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Sahasri Singar Academy

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Business Mathematics – Vol 1

CA | CMA

Foundation

CMA CS Yamuna Sridhar

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Content

1 Set Theory 1.1

2 Permutations and Combinations 2.1

3 Sequence & Series

[Arithmetic & Geometric Progression]

3.1

4 Ratios 4.1

5 Proportions 5.1

6 Indices 6.1

7 Logarithms 7.1

8 Time Value of Money

[Simple Interest & Compound Interest]

8.1

Annexure

A1 Number System A1.1

A2 Log & Antilog Tables A2.1

A3 Time Value of Money Tables A3.1

SSA Business Mathematics 1.1

1. Set Theory

Sets - Definition

A Group | well – defined (unique / distinct / condition - based) objects

Representation:

Rule Method

Set-Builder Method

Selection / Property

Method

Algebra Method

Defines the rule A:{𝑥: 𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙 𝑖𝑛 𝐸𝑛𝑔𝑙𝑖𝑠ℎ 𝐴𝑙𝑝ℎ𝑎𝑏𝑒𝑡}

A is a set of all x, such that x is a vowel

in English Alphabet”

A- Set of Vowels

Roster Method

Listing / Tabular Method

Lists the

elements

A = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}

Terms

Element / Member / Object

Notation Examples / Remarks

"𝜖" - Epsilon symbol

𝜖 - Belongs to | ∉ - Does not belongs to

1. Let A = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} ➔ a 𝜖 A | b∉A

2. A = {A} ➔ A(element) 𝜖 A(set)

Cardinal Number - The number of elements in a set

Notation Examples / Remarks

‘n’

Nature: +ve / 0 / Not –ve

1. A = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}

2. n(A) = 5 [Read as “n of A = 5”]


Types [Based on Cardinal Number]

Types Notation Examples / Remarks

0 Empty / Void / Null

Set

{ } , ɸ 1. A – A month having 32 days

2. A – Number of stupid

students in my class

1 Singleton set A = {1}

Finite Finite set Defines an end A = {a,e,i,o,u}

Infinite Infinite set Defines NO end N - Set of natural numbers

Subset (& other related sets)

Types & Notation Examples / Remarks

Subset

- A set contained in a set

Number of subsets = 2𝑛

𝐴𝑖 ⊆ A

Consider A= {1,2,3} ⟹ n(A) =3

Vary n ≤ 3

n=o: ɸ

n=1: 𝐴1 ={1}, 𝐴2 = {2}, 𝐴3 ={3}

n=2, 𝐴4 = {1,2}, 𝐴5 = {2,3}, 𝐴6 = {1,3}

n=3, 𝐴7 = {1,2,3}

Here,

Subsets - ɸ, 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5, 𝐴6, 𝐴7

Number of subsets, 2𝑛 = 23 = 8

Proper subsets –𝜙, 𝐴1, 𝐴2 ….𝐴6

Number of proper subset= 2𝑛- 1 = 7

Note: 𝐴7 =A itself,

hence not a proper subset

Superset – A

P(A) = {ɸ, 𝐴1 , 𝐴2, , 𝐴4, 𝐴5, 𝐴6 &𝐴7 }

Therefore, n(P(A)) = 2𝑛

Proper Subset

- A set properly contained in a set

Note: ɸ has no proper set

Number of Proper subsets = 2𝑛 − 1

Not Equal and Equivalent to Superset

𝐴𝑖 ⊂A

Superset

- A set contains a set

𝐴𝑖 ⊆ A (A - super set)

Power set

- Set of all subsets

Note: Power Set is never a null set

Complimentary Set & Universal Set

Other Sets


Other Sets:

Universal Set “ ∪ “[The whole Set]- The whole set, which contains all

Example: A - Set of English alphabet

Complimentary Set - The set which has elements not in A

Notation: A’/�̅�/𝐴𝑐

Example:

U - Set of English alphabet

A – Set of vowels

A’ – Set of Consonants

Comparison / Relationship between 2 sets

Types / Terms Notation Examples / Remarks

Equal sets

- The two sets

containing same

elements

Let A= {1,2,3}, n(A) =3,

B ={2,4,5}, n(B) = 3, C={1,2,3}, n(c) = 3

Here,

A and C are Equal Sets

A and B, B and C are Equivalent

Sets

Equivalent Sets

- The two sets

containing same

number of Elements

Equal set ⟹

(𝑐𝑜𝑟𝑟𝑒𝑐𝑡)Equivalent

set?

Equivalent set

⟹(wrong) Equal

Set?

Disjoint Sets

- There exists no

common elements in A

and B

A ∩ B = ɸ Let A = {1,2,3}, B= {a,c,d}

Then, A ∩ B = ɸ and n(A ∩ B) =0

Ordered / Product set:

A× 𝐵 = {a× 𝑏: 𝑎 ∈

𝐴&𝑏 ∈ 𝐵}

Read as “A cross B” Let A = {1,2}, B= {a, c, d}

A× 𝐵 =

{(1,a),(1,c),(1,d),(2,a),(2,c),(2,d)}

n(A) = 2 and n(B) =3

n(A×B) = n(A) ×n(B) = = 2×3 = 6


Note:

1. {∅} = singletion set with null set as the only element

2. ∅ is a subset of a Power set.

3. What is the power set of a null set?

For a null set, n(∅) =0

𝑝 (∅) = {∅} - Singleton Set

Operations on Sets / (& - X/or -+)

Operations Definitions Venn Diagram Example

A = {1,2} & B = {2,3}

Union

A∪B

Either A or B

A∪B = {𝑥: 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈

𝐴}

A∪B = {1,2,3}

Intersection

A∩B

Both A & B

A∩B = {𝑥: 𝑥 ∈ 𝐴 & 𝑥 ∈ 𝐵}

A∩B = {2}

Difference

A-B: only A,

but not B

B –A: only B,

but not

A - B = {𝑥: 𝑥 ∈ 𝐴 & 𝑥 ∉ 𝐵}

B - A = {𝑥: 𝑥 ∈ 𝐵 &𝑥 ∉ 𝐴}

A – B = {1}

B – A = {3}

Symmetric

Difference

A∆B

= (A-B)∪(B-A)

⟹ only A, but

not B or only

B, but not A

A△B

= {𝑥: 𝑥 ∈ (𝐴 − 𝐵) 𝑜𝑟 𝑥 ∈

(𝐵 − 𝐴)}

A△B = {1,3}

Complement

Set

A’ = {𝑥: 𝑥 ∉ 𝐴}

Let S = {1,2,3,4}

A’ = {3,4}


Points to Ponder:

1 A – B ≠ B – A

2 A – B in terms of other

operations

A – B = A – (A∩B) / (A∪B) – B

/ A∩B’ / (A∆B) – (B - A)

/ U – (A - B)’

3 A – B with 3 sets “only A” implies only A, but not B and C

[Notation: (A – B) ∩ (A – C)]

“only A, but not B” implies the set may

contain elements in C also [Notation: A –

B]

Neither A nor B⟹ (A∪B)’ = A’ ∩B’ = ∪ -

(A∪B)

Venn Diagram with different operations

4

Addition Theorem on Sets

On 2 Sets On 3 Sets

n(A∪B) = n(A) +n(B) –

n(A∩B)

n (A∪ 𝐵 ∪ 𝐶) = n(A) + n(B) + n(C) – n (A∩ 𝐵) – n (B∩ 𝐶) -

n(A∩ 𝐶) + n(A∩ 𝐵 ∩ 𝐶)

Points to Ponder

If A∩B = ɸ,

Then n(A∪B) = n(A) + n(B)

A∩ 𝐵 ∩ 𝐶 = ɸ ⟹ A∩B = ɸ, B∩C = ɸ , A∩C = ɸ,

Hence, n(A∪B∪C) = n(A) +n(B) + n(C)


Other Laws

De morgans law (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′

(𝐴 ∩ 𝐵)′ = A’∪B’

Commutative (2 sets 1 operation) A∪B = B∪A

A∩B = B∩A

Associative: (3sets 1 operation) (𝐴 ∪ 𝐵) ∪ 𝐶 = A ∪ (𝐵 ∪ 𝐶)

(𝐴 ∩ 𝐵) ∩ 𝐶 = A∩ (𝐵 ∩ 𝐶)

Distributive: 3sets 2 operation

(𝐴 ∪ 𝐵) ∩ C = (𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)

(𝐴 ∩ 𝐵) ∪ 𝐶 = (𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)

Identity A ∪A = A

A∩ 𝐴 = A

Simple Problems

1. Which of the following statements are correct /incorrect?

(a) 3 ∈ (1, 3, 5); (b) 3 ∉ (1, 5); (c) (3) ⊆ (1, 3, 5); (d) (3) ∈ (1, 3, 5)

2. Fill up the blanks by appropriate symbol ∈, ∉, , ⊆, ⊂, =

(i) 3 : …..(3, 4) ∪ (4, 5, 6)

