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# Ratio And Proportion-Indices-Logarithms– CA Foundation, CPT notes, PDF

## What we will study in this chapter: Ratio and Proportion-Indices-Logarithms

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 (non-zero) 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 kgHe reduces his weight in the ratio 7 : 6His 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 ratiosSo, the number of boys in the schoolAnd the number of girls in the school === 3 + 5(3 x 720)/8(5 x 720)/8 === 8270450

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 vice-versa.

1. A ratio a : b is said to be of greater inequality if a>b and of less inequality if a<b.
2. 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.

1. 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.

1. The sub-duplicate ratio of a : b is and the sub-triplicate ratio of a : b is

For example sub-duplicate ratio of 4 : 9 is = 2 : 3

And sub-triplicate ratio of 8 : 27 is = 2 : 3.

1. 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.
2. 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).

1. The inverse ratio of 11 : 15 is
 (a)  15 : 11 (b) (c)   121 : 225 (d)  None of these

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

1. 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
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

1. The duplicate ratio of 3 : 4 is
 (a) (b)  4 : 3 (c)   9 : 16 (d)  None of these

1. The sub-duplicate ratio of 25 : 36 is
 (a)  6 : 5 (b)  36 : 25 (c)   50 : 72 (d)  5 : 6

1. The triplicate ratio of 2 : 3 is
 (a)  8 : 27 (b)  6 : 9 (c)   3 : 2 (d)  None of these

1. The sub-triplicate ratio of 8 : 27 is
 (a)  27 : 8 (b)  24 : 81 (c)   2 : 3 (d)  None of these

1. 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

1. 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

1. 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

1. 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

1. 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

1. The angles of a triangle are in ratio 2 : 7 : 11. The angles are
 (a) (b) (c) (d)  None of these

1. 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

1. 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

1. 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

1. 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

1. If : = 3 : 4, the value of is
 (a)  13 : 12 (b)  12 : 13 (c)   21 : 31 (d)  None of these

1. If : is the sub-duplicate ratio of then  is
 (a) (b) (c) (d)  None of these

1. If 2s : 3t is the duplicate ratio of 2 – : 3  –  then
 (a) (b) (c) (d)  None of these

1. If = 2 : 3 and = 4 : 5, then the value of is
 (a)  71 : 82 (b)  27 : 28 (c)   17 : 28 (d)  None of these

1. 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

1. 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

1. 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 sub-duplicate ratio of a : b is a : b and the sub-triplicate 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 vice-versa.

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 1st 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
1. If a : b = c : d, then ad = bc

Proof. ;  ad = bc (By cross – multiplication)

1. If a : b = c : d, then b : a = d : c (Invertendo)

Proof. or , or

Hence, b : a = d : c.

1. If a : b = c : d, then a : c = b : d (Alternendo)

Dividing both sides by cd, we get

, or , i.e. a : c = b : d.

1. If a : b = c : d, then a + b : b = c + d : d (Componendo)

Proof. , or, =

Or, , i.e. a + b : b = c + d : d.

1. If a : b = c : d, then a – b : b = c – d : d (Dividendo)

Proof. ,

, i.e. a – b: b = c – d : d.

1. 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

1. 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

2nd difference = Profit by selling 1 kg of 2nd kind @ ₹ 7.26

= ₹ 7.77 – ₹ 7.26 = 51 Paise

1st 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 = (2nd diff.) : (1st. diff.) = 51 : 34 = 3 : 2.

EXERCISE 1(B)

Choose the most appropriate option (a) (b) (c) or (d).

1. The fourth proportional to 4, 6, 8 is
 (a)  12 (b)  32 (c)   48 (d)  None of these

1. The third proportional to 12, 18 is
 (a)  24 (b)  27 (c)   36 (d)  None of these

1. The mean proportional between 25, 81 is
 (a)  40 (b)  50 (c)   45 (d)  None of these

1. 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

1. The fourth proportional to 2a, , c is
 (a)  Ac/2 (b)  Ac (c)   2/ac (d)  None of these

1. 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

1. 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 =

1. 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

1. If a/3 = b/4 = c/7, then a + b + c/c is
 (a)  1 (b)  3 (c)   2 (d)  None of these

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

1. If , the value of is
 (a)  1 (b)  3/5 (c)   5/3 (d)  None of these

1. If = 3/4, the value of  is
 (a)  2 : 9 (b)  7 : 2 (c)   7 : 9 (d)  None of these

1. 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

1. If , then the value of  is
 (a)  6/23 (b)  23/6 (c)   3/2 (d)  17/6

1. If then  is
 (a)  2 : 3 : 4 (b)  4 : 3 : 2 (c)   3 : 2 : 4 (d)  None of these

1. 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

1. 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

1. 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
1. If , implies  , then the process is called
 (a)  Dividendo (b)  Componendo (c)   Alternendo (d)  None of these

