0  (c) 1  (d) None 
 If then the value of reduces to
(a) 1  (b) 1  (c) 0  (d) None 
 If then the value of reduces to
(a) 0  (b) 1  (c) 1  (d) None 
 If then the value of is
(a) 3  (b) 0  (c) 2  (d) 1 
 If then is
(a)  (b) 
(c) 2x  (d) 0 
 If and then the value of is
(a) 67  (b) 65  (c) 64  (d) 62 
 If and then is
(a) 5  (b)  (c)  (d) 4 
 If then the value of is given by
(a) 1  (b) 1  (c) 2  (d) 2 
 If then the value of P is
(a) 7/11  (b) 3/11  (c) 1/11  (d) 2/11 
 If then the value of is
(a)  (b)  (c)  (d) 
 If then the value of is
(a)  (b) 2  (c)  (d) 
 If then the value of is
(a) 0  (b) 1  (c) 5  (d) 1 
 If then the value of is
(a) 14  (b) 7  (c) 2  (d) 1 
 If then the value of is
(a) 10  (b) 14  (c) 0  (d) 15 
 If then the value of is
(a) 21  (b) 1  (c) 12  (d) None 
 The square root of
(a)  (b)  (c) Both the above  (d) None 
 If the value of x is given by
(a) 2  (b) 1  (c) 2  (d) 0 
 If then the value of a + b is
(a) 10  (b) 100  (c) 98  (d) 99 
 If then the value of is
(a) 10  (b) 100  (c) 98  (d) 99 
 If then the value of is
(a) 10  (b) 100  (c) 98  (d) 99 
 The square root of is given by
(a)  (b) 
(c)  (d) 
 The square root of is given by
(a)  (b)  (c)  (d) 
 log (1 + 2 + 3) is exactly equal to
(a) log 1 + log 2 + log 3  (b) log (1x2x3) 
(c) Both the above  (d) None 
 The logarithm of 21952 to the base of and 19683 to the base of are
(a) Equal  (b) Not equal 
(c) Have a difference of 2269  (d) None 
 The value of is is
(a) 0  (b) 1  (c) 2  (d) 1 
 has a value of
(a) 1  (b) 0  (c) 1  (d) None 
 is equal to
(a) 0  (b) 1  (c) 2  (d) 1 
 is equal to
(a) 0  (b) 1  (c) 3  (d) 1 
 is equal to
(a) 0  (b) 1  (c) 3  (d) 1 
 is equal to
(a) 0  (b) 1  (c) 1  (d) None 
 is equal to
(a) 0  (b) 1  (c) 1  (d) None 
 The value of is
(a) 0  (b) 1  (c) 1  (d) None 
 The value of is
(a) 0  (b) 1  (c) 1  (d) None 
 The value of
(a) 0  (b) 1  (c) 1  (d) None 
 The value of is
(a) 0  (b) 1  (c) 1  (d) None 
 if the value of a is given by
(a) 0  (b) 10  (c) 1  (d) None 
 the value of abc is
(a) 0  (b) 1  (c) 1  (d) None 
 the value of is given by
(a) 0  (b) 1  (c) 1  (d) None 
 If the value of is
(a) 0  (b) 1  (c) 1  (d) None 
 If the value of is
(a) 0  (b) 1  (c) 1  (d) None 
 If the value of is
(a) 0  (b) 1  (c) 1  (d) None 
 The value of is
(a) 0  (b) 1  (c) 1  (d) None 
 If then the value if z is given by
(a) abc  (b) a + b + c  (c) a(b + c)  (d) (a + b)c 
 If then the value of is
(a) 0  (b) 1  (c) 1  (d) 3 
 If then the value of is
(a)  (b) 
(c) 1 + 2 + 3 + 4  (d) None 
 The sum of the series is given by
(a)  (b)  (c)  (d) None 
 has a value of
(a) a (b) b  (b) b  (c) (a + b)  (d) None 
 The value of the following expression is given by
(a) t  (b) abcdt  (c) (a + b + c + d + t)  (d) None 
 For any three consecutive integers x y z the equation log ( 1+xz) – 2logy = 0 is
(a) True 
(b) False 
(c) Sometimes true 
(d) Cannot be determined in the cases of variables with cyclic order. 
 If then the value of is
(a) 2  (b) 5  (c) 7  (d) 3 
 If then the value of is
(a) 0  (b) 1  (c) 1  (d) 7 
 If then the value of is equal to
(a) 0  (b) 1  (c) 1  (d) 3 
 If then the value of xyz – x – y – z is
(a) 0  (b) 1  (c) 1  (d) 2

 On solving the equation we get the value of t as
(a) 5  (b) 2  (c) 3  (d) 0 
 On solving the equation we get the value of t as
(a) 8  (b) 18  (c) 81  (d) 6561 
 On solving the equation we get the value of t as
(a)  (b)  (c)  (d) None 
 If then the value of is
(a) 3  (b) 3  (c)  (d) 
 If then the value of is
(a) 3 log x  (b) log x  (c) 6 log x  (d) 5 log x 
 If then the value of n is
(a)  (b) 
(c)  (d) 
ANSWERS: –
1.  (b)  26.  (d)  51.  (d)  76.  (b) 
2.  (a)  27.  (a)  52.  (c)  77.  (b) 
3.  (c)  28.  (c)  53.  (b)  78.  (a) 
4.  (b)  29.  (d)  54.  (c)  79.  (b) 
5.  (a)  30.  (c)  55.  (a)  80.  (b) 
6.  (a)  31.  (a)  56.  (a)  81.  (a) 
7.  (d)  32.  (b)  57.  (c)  82.  (a) 
8.  (a)  33.  (a)  58.  (c)  83.  (a) 
9.  (a)  34.  (b)  59.  (a)  84.  (a), (c) 
10.  (a)  35.  (a)  60.  (b)  85.  (b) 
11.  (c)  36.  (c)  61.  (c)  86.  (a) 
12.  (a)  37.  (c)  62.  (a)  87.  (a) 
13.  (b)  38.  (c)  63.  (b)  88.  (c) 
14.  (a)  39.  (c)  64.  (a)  89.  (a) 
15.  (a)  40.  (a)  65.  (c)  90.  (a) 
16.  (c)  41.  (b)  66.  (b)  91.  (d) 
17.  (d)  42.  (b)  67.  (a)  92.  (a) 
18.  (b)  43.  (a)  68.  (b)  93.  (d) 
19.  (c)  44.  (c)  69.  (b)  94.  (c) 
20.  (b)  45.  (d)  70.  (b)  95.  (c) 
21.  (d)  46.  (c)  71.  (b)  96.  (a) 
22.  (b)  47.  (a)  72.  (a)  97.  (a) 
23.  (a)  48.  (c)  73.  (a)  
24.  (b)  49.  (b)  74.  (b)  
25.  (c)  50.  (a)  75.  (b) 
*This article contains all topics about Ratio And ProportionIndicesLogarithms Business Mathematics and Logical Reasoning & Statistics
For notes on all CA foundation topics, you can visit this article CA foundation note