(ii) (6) ….. (5, 6) ∩ (6, 7, 8)

(iii) {3, 4, 5}…{2, 3, 4}∪{3,4,5}

(iv) (a, b)…..(a)

(v) 4….(3, 5) ∪ (5, 6, 7)

(vi) (1, 2, 2, 3) ….(3, 2, 1)

3. A∪A is equal to

a. A

b. E

c. ∅

d. None of these

+

4. A∩A is equal to

a. ∅

b. A

c. E

d. None of these

5. (A∪B)’ is equal to

a. (A∩ 𝐵)’


b. A∪B’

c. A’ ∩ B’

d. None of these

6. (A∩ 𝐵)’ is equal to

a. (A’∪ 𝐵)’

b. A’∪B’

c. A’∩B’

d. None of these

7. A∪E is equal to (E is a superset of A)

a. A

b. E

c. ∅

d. None of these

8. A∩E is equal to

a. A

b. E

c. ∅

d. None of these

9. E∪E is equal to

a. E

b. ∅

c. 2E

d. None of these

10. A∩E’ is equal to

a. E

b. ∅

c. A

d. None of these

11. A∩ ∅ is equal to

a. A

b. E

c. ∅

d. None these

12. A∪A’ is equal to


a. E

b. ∅

c. A

d. None of these

13. The set { 2𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}

a. An infinite set

b. A null set

c. A finite set,

d. None of these

14. {1 − (−1)𝑥} for all integral 𝑥 is the set

a. {0}

b. {2}

c. {0,2}

d. None of these

15. E is a set of positive even number

and O is a set of positive odd

numbers, then E∪O is a

a. Set of whole numbers

b. N

c. A set of rational number

d. None of these

16. If R is the set of positive rational number and E is the

set of real numbers then

a. R⊆E

b. R⊂E

c. E⊂R

d. None of these

17. If N is the set of natural numbers

and I is the set of positive integers, then

a. N= 1

b. N⊂1

c. N⊆I

d. None of these


18. If I is the set of isosceles triangles

and E is the set of

equilateral triangles, then

a. I⊂E

b. E⊂I

c. E=I

d. None of these

19. If R is the set of isosceles right angled triangles and I is set of isosceles triangles, then

a. R=I

b. R⊃I

c. R⊂1

d. None of these

20. {𝑛(𝑛+1)

2: 𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}is

a. A finite set

b. An infinite set

c. Is an empty set

d. None of these

Word Problems

Question 1: A survey shows that 40% of the Indians like grapes, whereas 68% like bananas.

What percentage of the Indians like both grapes and bananas?

Question 2: In a class of 60 students, 40 students like Maths, 36 like Science, and 24 like both

the subjects. Find the number of students who like

a. Maths only b. Science only c. Either Maths or Science d. Neither Maths

nor Science

Question 3: In a class of 50 students appearing for an examination of ICWA, from a centre,

20 failed in Accounts, 21 failed in Mathematics and 27 failed in Costing, 10 failed both in

Accounts and Costing, 13 failed both in Mathematics and Costing and 7 failed both in

Accounts and Mathematics. If 4 failed in all the three, find the number of (i) Failures in

Accounts only (ii) Students who passed in all the three subjects.


2. Permutations and Combinations

Introduction

Arrangement – The sequence / order of things is taken on account

Grouping – Sequence/order is not necessary

Why Permutations & Combinations?

Illustration

The manager of a large bank has a difficult task of filling two important positions from a

group of five equally qualified employees. Since none of them has had actual experience, he

decides to allow each of them to work for one month in each of the positions before he makes

the decision. How long can the bank operate before the positions are filled by permanent

appointments?

Answer to above – cited situation requires an efficient counting of the possible ways in which

the desired outcomes can be obtained. A listing of all possible outcomes may be desirable, but

is likely to be very tedious and subject to errors of duplication or omission. We need to devise

certain techniques which will help us to cope with such problems. The techniques of

permutation and combination will help in tackling problems such as above.

Permutations: A group of persons want themselves to be photographed. They approach the

photographer and request him to take as many different photographs as possible with

persons standing in different positions amongst themselves. The photographer wants to

calculate how many films does he need to exhaust all possibilities? How can he calculate the

number?

Combinations: There are situations in which order is not important. For example, consider

selection of 5 clerks from 20 applicants.

Fundamental Principles of counting

Rule Explanation Example

Multiplication

Rule AND⇒m×n

multiply

If certain thing may be done in ‘m’

different ways and when it has been

done, a second thing can be done in ‘n’

different ways then total number of

ways of doing both things

simultaneously = m×n.

If one can go to school by 5

different buses and then

come back by 4 different

buses then total number of

ways of going to and coming

back from school = 5×4 =20

Addition Rule OR

⇒ Add

If there are two alternative jobs which

can be done in ‘m’ ways and in ‘n’ ways

respectively then either of two jobs can

be done in (m + n) ways.

If one wants to go school by

bus where there are 5 buses

or to by auto where there are

4 autos, then total number of

ways of going school = 5 + 4

=9.


Factorial notation

Definition – The Factorial n, represents the product of all integers from 1 to n both inclusive

Notation – n!

n! = n× (𝑛 − 1) × (𝑛 − 2) × … . .× 3 × 2 × 1

Justification: 0! = 1

1. As per definition

2. Default case - In the case where no arrangement is needed, there is an arrangement in

default, hence,

3. On reverse working

0! = 1

1! = 1

2! =2×1

3! = 3×2×1

4! = 4×3×2× 1

1!

1 = 1

2!

2 = 2

3!

3 = 2!

4!

4 = 3!

4. Apply r =n in the formula for nPr = 𝑛!

(𝑛−𝑟)!

, we get nPn = n!/(n-n)! = n!/0!

We know that nPn = n! Therefore, by applying this we drive, 0! = n!/n! =1

Permutations and Combinations - on the whole

Permutations Combinations

Meaning Arrangement Grouping

Definition The ways of arranging

or selecting smaller or

equal number of persons

or objects from a group

of persons or collection

of objects with due

regard being paid to the

order of arrangement or

selection

The number of ways in which smaller or

equal number of things are arranged or

selected from a collection of things where

the order of selection or arrangement is

not important

Formulae nPr = 𝑛!

(𝑛−𝑟)! nCr =

𝑛!

𝑟!(𝑛−𝑟)! =

nPr

𝑟! ➔ 𝑛𝑃𝑛 =n!/𝑛𝐶𝑛

Repetitions 𝑛!

𝑃! 𝑞! 𝑟!

No such in combinations

Circular

Case

1. Circular / Ring - (n-1) !

2. circular + no

difference in order

No such in combinations


(clockwise and anti-

clockwise) (𝑛−1)!

2

Permutations Combinations

Restricted Case

Should

occur

(𝑛 − 𝑝)𝑃(𝑟−𝑝)

× 𝑟𝑃𝑝 (𝑛 − 𝑝)𝐶

(𝑟−𝑝)

Should not

occur

(𝑛 − 𝑝)𝑃𝑟 (𝑛 − 𝑝)𝐶

𝑟

Cases at different r values

r = 0 𝑛!

[𝑛 − 0]!=

𝑛!

𝑛!= 1

𝑛!

(𝑛−0)0!= 1 (𝑁𝑜𝑡𝑒: 𝑛𝑃𝑟 = 𝑛𝐶𝑟)

r = 1 𝑛!

(𝑛 − 1)!=

𝑛 × (𝑛 − 1)!

(𝑛 − 1)!

= 𝑛

𝑛!

(𝑛−1)!1! = n (𝑁𝑜𝑡𝑒: 𝑛𝑃𝑟 = 𝑛𝐶𝑟)

r = n-1 𝑛!

[𝑛 − (𝑛 − 1)]!=

𝑛!

1

𝑛!

(𝑛−𝑛+1)(𝑛−1)!

𝑛! × 𝑛×(𝑛−1)!

1(𝑛−1)! = n

r = n 𝑛!

(𝑛−𝑛)! =

𝑛!

0! = n! 𝑛!

(𝑛 − 𝑛)𝑛!= 1

Remarks At, r = n and r= n-1,

𝑛𝑃𝑛 = 𝑛𝑃𝑛−1

𝑛𝐶𝑟 = 𝑛𝐶𝑛−𝑟

(𝐶𝑜𝑚𝑝𝑙𝑖𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐶𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠)

If we form a group of r things out

of n different things, (n-r) remaining

things which are not included in this

group form another group of rejected

things.

Points to ponder

Relation between Permutations and Combination

(𝑛 + 1)𝐶𝑟 = 𝑛𝐶𝑟+ 𝑛𝐶𝑟−1 and 𝑛𝑃𝑟 = (𝑛 − 1)𝑃𝑟 + (𝑛 − 1)𝑃(𝑟−1)

Grouping & Order = Arranging

𝑛𝐶𝑟. 𝑟𝑃𝑟 = 𝑛𝑃𝑟

⇒ 𝑛𝐶𝑟 = 𝑛𝑃𝑟

𝑟𝑃𝑟 =

𝑛!(𝑛−𝑟)!