1. If p/q = r/s = p – r/q – s, the process is called
 (a)  Subtahendo (b)  Addendo (c)   Invertendo (d)  None of these

1. 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

1. 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

1. 12, 16, *, 20 are in proportion. Then * is
 (a)  25 (b)  14 (c)   15 (d)  None of these

1. 4, *, 9, 13 ½ are in proportion. Then * is
 (a)  6 (b)  8 (c)   9 (d)  None of these

1. 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

1. If then  is
 (a)  4 (b)  2 (c)   7 (d)  None of these

1. 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

1. If then
 (a) (b) (c) (d)  None of these

1. If then  is
 (a)  5/2 (b)  4 (c)   5 (d)  None of these

1. 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

 (a) (b) (c) (d) None of these

2. The value of is

 (a) (b) (c) (d) None of these

3. The value of is

 (a) (b) (c) (d) None of these

4. The value of is

 (a) (b) (c) (d) None of these

5. The value of is

 (a) (b) (c) (d) None of these

6. The value of is

 (a) (b) (c) (d) None of these

7. is equal to

 (a) a fraction (b) a positive integer (c) a negative integer (d) None of these
8. has simplified value equal to

 (a) (b) (c) (d) None of these

9. is equal to

 (a) (b) (c) (d) None of these

10. The value of where i s equal to

 (a) (b) (c) (d) None of these

11. is

 (a) (b) (c) (d)

12. Which is True?

 (a) (b) (c) (d) none of these

13. If and then the value of  is

 (a) (b) (c) (d) none of these

14. The value of

 (a) (b) (c) (d)

15. The True option is

 (a) (b) (c) (d)

16. The simplified value of  is

 (a) (b) (c) (d) none of these

17. The value of  is

 (a) (b) (c) (d) none of these

18. The value of  is

 (a) (b) (c) (d) none of these

19. Simplified value of  is

 (a) (b) (c) (d) none of these

20.  is

 (a) A fraction (b) an integer (c) (d) none of these

21.  is equal to

 (a) x (b) (c) (d) none of these

22.  is equal to

 (a) (b) (c) (d) none of these

23. If then the simplified form of

 (a) (b) (c) (d) none of these

24. Using tick the correct of these when

 (a) (b) (c) (d) none of these

25. On simplification is equal to

 (a) (b) (c) (d) 1/a

26. The value of

 (a) (b) (c) (d) none of these

27. If , then is

 (a) (b) (c) (d) none of these

28. If , then is

 (a) (b) (c) (d) none of these

29. The value of

 (a) (b) (c) (d) none of these

30. If , is

 (a) (b) (c) (d) none of these

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

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

1. The two equations and are only transformations of each other and should be remembered to change one form of the relation into the other.
2. The logarithm of 1 to any base is zero. This is because any number raised to the power zero is one.

since

1. 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:

1. If find the value of a.

We have

1. 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

1. 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

1. 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:

1. 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 74 6 2 36.21 130 [One less than the number of digits tothe left of the decimal point ] Number Characteristic .8.07.00507.000670 -1-2-3-4 [One more than the number of zeros onthe 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.

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).

1. Log 6+ log 5 is expressed as
 (a) log 11 (b) log 30 (c) log 5/6 (d) none of these

1. is equal to
 (a) 2 (b) 8 (c) (d) none of these

1. log 32/4 is equal to
 (a) log 32/log 4 (b) log 32 – log 4 (c) log 5/6 (d) none of these