𝑟!(𝑟−1)!

= 𝑛!

𝑟!(𝑛−𝑟)! (since (r-r)! =

0! = 1)

Prove that nCr = n-2Cr-2 + 2 n-2Cr-1 + n-2Cr

RHS = n-2Cr-2 + 2 n-2Cr-1 + n-2Cr

= n-2Cr-2 + n-2Cr-1+ n-2Cr-1 + n-2Cr (Refer Note 1

and 2)

= n-1Cr-1 + n-1Cr (Refer Note 3)

= (n-1) + 1Cr = nCr = L.H.S


Note

nCr-1 + nCr = n+1Cr

Take n = n-2 & r = r-1

∴n-2C(r-1) -1 + n-2Cr-1 = n-2+1Cr-1

n-2Cr-2 + n-2Cr-1 = n-1Cr-1

Take n = n-2 & r = r

∴ n-2Cr-1 + n-2Cr = (n-2)+1Cr = n-1Cr

Take n = n-1 & r = r

∴ n-1Cr-1 + n-1Cr = (n-1)+1Cr= nCr

Theorem: The number of permutations of n things chosen r at a time is given by

nPr = 𝑛(𝑛 − 1)(𝑛 − 2) … (𝑛 − 𝑟 + 1), where the product has exactly r factors.

Results

Number of permutations of n

different

things taken all n things at a time is

given by

nPn = 𝑛(𝑛 − 1)(𝑛 − 2) … (𝑛 − 𝑛 + 1)

= 𝑛(𝑛 − 1)(𝑛 − 2) … .2.1 = n!

nPr = 𝑛. (𝑛 − 1)((𝑛 − 2) … … . (𝑛 − 𝑟 + 1)

= 𝑛. (𝑛 − 1)((𝑛 − 2) … … . (𝑛 − 𝑟 + 1) ×(𝑛−𝑟)(𝑛−𝑟−1)2.1

1.2…(𝑛−𝑟−1)(𝑛−𝑟)

= n!/(n-r)!

Thus, nPr = 𝑛!

(𝑛−𝑟)!

Standard Results

I. Permutations when some of the things are alike, taken all at a time, the number of ways p

in which n things may be arranged among themselves, taking them all at a time, when 𝑛1 of

the things are exactly alike of one kind, 𝑛2 of the things are exactly alike of another kind, 𝑛3

of the things are different is given by, P = 𝒏!

𝒏𝟏!𝒏𝟐!𝒏𝟑!

II. Permutations when each thing may be repeated once, twice ,…… up to r times in any

arrangement, 𝒏𝒓

III. Combinations of n different things taking some or all of n things at a time.

∑ 𝒏𝑪𝒓 = 𝟐𝒏 − 𝟏𝒏𝒓=𝟏 (that is one or more) [No or more = ∑ 𝒏𝑪𝒓

𝒏𝒓=𝟎 = 𝟐𝒏]

IV. Combinations of n things taken some or all at a time when 𝑛1 of the things are alike of

one kind, 𝑛2 of the things are alike of another kind 𝑛3 of the things are alike of a third kind.

{(𝒏𝟏 + 𝟏)(𝒏𝟐 + 𝟏)(𝒏𝟑 + 𝟏) … . . ) − 𝟏}

V. The notion of Independence in Combinations

VI. The value of ∑ 𝑟10𝑟=1 . rPr [Note: (𝑟 + 1)! - r! = (𝑟 + 1)𝑟! − 𝑟! = r! (𝑟 + 1 − 1) = r×r! = r× 𝑟𝑃𝑟]

∑ 𝑟 × 𝑟𝑃𝑟10𝑟=1 = ∑ r × r!10

r=1

= ∑ (𝑟 + 1)!10𝑟=1 − 𝑟! = ( 2! – 1!) + (3! – 2!) + ....... + (10! – 9!) + (11! – 10!)

= 11! – 1!

Result: The combinations of selecting 𝑟1 things from a set having 𝑛1 objects and 𝑟2 things

from a set having 𝑛2 objects where combination of 𝑟1 things, 𝑟2 things are independent is

given by n1 𝐶𝑟1 × n2 𝐶𝑟2


Note: This result can be extended to more than two sets of objects by a similar reasoning.

𝑛1𝐶𝑟1× 𝑛2𝐶𝑟2

Simple Problems

Permutations

Sl.

No

Question Solution

1 Prove that C for “CALCUTTA” is twice of A for

“AMERICA” in respect of number

of arrangements of letters.

For C, 8!

2!2!2! =

8×7!

2×2×2 = 7!

For A, 7!

2! =

7!

2!

∴ C = 2 × A

7! = 2 ×7!

2! ➔ 7! = 7!

2 Four travellers arrive in a town where there are six

hotels. In how many ways can they take their

quarters each at a different hotel?

6 hotels & 4 travellers

6𝑃4 = 6!

2! =

720

2 = 360s

3 In how many ways can 8 mangoes of different

sizes be distributed amongst 8 boys of different

ages so that the largest one is always given to the

youngest boy?

𝑛𝑃𝑛 = 7𝑃𝑟 = 7! ×1!

4 How many different odd numbers of 4 digits can

be formed with the digits 1, 2, 3, 4, 5, 6, 7; the digits

in any number being all different?

4𝑃1 × 6𝑃3 = 4× 6!

3! =

4 ×720

6 = 480

(4𝑃1 is the ways for

fixing the odd number

in the ones place)

5 In how many ways can the colours of a rainbow

(VIBGYOR) be arranged, so that the red and the

blue colours are

i. always together.

ii. always separated.

Total: 7!

Together: 6! × 2! =720 ×

2 = 1440

Seperated = Total -

Together

= 7! – (6! × 2!) = 5040 –

1440 = 3600

6 i. In how many ways can 5 boys form a ring?

ii. In how many ways 5 different beads be strung

on a necklace?

i. (𝑛 − 1)! = (5 − 1)! = 4!

= 24

ii.(𝑛−1)!

2 =

(5−1)!

2 =

24

2 = 12


7 If the letters word ‘DAUGHTER’ are to be

arranged so that vowels occupy the odd places,

then number of different

words are?

No. of vowels = 3(A,

U,E)

No. of Odd places = 4

Places

Hence, 4𝑃3ways for

vowels

5 Consonants in

remaining 5 places, i.e.

5𝑃5

∴ 4𝑃3 × 5𝑃5 = 2880

ways

8 The total number of ways in which six ‘+’ and four

‘-‘ signs can be arranged in a line such that no two

‘- ‘ signs occur together is?

_ + _ + _ + _ + _ + _

𝟔!×𝟕𝑷𝟒

𝟔!×𝟒! =

𝟔!×𝟕!

𝟒!

𝟔!×𝟒!

Combination

No Question Solution

1 In an examination paper, 10 questions are set. In how many

different ways can you choose 6 questions to answer. If

however no. 1 is made compulsory in how many ways can

you select to answer 6 questions in all?

10𝐶6 = 7

9𝐶5 × 1𝐶1 =

126

2 In a meeting after every one had shaken hands with

everyone else, it was found that 66 handshakens were

exchanged. How many members were present at the

meeting?

66 = n𝐶2

3 A man has 3 friends. In how many ways can be invite one or

more of them to dinner?

2𝑛 − 1 = 7

4 n point are in space, no three of which are collinear. If the

number of straight lines and triangles with the given points

only as the vertices, obtained by joining them are equal, find

the value of n.

nC3 = nC2

𝑛×(𝑛−1)×(𝑛−2)

3×2×1 =

𝑛×(𝑛−1)

2×1

n -2 = 3 ➔ n =

5

5 How many different triangles can be formed by joining the

angular points of a decagon? Find also the number of

diagonals of the decagon?

10C3 → no.of

Triangles

10C2 -10 → no .of

diagonals


Note: In general, n points. nC3 → no.of Triangles and nC2 – n

→ no.of Diagonal

Practice Problems

1. There are 20 stations on a railway line. How many different kinds of single first-class

tickets must be printed so as to enable a passenger to go from one station to another?

Answer:

20𝑃2 = 19×20 = 380

2. How many number lying between 1000 and 2000 can be formed from the digits 1, 2, 4, 7, 8,

9 ; each digit not occurring more than once in the number?

Answer:

1𝑃1 × 5𝑃3 = 1× 5!

2! = 1 ×

120

2 = 60

(4𝑃1 is the ways for fixing the odd number in the ones place)

3 In how many ways can 7 papers be arranged so that the best and the worst papers never

come together?

Answer:

Total: 7!

Together: 6! × 2! =720 × 2 = 1440

Seperated = Total - Together = 7! – (6! × 2!) = 5040 – 1440 = 3600

4. Show that the number of ways in which 16 different books can be arranged on a shelf so

that two particular books shall not be together is 14 (15!).

Total: 16!

Together: 15! × 2!

Seperated = Total - Together

= 16! – (15! × 2!) = 14(15!)

5. Out of 5 ladies and 3 gentlemen, a committee of 6 is to be selected. In how many ways can

this be done : (i) when there are 4 ladies? (ii) when there is a majority of ladies?

Answer:

When there are 4 ladies,

5𝐶4 × 3𝐶2 = 5 × 3 = 15

When there are majority of ladies,

(5𝐶5 × 3𝐶1)+(5𝐶4 × 3𝐶2)

=(1 × 3) + (15)


=18(3+15)

6. Out of 16 men, in how many ways a group of 7 men may be selected so that (i) particular 4

men will not come and (ii) particular 4 men will always come?

Answer: i) 12𝐶7 (ii) 12𝐶3

Comprehensive problem

Question 1: Find the number of ways of selecting 4 letters from the word

“EXAMINATION”.

Answer:

There are 11 letters in the word of which A, I,

N

are repeated twice.

Thus we have 11 letters of 8 different kinds

(A,A), (I,I), (N,N), E, X, M, T, O.

The group of four selected letters may take

any of the following forms:

a. Two alike and other two alike, 3C2 =

3

b. Two alike and other two different,

3C1×7C2 =3×21 = 63

c. All four different, 8C4 = 8×7×6×5

1×2×3×4 =70

Hence, the required number of ways

= 3 + 63 +70 = 136 ways

Question 2: The number of parallelograms that can be formed from a set of four parallel

lines intersecting another set of three parallel lines is?

Answer: Select 2 lines from Set 1 (containing 4 parallel lines), 4C2 = 6

Select 2 lines from Set 2 (containing 3 parallel lines), 3C2 = 3

Hence, 6 × 3 = 18 parallelograms can be formed

Question 3: A boat’s crew consist of 8 men, 3 of whom can row only on one side and 2 only

on the other. The number of ways in which the crew can be arranged is __________

Answer:

Out of 8 persons, 5 are fixed and two persons are to be choosed from the remaining 3.

Hence, 3𝐶2

The 4 persons in both the sides could be arranged in 4! × 4!

Therefore, 3𝐶2 × 4! × 4! = 3𝐶2 × (4!)2 = 1728

Question 4: If all the permutations of the letters of the word ‘CHALK’ are written in a

dictionary the rank of this word will be__________

Answer:


1. A _ _ _ _ = 1 × 4! = 24

2. C A _ _ _ = 1 × 1 × 3! = 6

3. C H A K L = 1 × 1 × 1 × 1 × 1 = 1

4. C H A L K = 1 × 1 × 1 × 1 × 1 = 1

Hence, 24 + 6 + 1 +1 = 32, 32nd rank

Question 5: A family of 4 brothers and three sisters is to be arranged for a photograph in one

row. In how many ways can they be seated if, 1. All the sisters sit together, 2. No two sisters

sit together?

Answer:

1. Consider the sisters as one unit and each brother as one unit. 4 brothers and 3 sisters make

5 units which can be arranged in 5! Ways. Again 3 sisters may be arranged amongst

themselves in 3! Ways

Therefore, total number of ways in which all the sisters sit together = 5!×3! = 720 ways.

2. In this case, each sister must sit on each side of the brothers.

4 brothers may be arranged among themselves in 4! Ways. For each of these arrangements 3

sisters can sit in the 5 places in 5P3 ways.

Thus the total number of ways = 5P3 ×4! = 60×24 = 1,440

Question 6: The Supreme Court has given a 6 to 3 decision upholding a lower court; the

number of ways it can give a majority decision reversing the lower court is a. 256 b. 276 c.

245 d.115

Answer:

It can by 5 votes for or 6,7,8 and the whole 9 to get majority so,

9C9+9C8+9C7+9C6+9C5 =1+9+36+84+126 =256

Question 7: Five bulbs of which three are defective are to be tried in two bulb points in a dark

room. Number of trials the room shall be lighted is a. 6 b. 8 c. 5 d. 7

Answer:

The number of ways in choosing 2 bulbs to fit in the bulb points are C(5,2) = 10

The number of ways it will not light, due to choosing of defective bulbs are C(3,2) = 3

Hence, 10-3 = 7 ways


3. Sequence and Series – Arithmetic and Geometric progressions

Sequence

An ordered collection of numbers

𝑎1, 𝑎2, 𝑎3, 𝑎4, … 𝑎𝑛 is the term or element

of the sequence, corresponding to any

value of the natural number n.

Terms: 𝑎1, 𝑎2, 𝑎3,….., 𝑎𝑛, …..

(1) 28,2,25,27 ___________________

(2) 2,7,11,19,31,51, _______________

(3) 1,2,3,4,5,6,_________________

(4) 20,18,16,14,12,10 _____________

Types

Types Finite sequence Infinite sequence

Definition n-finite/𝒂𝟏, 𝒂𝟐, … . 𝒂𝒏 n- Infinite 𝑎1, 𝑎2, 𝑎3……

Examples 1. A Sequence of even

positive

integers within 12 ➔ 2,4,6,10

2. A sequence of odd positive

integers within 11 ➔1,3,5,7,9.

1. The Sequence {1

𝑛} is 1,

1

2,

1

3,

1

4……

2. The Sequence {(−1)𝑛𝑛} is -1, 2, -3,4-

5,……

3. The Sequence {𝑛

𝑛+1} is

1

2,

2

3,

3

4,

4

5, ………

4. A Sequence of even positive integers is

2, 4, 6,……….

Series

Form: 𝑎1 + 𝑎2 + ⋯ + 𝑎𝑛, 𝑆𝑛 = ∑ 𝑢𝑟𝑛𝑟=1

Finite Series Infinite series

n-finite 𝑎1 + 𝑎2 + ⋯ + 𝑎𝑛 n-infinite, 𝑎1, 𝑎2 ….

AP GP

Definition A sequence

𝑎1, 𝑎2, 𝑎3, 𝑎4……,𝑎𝑛 is called

an Arithmetic Progression

A sequence of terms each term is constant

multiple of the proceeding term, then the


(A.P) when 𝑎2 − 𝑎1 = 𝑎3 − 𝑎2

= …. =𝑎𝑛 − 𝑎𝑛−1

sequence is called a Geometric Progression

(G.P).

Terms 1st term →a

common difference →d

Number of terms →n+ve

nth term → 𝑡𝑛 = a+(n-1)𝑑

1st term →a

common ratio → r

Number of terms → n→+ve

nth term→ 𝑡𝑛 = 𝑎𝑟−1

Note 𝑎2 − 𝑎1 = 𝑎3 − 𝑎2 = …..= 𝑎𝑛 −

𝑎𝑛−1 = d

𝑎2

𝑎1 =

𝑎3

𝑎2 = ….. =

𝑎𝑛

𝑎𝑛−1 = r

Illustration

1) 2, 5, 8, 11, 14, 17, …..

2) 15,13,11,9,7,5,3,1, -1,…..

1) 1,4,6 ……

2) 1

3 ,

1

𝑎,

1

27, ….

Means–the

sequence

a,b,c

Arithmetic Mean = b = 𝑎+𝑐

2 Geometric Mean = b = √𝑎 × 𝑐

Series S = n(𝑎+𝑙

2)

S = 𝑛

2 [2𝑎 + (𝑛 − 1)𝑑]

𝑆𝑛 = {𝑎(1−𝑟𝑛)

1−𝑟, 𝑟 < 1

𝑎(𝑟𝑛−1)

𝑟−1 , 𝑟 > 1

Note: 𝑆⋈ = 𝑎

1−𝑟, −1 < 𝑟 < 1

if n→⋈, r = 1

𝑅 ,

1

𝑅𝑛 →0

Nature of d:

1. Highest to Lowest: [d = -ve]

Ex: 75, 70, 65, 60, 55, ……

2. Lowest to Highest: [d = +ve]

Ex: 75, 80, 85, 90, 95, …..

3. Constant [d = 0]

Ex: 75, 75, 75, 75, 75, …….


Nature of r:

1. Highest to Lowest: [division | r= +ve]

Ex: 10, 5, 2.5, 1.25, ….. [r = 1/2]

Special Case: r >1

Ex: 10, 15, 45/2, 135/2, ……. [r = 3/2]

2. Lowest to Highest: [multiplication | r= +ve]

Ex: 10, 20, 40, 80, 160, …. [r = 2]

Special Case: r <1

Ex: 10, 5, 2.5, 1.25, …. [r = ½ or 0.5]

3. Constant [r = 1]

Ex: 10, 10, 10, ……

4. Subsequent Values “0” [r = 0]

Ex: 10, 0, 0, 0, 0, ……

5. Zig – Zag Sequence [r = -ve]

Ex: 10, -20, 40, -60, -160, …..