1. 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

1. The value of log 0.0001 to the base 0.1 is
 (a) -4 (b) 4 (c) 1/4 (d) none of these

1. If 2 log x = 4 log 3, the is equal to
 (a) 3 (b) 9 (c) 2 (d) none of these

1. 64 is equal to
 (a) 12 (b) 6 (c) 1 (d) none of these

1. 1728 is equal to
 (a) (b) 2 (c) 6 (d) none of these

1. log (1/81) to the base 9 is equal to
 (a) 2 (b) ½ (c) -2 (d) none of these

1. log 0.0625 to the base 2 is equal to
 (a) 4 (b) 5 (c) 1 (d) none of these

1. 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

1. The value of 16
 (a) 0 (b) 2 (c) 1 (d) none of these

1. The value of log to the base 9 is
 (a) – ½ (b) ½ (c) 1 (d) none of these

1. 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

1. The value of [ { ( )}] is equal to
 (a) 1 (b) 2 (c) 0 (d) none of these

1. , these is equal to
 (a) 8 (b) 4 (c) 16 (d) none of these

1. Given that the value of is expressed as
 (a) x – y + 1 (b) x + y + 1 (c) x – y – 1 (d) none of these

1. 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

1. 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

1. The simplified value of 2 is
 (a) ½ (b) 4 (c) 2 (d) none of these

1. log can be written as
 (a) (b) (c) log 1/ (d) none of these

1. The simplified value of is
 (a) log 3 (b) log 2 (c) log ½ (d) none of these

1. The value of is equal to
 (a) 3 (b) 0 (c) 1 (d) none of these

1. The logarithm of 64 to the base is
 (a) 2 (b) (c) ½ (d) none of these

1. The value of given log 2 = 0.3010 is
 (a) 1 (b) 2 (c) 1.5482 (d) none of these

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)

1. The value of is
 (a) 0 (b) 252 (c) 250 (d)248

1. The value of is
 (a) 1 (b) -1 (c) 0 (d) None

1. On simplification reduces to
 (a) -1 (b) 0 (c) 1 (d) 10

1. If then x – y is given by
 (a) -1 (b) 1 (c) 0 (d) None

1. Show that is given by
 (a) 1 (b) -1 (c) 3 (d) 0

1. Show that is given by
 (a) 1 (b) -1 (c) 4 (d) 0

1. Show that is given by
 (a) 0 (b) -1 (c) 3 (d) 1

1. Show that reduces to
 (a) 1 (b) 0 (c) -1 (d) None

1. Show that reduces to
 (a) 1 (b) 3 (c) -1 (d) None

1. Show that reduces to
 (a) 1 (b) 3 (c) 0 (d) 2

1. Show that reduces to
 (a) -1 (b) 0 (c) 1 (d) None

1. Show that is given by
 (a) 1 (b) -1 (c) 0 (d) 3

1. Show that is given by
 (a) 0 (b) 1 (c) -1 (d) None

1. Show that is given by
 (a) 1 (b) 0 (c) -1 (d) None

1. Show that reduces to
 (a) 1 (b) (c) (d)

1. would reduce to zero if a + b + c is given by
 (a) 1 (b) -1 (c) 0 (d) None

1. The value of z Is given by the following if
 (a) 2 (b) 3/2 (c) -3/2 (d) 9/4

1. would reduce to one if a + b + c is given by
 (a) 1 (b) 0 (c) -1 (d) None

1. would reduce to
 (a) (b) (c) 1 (d) 0

1. If then
 (a) (b) (c) (d) None

1. If prove that is given by
 (a) 12 (b) 13 (c) 15 (d) 17

1. If prove that is given by
 (a) 25 (b) 26 (c) 27 (d) 30

1. If then the value of is given by
 (a) (b) (c) (d)

1. If then the value of is given by
 (a) 0 (b) 1 (c) -1 (d) None

1. If and the value of q(p + r)/pr is given by
 (a) 1 (b) -1 (c) 2 (d) None

1. On simplification reduced to
 (a) 1 (b) -1 (c) 0 (d) None

1. On simplification reduces to
 (a) (b) (c) (d)

1. On simplification reduces to
 (a) (b) (c) (d)

1. On simplification , reduces to
 (a) 3 (b) -3 (c) -1/3 (d) 1/3

1. The value of is given by
 (a) -1 (b) 0 (c) 1 (d) None

1. If then the value of is
 (a) 1 (b) 0 (c) 2 (d) None

1. If then reduces to
 (a) 1 (b) 0 (c) 2 (d) None

1. If then the value of reducing to
 (a) 0 (b) 2 (c) 3 (d) 1

1. If , then the value of ab-c(2a+b) reduces to
 (a) 1 (b) 0 (c) 3 (d) 5

1. , then the value of ab-c(a+2b) reduces to
 (a) 0 (b) 1 (c) 2 (d) 3

1. If and abc=288 then the value of is given by
 (a) 1/8 (b) -1/8 (c) 11/96 (d) -11/96

1. If and ab = cd then the value of reduces to
 (a) 1/a (b) 1/b (c) 0 (d) 1

1. If , then the value of reduces to
 (a) a (b) b (c) 0 (d) None

1. If and the value of is given by
 (a) -1 (b) 0 (c) 1 (d) None
1. If then the value of reduces to
 (a) 1 (b) -1 (c) 0 (d) None