Results

AP GP

1. Sum of 1st n natural or counting numbers, S = 𝑛(𝑛+1)

2

2.Sum of 1st n odd numbers, S = 𝑛2

3. Sum of the Squares of the first, n natural numbers,

S = 𝑛(𝑛+1)(2𝑛+1)

6

In general,

X + xx + xxx +…. to n terms

= 𝑥

9 [{10(10𝑛 − 1/(10 − 1)} − 𝑛]

&

0.x + 0.xx + …… to n term


4. Sum of the cubes of the first n natural numbers,

{𝑛(𝑛+1)

2}

2

= 𝑥

9 [𝑛 −

1

10(1 −

1

10𝑛) / (1 −1

10)]

Problems

Arithmetic Progression

1. Find the 7th term of the A.P. 8,5,2,-1,-4, ……

Solution: Here

a= 8, d = 5-8 = -3

Now 𝑡7 = 8 + (7-1) d

= 8 + (7-1) (-3)

= 8 + 6(-3)

= 8 -18

= -10

2. Which term of the AP

3

√7 ,

4

√7 ,

5

√7….. is

17

√7?

Solution:

a = 3

√7, d=

4

√7 -

3

√7 =

1

√7, 𝑡𝑛 =

17

√7

We may write, 17

√7 =

3

√7 +(n-1) ×

1

√7

Or, 17 = 3 +(n-1) ➔ n = 17-2 = 15

Hence, 15th term of the A.P. is 17

√7

3. If 5th and 12th terms of an A.P. are 14 and 35 respectively, find the A.P.

Solution: Let a - first term & d - common difference of A.P, then

𝑡5 = a + 4d = 14 and 𝑡12 = a + 11d = 35

On solving the above two equations,

7d =21 = i.e., d =3

And a =14 – (4 × 3) = 14 -12 = 2

Hence, the required A.P. is 2,5,8,11,14, …


4. Find the arithmetic mean between 4 and 10.

Solution:

We know that the A.M. of a & b is = (a + b) / 2

Hence, The A. M between 4&10 = (4 + 10) / 2 = 7

5. Divide 69 into three parts which are in A.P. and are such that the product of the first two

parts is 483.

Solution: Given that the three parts are in A.P., let the three parts which are in A.P. be a-d, a,

a+d.

Thus (a-d) + a + (a+d) = 69

Or 3a = 69

Or a = 23

So the three parts are 23 – d, 23,23+d

Since the product of first two parts is 483,

23 (23-d) =483

Or 23 –d = 483/23 =21➔ Or d = 23-21 =2

Hence the three parts are 21,23,25

6. Insert 4 arithmetic means between 4 and 324.

Solution:

Here a = 4, d= ? n=2 + 4 =6, 𝑡𝑛 =324

Now 𝑡𝑛 = a + (n-1)d

Or 324 = 4+ (6-1) d

Or 320 = 5d i.e., d = 320/5 = 64

So the 1st AM = 4+64 =68

2nd AM = 68 +64 = 132

3rd AM = 132+64 = 196

4th AM = 196+64 = 260

Geometric Progression

1. Which term of the progression 1,2,4,8,… if 256?

Solution: a=1, r = 2/1=2, n =? 𝑡𝑛 =256

𝑡𝑛 = 𝑎𝑟𝑛−1 ➔ 256 = 1× 2𝑛−1 or, n-1 = 8 i.e., n=9

Thus 9th term of the G.P. is 256

2. Insert 3 geometric means between 1/9 and 9.

Solution: 1/9, -, -, -, 9

a = 1/9 r, =?, n=2+3 =5, 𝑡𝑛 = 9


we know,𝑡𝑛 = 𝑎𝑟𝑛−1

or 1/9× 𝑟5−1 = 9

or 𝑟4 = 81 = 34 ⇒ r =3

Thus, 1st G.M = 1/9 ×3 = 1/3

2nd G.M = 1/3 ×3 = 1

3rd G.M = 1×3 =3

3. Find the G.P where 4th term is 8 and 8th term is 128/625

Solution: Let a be the 1st term and r be the common ratio.

By the question 𝑡4 = 8 and 𝑡8 = 128/625

So 𝑎𝑟3 = 8 and 𝑎𝑟7 = 128/625

Therefore, 𝑎𝑟7/𝑎𝑟3 = 128

625×8 ⇒ 𝑟4 = 16/625 =(± 2 5⁄ )4 ⇒r = 2/5 and -2/5

Now 𝑎𝑟3 = 8⇒ a × (2 5⁄ )3 = 8⇒ a = 125

Thus the G.P is 125,50,20,8,1685,……..

When r = -2/5, a = -125 and the G.P is -125, 50, -20,8, -16/5,…….

Finally, the G.P. is 125,50,20,8,16/5,……. or, -125,50,-20,8, -16/5, ……

4. Find the 3 numbers in G.P whose sum is 14 and product is 64.

Solution: Let the 3 numbers be a/r, a, ar.

According to the question a/r× 𝑎 × 𝑎𝑟 = 64

Or 𝑎3 = 43 ⇒a=4

So the numbers are 4/r, 4,4r

Again 4/r + 4 +4r =14

Or 4/r + 4r = 10

Or 4 + 4𝑟2 = 10r

Or 4𝑟2 - 10r + 4 =0

Or 4𝑟2- 8r -2r + 4 =0

Or 4r(r-2) – 2(r-2) = 0

Or (4r – 2) (r -2) = 0

Or r =2 or, 1/2


So the numbers are 2,4,8 or 8,4,2

Note: Find the 5 numbers in G.P whose sum is 31 and product is 1024.

Series

1. Find the sum of 1 + 2 + 3 + 4 + ….. to 8 terms.

Solution: Here a =1, d = 1, n=8

S = 𝑛

2 [2𝑎 + (𝑛 − 1)𝑑] = 36

or

Sum of 1st n natural or counting numbers, S = 𝑛(𝑛+1)

2 = 36

2. Find the sum of 1 + 2 + 4 + 8 + …. to 8 terms.

Solution: Here a =1, r = 2/1 = 2, n=8

Let S = 1 + 2+ 4 + 8 + ….. to 8 terms

= 1(28 − 1/(2 − 1)) = 28 − 1 = 255

3. Find the sum to n terms of 6+27+128+629 + ……

Solution: Required Sum

= (5+1) + (52 + 2) + (53 + 3) + (54 + 4) + ⋯to n terms

=(5 + 52 + ⋯ + 5𝑛) + (1 + 2 + 3+. . +𝑛 𝑡𝑒𝑟𝑚𝑠)

= {5(5𝑛 − 1)/(5 − 1)} + {𝑛(𝑛 + 1)/2}

= {5(5𝑛 − 1)/4} + {𝑛(𝑛 + 1)/2}

4. Find the sum to n terms of the series

3 + 33 +333 +…..

Solution: Let S denote the required sum.

i.e. S = 3 + 33 +333 + ……. to n terms

= 3(1+11+111+…… to n terms)

= 3

9 (9 + 99 + 999 + …. to n terms)

5. Find the sum of n terms of the series

0.7 + 0.77 + 0.777+ …. to n terms

Solution: Let S denote the required sum.

i.e. S = 0.7+0.77+0.777+….to n terms

= 7(0.1+0.11+0.111+….to n terms)

=7

9{(1 −

1

10) + (1 −

1

102) + (1 −1

103) + ⋯ + (1 −

1

10𝑛)}


=3

9 {(10 − 1) + (102 − 1) + (103 − 1) + ⋯ +

(10𝑛 − 1)}

= 3

9{(10 + 102 + 103 + ⋯ + 10𝑛) − 𝑛}

= 3

9{10(1 + 10 + 102 + ⋯ + 10𝑛−1) − 𝑛}

= 3

9[{10(10𝑛 − 1)/(10 − 1)} − 𝑛]

= 3

81 (10𝑛−1 − 10 − 9𝑛)

= 1

27(10𝑛+1 − 9𝑛 − 10)

= 7

9{𝑛 −

1

10(1 +

1

10+ 1 +

1

102 + ⋯ +1

10𝑛−1)}

So S = 7

9 {𝑛 −

1

10(1 −

1

10𝑛)/(1 −1

10)}

= 7

9 {𝑛 −

1−10−𝑛

9}

= 7

81 {9𝑛 − 1 + 10−𝑛}

5. Evaluate 0.2175 using the sum of an infinite geometric series.

Solution: 0.2175 = 0.217575….

0.2175 = 0.21 + 0.00075 + 0.000075 + ….

= 0.21 + 75(1+1/102 +1

104 + ⋯ )/104

= 0.21 + (75/104) × 102/99

= 21/100 +(3

4) × (

1

99)

= 21/100 + 1/132

= (639+25)/3300 = 718/3300 = 359/1650

Explanations

Sum of the first n terms

Let S be the Sum, a be the 1st term and l the

last term of an A.P. if the number of term

are n,then 𝑡𝑛 =l.Let d be the common

difference of the A.P.

Now S = a +(a+d) + (a+2d) + … +(l-2d) + (l-d)

+ l

Again S = l +(l-d) + (l-2d) + … + (a+2d) +(a+d)

+ a

On adding the above, we have

2S = (a+l) + (a+l) + (a+l) + …. + (a+l)

= n(a+l)

Sum of first n terms of a G.P

Let a be the first term and r be the common

ratio. So the first n terms are a, ar, a𝑟2,….

a𝑟𝑛−1.

If S be the sum of n terms,

𝑆𝑛 = a + ar + a𝑟2 + ⋯ + 𝑎𝑟𝑛−1-------(i)

Now 𝑟𝑆𝑛 = ar + a𝑟2+…..+ 𝑎𝑟𝑛−1+𝑎𝑟𝑛 -------

(ii)

Sum of infinite geometric series

S = a (1 - 𝑟𝑛)/(1 − 𝑟) when r<1

= a (1-1/𝑅𝑛)/(1-1/R) Since r<1, we take r =

1/R).


Or S = n(a+l)/2

Note: The above formula may be used to

determine the sum of n terms of an A.P.

when the first term a and the last term is

given.

Now l = 𝑡𝑛 = a + (n-1) d

∴ S = 𝑛{𝑎+𝑎+(𝑛−1)𝑑}

2

Or S = 𝑛

2 2a +(n-1)d

Note: The above formula may be used

when the first term a, common difference d

and the number of terms of an A.P. are

given.

If n →∝ , 1/𝑅𝑛 →0

Thus 𝑆∞ = 𝑎

1−𝑟, 𝑟 < 1

i.e. sum of G.P. upto infinity is = 𝑎

1−𝑟 , where

r<1

Also, 𝑆∝ = 𝑎

1−𝑟, if -1<r<1.


4. Ratios

Ratio – A Comparison of the sizes of two or more quantities of the same kind (units by

division)

𝑅𝑎𝑡𝑖𝑜 𝑜𝑓 𝑎 𝑡𝑜 𝑏 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑒𝑑 𝑎𝑠 𝑎: 𝑏 𝑜𝑟𝑎

𝑏

Here 𝑎 First term / antecedent

𝑎, 𝑏 Term of the ratio 𝑏 Second term / consequent

Remarks

Sl.no Remarks Explanation Expression

𝒂: 𝒃

Example

1 Expressed in

lowest

terms/simplest

form

Both terms

can be

divided with

the same

(non-zero)

number

𝑎

𝑏=

2 × 𝑎

2 × 𝑏

=𝑎/2

𝑏/2

12: 16 =12

16

=12 ÷ 4

16 ÷ 4=

3

4= 3: 4

2 Importance of

order of terms

order of

terms should

not be

changed

𝑎: 𝑏 ≠ 𝑏: 𝑎 2: 3 ≠ 3: 2

3

Ratio does not

exist as the

quantities not

of same kind

necessity to

be in same

kind 𝑎 (𝑘𝑔): 𝑏 (𝑦𝑒𝑎𝑟𝑠)

Ratio exists between

number of students &

number of teachers, but

not between number of

students & salary of

teacher

2 𝑘𝑔: 3 𝑦𝑒𝑎𝑟𝑠

4 Quantities to

be compared

must be in the

same units

(expressed in

equivalent

terms)

quantities to

be compared

(by division)

in the same

units in

equivalent

terms

Note: it is

divided to

bring it in the

lowest form

𝑎 (𝑘𝑔): 𝑏 (𝑔𝑚𝑠)

2 𝑘𝑔: 500 𝑔𝑚𝑠

Hence it should be

2,000 𝑔𝑟𝑎𝑚𝑠: 500 𝑔𝑟𝑎𝑚𝑠

And in simplest form 4: 1

5 Comparing–

fractions

Convert into

equivalent

fractions

21

3: 3

1

3 =

7

3:

10

3 = 7:10 =

7

10


6 Increase /

decrease in

ratio

Hint:

a:b = Old :

New

𝑛𝑒𝑤 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦

= 𝑏 × 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑞𝑢𝑎𝑙𝑖𝑡𝑖𝑡𝑦

𝑎

Here, 𝑏

𝑎 - factor

multipling

ratio

Rounaq weighs 56.7kg.

If he reduces his weight

in the ratio 7:6, find his

new weight.

Original weight of

Rounaq = 56.7kg.

He reduces his weight in

the ratio 7:6

His new weight = (6×

56.7)/7 = 6×8.1 = 48.6kg.

Note:

1. 𝐼𝑓 𝑎 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 / 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑖𝑛 𝑟𝑎𝑡𝑖𝑜 𝑎: 𝑏 𝑡ℎ𝑒𝑛 𝑛𝑒𝑤 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 =

𝑏

𝑎𝑜𝑓 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦

Ratio Increases, Quantity Increases | Ratios Decreases, Quantity Decreases

2. To compare two ratios, convert them into equivalent like fractions

Other Ratios

Ratio Forms Formula Example Description

1 Ratio 𝑎: 𝑏 2: 3 Expressed at its simplest

form Duplicate ratio 𝑎2: 𝑏2 4: 9

Triplicate ratio 𝑎3: 𝑏3 8: 27

Sub-duplicate ratio √𝑎: √𝑏 √2: √3

Sub- triplicate ratio √𝑎3

: √𝑏3

√23

: √33

Inverse ratio 𝑏: 𝑎 3: 2 𝑅𝑎𝑡𝑖𝑜 × 𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑅𝑎𝑡𝑖𝑜 = 1

2 Ratio 𝑎: 𝑏 1: 1 Ratio of equality as 𝑎 = 𝑏

2: 3 Ratio of lesser inequality as 𝑎 < 𝑏

3: 2 Ratio of greater inequality

as 𝑎 > 𝑏

3 Ratio - Commensurable 𝑎: 𝑏 2: 3 a, b are integers (Ratio of

integers)

- In

Commensurable

√2: √3 a, b are Non - integers (Ratio

of non - integers)

4 Ratio 𝑎: 𝑏: 𝑐 2: 3: 4 Continued ratio (relation

with 3 or more)

5 Compound Ratio a:b & c:d ⟹

ac:bd

3:4 & 5:7

⟹ (3 × 5): (4× 7) = 15: 28

Product of antecedents:

Product of consequents

Note: Duplicate Ratio is a

compound ratio


Simple Problems

1. Mr Singar weighs 60 kg if he reduces his weight in the ratio 12: 11 find his new weight.

Answer:

Formula Calculation Answer

New Weight 𝑏

𝑎𝑜𝑓 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦

11

12× 60

55 kg

2. Find out the greater ratio:21

3: 3

1

3; 3.6: 4.8

Answer:

Convert in to like fractions Fraction After LCM

First ratio 21

3: 3

1

3=

7

3:10

3= 7: 10

7

10

7 × 2

10 × 2

=14

20

Second ratio 3.6: 4.8 =3.6

4.8=

36

48

3

4

3 × 5

4 × 5=

15

20

As 𝟏𝟓 > 14

Hence 15

20>

14

20 𝑜𝑟 3.6: 4.8 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑟𝑎𝑡𝑖𝑜

3. Simplify the ratio:1

3:

1

8:

1

6

Answer: L.C.M of 3,8 and 6 is 24.

1

3:1

8:1

6

1 ×24

3: 1 ×

24

8: 1

×24

6

8: 3: 4

4. Find out the continued ratio of ₹100, ₹150 and ₹250.

Answer: ₹100:₹150:₹250 = 2:3:5.

5. Anand earns ₹80 in 7 hours and Promote ₹90 in 12 hours. The ratio of their earnings is1

(a) 32: 21 (b) 23: 12

(c) 8: 9 (d) None of these

Answer: 80

7:

90

12 = 960: 630 = 32: 21

6. 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 & R is

(a) 22: 27 (b) 27: 22

© 32: 33 (d) None of these

1 a


Answer: P:Q = 11:12 ➔ Q:P = 12:11=108: 99

P:R = 9:8 = 99: 88

Q:P and P:R ➔ Q:R = 108:88 = 27:22

Conceptual Problems

7. 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 no. of boys to the no. of girls

= 3:5

Sum of the ratios = 3 + 5 = 8

So, the no.of boys in the school = (3×720)

8= 270

And the no.of girls in the school = (5×720)

8= 450

Let the no. of new boys admitted be x, then

the number of boys become (270 + x).

After admitting 18 new girls, the no. of

girls become 450 + 18 = 468

According to given description of the

problem,(270 + 𝑥)/468 = 2/3

Or, 3(270 + 𝑥) = 2× 468

Or, 810 + 3𝑥 = 936 or, 3𝑥 = 126 or, 𝑥 = 42.

Hence the number of new boys admitted

=42.