1. If then the value of  reduces to
 (a) 0 (b) 1 (c) -1 (d) None

1. If then the value of  is
 (a) 3 (b) 0 (c) 2 (d) 1

1. If then  is
 (a) (b) (c) 2x (d) 0

1. If and  then the value of  is
 (a) 67 (b) 65 (c) 64 (d) 62

1. If and  then  is
 (a) 5 (b) (c) (d) 4

1. If then the value of  is given by
 (a) 1 (b) -1 (c) 2 (d) -2

1. If then the value of P is
 (a) 7/11 (b) 3/11 (c) -1/11 (d) -2/11

1. If then the value of  is
 (a) (b) (c) (d)
1. If then the value of  is
 (a) (b) 2 (c) (d)

1. If  then the value of  is
 (a) 0 (b) 1 (c) 5 (d) -1

1. If then the value of  is
 (a) 14 (b) 7 (c) 2 (d) 1

1. If then the value of  is
 (a) 10 (b) 14 (c) 0 (d) 15

1. If then the value of  is
 (a) 21 (b) 1 (c) 12 (d) None

1. The square root of
 (a) (b) (c) Both the above (d) None

1. If the value of x is given by
 (a) -2 (b) 1 (c) 2 (d) 0

1. If then the value of a + b is
 (a) 10 (b) 100 (c) 98 (d) 99

1. If then the value of  is
 (a) 10 (b) 100 (c) 98 (d) 99

1. If then the value of  is
 (a) 10 (b) 100 (c) 98 (d) 99

1. The square root of is given by
 (a) (b) (c) (d)

1. The square root of is given by
 (a) (b) (c) (d)

1. log (1 + 2 + 3) is exactly equal to
 (a) log 1 + log 2 + log 3 (b) log (1x2x3) (c) Both the above (d) None

1. 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

1. The value of is is
 (a) 0 (b) 1 (c) 2 (d) -1

1. has a value of
 (a) 1 (b) 0 (c) -1 (d) None

1. is equal to
 (a) 0 (b) 1 (c) 2 (d) -1

1. is equal to
 (a) 0 (b) 1 (c) 3 (d) -1

1. is equal to
 (a) 0 (b) 1 (c) 3 (d) -1

1. is equal to
 (a) 0 (b) 1 (c) -1 (d) None

1. is equal to
 (a) 0 (b) 1 (c) -1 (d) None

1. The value of is
 (a) 0 (b) 1 (c) -1 (d) None

1. The value of is
 (a) 0 (b) 1 (c) -1 (d) None

1. The value of
 (a) 0 (b) 1 (c) -1 (d) None

1. The value of is
 (a) 0 (b) 1 (c) -1 (d) None

1. if the value of a is given by
 (a) 0 (b) 10 (c) -1 (d) None

1. the value of abc is
 (a) 0 (b) 1 (c) -1 (d) None

1. the value of  is given by
 (a) 0 (b) 1 (c) -1 (d) None

1. If the value of  is
 (a) 0 (b) 1 (c) -1 (d) None

1. If the value of  is
 (a) 0 (b) 1 (c) -1 (d) None

1. If the value of  is
 (a) 0 (b) 1 (c) -1 (d) None

1. The value of is
 (a) 0 (b) 1 (c) -1 (d) None

1. If then the value if z is given by
 (a) abc (b) a + b + c (c) a(b + c) (d) (a + b)c

1. If then the value of  is
 (a) 0 (b) 1 (c) -1 (d) 3

1. If then the value of  is
 (a) (b) (c) 1 + 2 + 3 + 4 (d) None

1. The sum of the series is given by
 (a) (b) (c) (d) None

1. has a value of
 (a) a (b) b (b) b (c) (a + b) (d) None

1. The value of the following expression is given by
 (a) t (b) abcdt (c) (a + b + c + d + t) (d) None

1. 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.

1. If then the value of  is
 (a) 2 (b) 5 (c) 7 (d) 3

1. If then the value of is
 (a) 0 (b) 1 (c) -1 (d) 7

1. If then the value of  is equal to
 (a) 0 (b) 1 (c) -1 (d) 3

1. If then the value of xyz – x – y – z is
 (a) 0 (b) 1 (c) -1 (d) 2
1. On solving the equation we get the value of t as
 (a) 5 (b) 2 (c) 3 (d) 0

1. On solving the equation we get the value of t as
 (a) 8 (b) 18 (c) 81 (d) 6561

1. On solving the equation we get the value of t as
 (a) (b) (c) (d) None
1. If then the value of  is
 (a) 3 (b) -3 (c) (d)

1. If then the value of  is
 (a) 3 log x (b) log x (c) 6 log x (d) 5 log x

1. If then the value of n is
 (a) (b) (c) (d)