8. 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 income of two persons be ₹4𝑥 and ₹5𝑥 so that the ratio is ₹4𝑥:5𝑥 =

4:5. If each saves ₹50 per month, then the expenditures of two persons are ₹(𝑅𝑠. (4𝑥 − 50) and

₹(5𝑥-50). 4𝑥−50

5𝑥−50=

7

9 or 36𝑥450 = 35𝑥350

Or, 36𝑥 - 35𝑥 = 450 – 350, or 𝑥 = 100

Hence, the monthly incomes of the two persons are ₹4×100 and ₹5×100, i.e.₹400 and ₹500.

9. 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 ₹477, 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 ₹16𝑥 and ₹23𝑥 respectively. Then by the

given conditions, 16𝑥 + 10% 𝑜𝑓16𝑥

23𝑥 + 477 =

11

20

16𝑥 + 1.6𝑥

23𝑥 + 477 =

11

20

320𝑥 + 32𝑥 = 253𝑥 + 5247

352𝑥 -253𝑥 = 5247

99𝑥 = 5247;


∴ 𝑥= 53

Hence, the original prices of two houses are ₹ 848 and ₹1,219.

10. 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.

Answer: 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 = (11

15𝑥) × (₹

25

22𝑦) = ₹

5𝑥𝑦

6

Thus, the total wages of workers get decreased from ₹𝑥𝑦 𝑡𝑜 ₹5𝑥𝑦/6

Hence, the required ratio in which the total wages decrease is 𝑥𝑦:5𝑥𝑦

6= 6: 5


5. Proportions

Proportions – An equality of two ratios

Points to Ponder:

Sl. No

1 a, b, c, d are in proportion – a:b = c:d or a:b : : c:d

2 Cross product rule

If 𝑎

𝑏 =

𝑐

𝑑 ⇒ ad = bc ⇒ 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑠 = 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛𝑠

3 a / b / c / d – first / second / third / fourth proportional

a / d – extremes and c / d – mean/middle terms

4 Continuous proportion

a, b, c of same kind(in same units), if a:b=b:c

i.e 𝑏2 = ac, ⇒b =√𝑎𝑐 ➔i.e b = Geometric Mean of a and c

⇒ Middle term is the mean proportional between a and c

a – first proportional and c – third proportional

5 Continued proportion → 𝑥

𝑦 =

𝑦

𝑧 =

𝑧

𝑤 =

𝑤

𝑝 =

𝑝

𝑞

6 Inverse / Reciprocal proportion – A ratio equal to the reciprocal of the other.

Example: 5

4 &

4

5 are inverse proportion.

Note: In the case of ratio a:b, a & b must be of same kind, whereas in proportions (a:b :: c:d), a

& b must be of same kind and c& d must be of same kind(need not be the same kind as a and

b)

Properties / Laws

Properties / Laws Proof

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

𝑎

𝑏=

𝑐

𝑑; ∴ ad = bc (By cross –

multiplication)

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

(Inverting the ratios)

𝑎

𝑏 =

𝑐

𝑑 or

1𝑎

𝑏

= 1𝑐

𝑑

, or, 𝑏

𝑎 =

𝑑

𝑐

Hence, b : a = d : c.

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

(Alter / Interchange the middle values)

𝑎

𝑏 =

𝑐

𝑑 or, ad =bc

Dividing both sides by cd, we get 𝑎𝑑

𝑐𝑑

=𝑏𝑐

𝑐𝑑, or

𝑎

𝑐 =

𝑏

𝑑, i.e. a:c = b:d.

4 If a:b = c:d, then a + b : b = c + d : d

(Componendo)

𝑎

𝑏 =

𝑐

𝑑 , or,

𝑎

𝑏 + 1 =

𝑐

𝑑 + 1


(Addition – New Nr / Dr = Antecedent +

Consequent)

or, 𝑎 + 𝑏

𝑏 =

𝑐 + 𝑑

𝑑 , i.e. a + b:b

= c + d:d.

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

(Subtraction – New Nr / Dr= Antecedent +

Consequent)

𝑎

𝑏 =

𝑐

𝑑, ∴

𝑎

𝑏− 1 =

𝑐

𝑑− 1

𝑎−𝑏

𝑏 =

𝑐−𝑑

𝑑, i.e a – b :b = c – d:d.

6 If a:b = c:d, then a + b : a – b = c + d : c – d

(Componendo and Dividendo)

(New Antecedent= Componendo Rule

New Consequent = Dividendo Rule)

𝑎

𝑏 =

𝑐

𝑑 , or

𝑎

𝑏 + 1=

𝑐

𝑑 + 1, or

𝑎 + 𝑏

𝑏 =

𝑐 + 𝑑

𝑑 -----------------1

Again 𝑎

𝑏− 1,

𝑐

𝑑− 1, or

𝑎−𝑏

𝑏 =

𝑐−𝑑

𝑑 ------------------------------2

Dividing (1) and (2) we get

𝑎 + 𝑏

𝑎−𝑏 =

𝑐 + 𝑑

𝑐−𝑑 , i.e a + b : a –b = c + d : c – d

7 If a : b = c : d = e : f = …………………., then each

of these ratios (Addendo) is equal

(a + c + e + ……) : (b + d + f + ……)

(New Antcedent = Addition of all antecedent

values

(New Consequent = Addition of all Consequent

values)

Note: The new ratio is equal to the other ratios

𝑎

𝑏 =

𝑐

𝑑 =

𝑒

𝑓 = …………….(say)k,

∴ a = bk, c= dk, e=fk,………

Now a + c + e…… = k (b + d + f) ……..

or 𝑎 + 𝑐 + 𝑒……

𝑏 + 𝑑 + 𝑓…. = k

Hence, (a + c + e + …..): (b + d + f + ……)

is equal to each ratio

8 If a: b = c:d = e:f = ….., then each of these ratios

(subtrahendo) is equal to

(a – c – e – c ……): (b – d – f - …..)

(New Antcedent = Subtraction of all antecedent

values

(New Consequent = Subtraction of all

Consequent values)

Note: The new ratio is equal to the other ratios

𝑎

𝑏 =

𝑐

𝑑 =

𝑒

𝑓 = ……………. (say)k,

∴ a = bk, c= dk, e=fk ,………

Now a - c - e…… = k (b - d – f - ...) or 𝑎− 𝑐− 𝑒……

𝑏− 𝑑− 𝑓…. = k

Hence, (a - c - e -…): (b - d - f - ……) is

equal to each ratio

Points to Ponder: Applications

1. To find out new income – if the income of a man is increased in the given ratio & if the

increase in his income is given


2. To find out the age – if the ages of two men are in the given ratio & if the age of one man is

given.

Simple Problems

Sl.

No

Question Solution

1 Check whether the 2.4, 3.2, 1.5, 2

are in proportion

Here 2.4×2 = 4.8 and 3.2 ×1.5 = 4.8

𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑠= 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛𝑠

Hence in proportion

2 Find the value of x if

10/3 : x :: 5/2 : 5/4

10/3: 𝑥 =5/2:5/4

Using cross product rule, 𝑥 ×5/2 = (10/3) × 5/4

Or, 𝑥 = (10/3) × 5/4 × (2/5) = 5/3

3 Find the fourth proportional to

2/3, 3/7,4

If the fourth proportional be x, then 2/3, 3/7, 4, x

are in proportion. Using cross product rule,

(2/3) × 𝑥 = (3 × 4)/7

➔ 𝑥 = (3 × 4 × 3)/(7 × 2) = 18/7.

4 Find the third proportion to

2.4kg, 9.6kg.

Let the third proportion to 2.4 kg, 9.6kg be 𝑥kg.

Then 2.4kg, 9.6 kg and 𝑥kg are in continued

proportion since 𝑏2 = ac

So, 2.4/9.6 = 9.6/ 𝑥 or, 𝑥 = (9.6 × 9.6)/2.4 = 38.4

Hence the third proportional is 38.4 kg.

5 Find the mean proportion

between 1.25 and 1.8

Mean proportion between 1.25 and 1.8 is

√(1.25 × 1.8) = √2.25 = 1.5.

Conceptual Problems

1. If a:b = c:d = 2.5: 1.5, what are the values of ad : bc and a + c : b + d?

Solution:

We have 𝑎

𝑏 =

𝑐

𝑑 =

2.5

1.5 ……………(1)

From (1) ad = bc, or, 𝑎𝑑

𝑏𝑐 =1, i.e. ad:bc = 1:1

Again from (1) 𝑎

𝑏 =

𝑐

𝑑 =

𝑎 + 𝑐

𝑏 + 𝑑

∴ 𝑎 + 𝑐

𝑏 + 𝑑 =

2.5

1.5 =

25

15 =

5

3 , 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.

2. If 𝑎

3 =

𝑏

4 =

𝑐

7, then Prove that

𝑎 + 𝑏 + 𝑐

𝑐 = 2

Solution:


We have 𝑎

3 =

𝑏

4 =

𝑐

7 =

𝑎 + 𝑏 + 𝑐

3 + 4 + 7 =

𝑎 + 𝑏 + 𝑐

14

∴ 𝑎 + 𝑏 + 𝑐

14 =

𝑐

7 or

𝑎 + 𝑏 + 𝑐

𝑐 =

14

7 = 2

3. The sum of the ages of 3 persons is 150 years. 10 years ago their ages were in the ratio

7:8:9. What is the ration of their present ages?

Solution: 150-10-10-10 = 120

120 * 7/24 = 35 | 120 * 8/24 = 40 | 120 * 9/24 = 45

Their ages are 45, 50, 55

4. A dealer mixes Tea A costing ₹6.92 per kg with Tea B costing ₹7.77 per kg. and sells the

Tea mixture at ₹8.80 per kg. and earns a profit of 17.5 % on his sale price. In what

proportion does he mix them?

Solution:

Step 1: To find the Cost Price of the Mixture

Since, the price of Tea A and Tea B is given as Cost Price

Sale Price ₹ 100 (Assumption) ₹ 8.8 (Given)

- Profit ₹ 17.5 (17.5% on Sale Price) ₹ 1.54

Cost Price ₹ 82.5 ₹ 7.26

C.P of the mixture per kg = ₹7.26

Step 2:

We have to mix the two kinds in such a ratio that the amount of profit from Tea A must

balance the amount of loss from Tea B.

Difference between the Cost Price of Tea Mixture and Tea A : (₹7.26 – ₹ 6.92) = ₹ 0.34

Difference between the Cost Price of Tea Mixture and Tea B : (₹7.26 – ₹ 7.77) = (- ₹ 0.51)

Therefore, 1st Difference: 2nd Difference = 0.34: (-0.51) = 2: (-3) = 3:2

Note:

1. Check:

3 parts of Tea A costs = 3 * ₹ 6.92 = 20.74

2 Parts of Tea B costs = 2 * ₹ 7.77 = 15.54

Therefore, Total Cost = ₹ 20.74 + ₹ 15.54 = ₹ 36.28

5 Parts (3 parts of Tea A and 2 Parts of Tea B) of Mixture Costs = 5 * 7.26 = 36.3

2. Note: Since, 2: (-3) cannot exists as (-3) of Tea B cannot be mixed, the ratio is inverted

to 3:2

5. If𝑥

𝑏+𝑐−𝑎 =

𝑦

𝑐+𝑎−𝑏=

𝑧

𝑎+𝑏−𝑐, then,(b-c)x+(c-a)y + (a-b)z is,


(a)1 (b) 0

3. (c) 5 (d) None of these

Solution:

x + y + z

b + c – a + c + a – b + a + b − c=

𝑥+𝑦+𝑧

𝑎+𝑏+𝑐

➔ 𝑥

𝑎=

𝑦

𝑏=

𝑧

𝑐

➔ 𝑦−𝑧

𝑏−𝑐=

𝑧− 𝑥

𝑐−𝑎=

𝑥−𝑦

𝑎−𝑏

➔ 𝑏 − 𝑐 = 𝑘(𝑦 − 𝑧), 𝑐 − 𝑎 = 𝑘(𝑧 − 𝑥), 𝑎 − 𝑏 = 𝑘(𝑥 − 𝑦)

➔ (b-c)x+(c-a)y + (a-b)z

= k[ (y-z)x+(z-x)y+ (x-y)z ]

= xy – xz+yz – xy+xz – yz=0


6. Indices

Indices – A higher order operation with powers and roots

𝑎𝑛 = 𝑎 × 𝑎 × … . .× 𝑎(𝑛 𝑡𝑖𝑚𝑒𝑠) Example: 3× 3 × 3 × 3 = 34

a – Base and n – Index / Power Base, a = 3 and Index / Power, n = 4

Note:

Multiplication is the strong form of Addition and Indices is the strong form of

Multiplication.

Sl. No Laws Example

Base is Same

1 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛

32 × 33 = (3×3)× (3 × 3 × 3)

= 3× 3 × 3 × 3 × 3 = 35 = 32+3

Check 32 × 33 = 9×27 = 243 = 35

2 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛 33

32 = 3×3×3

3×3 = 31 = 33−2

Check:

33

32 = 27

9 = 31

3 (𝑎𝑚)𝑛 = 𝑎𝑚𝑛 (33)2 = (3 × 3 × 3)2= (3 × 3 × 3) × (3 × 3 × 3) = 36 = 33×2

Check (33)2 = (27)2 = 729 = 36

Different Base, but power is ‘n’

4 (𝑎𝑏)𝑛 = 𝑎𝑛 × 𝑏𝑛 (10)2 = (2 × 5)2= (2 × 5) × (2 × 5)

= (2 × 2) × (5 × 5) = 22 × 52

Check: (10)2 = 100

= 4 × 25 = 22 × 52

Points to Ponder

Example

1 𝒂𝟎 = 1

1 = 𝑎𝑚

𝑎𝑚 = 35

35 = 35−5 = 30 (Applying 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛)

2 (i) 𝑎−𝑚 = 1

𝑎𝑚

𝑎0−𝑚 = 30−𝑚 = 30

3𝑚 = 1

3𝑚 ( from 1)

(ii) 1

𝑎−𝑚 = 𝑎𝑚 1

3−𝑚 = 30

3−𝑚 = 30−(−𝑚) (Applying 𝑎𝑚

𝑎𝑚 = 𝑎𝑚−𝑛) = 30+𝑚 =

3𝑚

3 If Bases are same, power is same

i.e if 𝑎𝑥 = 𝑎𝑦, then x = y


4 If powers are same, base is

same

i.e if 𝑥𝑎 = 𝑦𝑎 , then x =y

5 √𝑎𝑚

= 𝑎1 𝑚⁄

√83

= √233= (23)1 3⁄ = 2

Simple Problems

No Question Answer

1 Simplify 2𝑥1/2 3𝑥−1 If x=4 6𝑥−1/2 =

6

√𝑥=

6

√4= 3

2 Simplify 6𝑎𝑏2𝑐3 × 4𝑏−2𝑐−3𝑑 24𝑎𝑏2−2𝑐3−3𝑑 = 24ad

3 Find the value of4𝑥−1

𝑥−1/3 4𝑥−1+1

34𝑥−2/34

𝑥2/3

4 Simplify2𝑎1/2 × 𝑎2/3 × 6𝑎−7/3

9𝑎−5/3 × 𝑎3/2 , ifa= 4 12𝑎−7/6

9𝑎−1/6 = 4𝑎−7/6+1/6

3=

4𝑎−1

3=

4

3 ×

1

𝑎=

4

3 ×

1

4=

1

3

5 Simplify(𝑥𝑎 . 𝑦−𝑏)3 . (𝑥3. 𝑦2)−𝑎 𝑥3𝑎 𝑦−3𝑏𝑥−3𝑎𝑦−2𝑎 = 𝑦−3𝑏−2𝑎 =

1

𝑦3𝑏+2𝑎

6 Simplify √𝑎4𝑏𝑥66 . (𝑎2/3𝑥−1)−𝑏 (𝑎4𝑏𝑥6)

1

6 . (𝑎2

3𝑥−1)−𝑏 = 𝑎4𝑏/6 x 𝑎−2

3𝑏 𝑥𝑏 = 𝑥𝑏+1

7 Find x, if 𝑥√𝑥 = (𝑥√𝑥)𝑥 𝑥3/2 = 𝑥𝑥 . 𝑥𝑥/2

𝑥3/2 = 𝑥3/2𝑥 (If base is equal, power is also equal) 3

2=

3𝑥

2 (or) x=

3

2 ×

2

3= 1

8 Find the value of k from

((√9)−7 × (√3)−5 = 3𝑘

3−7 × 3 1

2 ×−5 = 3𝑘

3−7 × 3−5/2 = 3𝑘

3−7−5

2 = 3𝑘 → 3−19

2 = 3𝑘→𝑘 =

−19

2

Conceptual Problems

Question 1: If 𝑥1/𝑝 = 𝑦1/𝑞 = 𝑧1/𝑟 and 𝑥𝑦𝑧 = 1, then the value of p + q + r is

Solution:

Let 𝑥1

𝑝 = 𝑦1

𝑞 = 𝑧1

𝑟 = k

Then 𝑥1

𝑝 = k ➔ x = 𝑘𝑝

𝑎𝑛𝑑 𝑦1

𝑞 =k ➔ y = 𝑘𝑞

and 𝑧1

𝑟 = k ➔ 𝑧 = 𝑘𝑟

xyz =1

𝑘𝑝 × 𝑘𝑞 × 𝑘𝑟 = 1

𝑘𝑝+𝑞+𝑟 = 𝑘𝑜

𝑝 + 𝑞 + 𝑟 = 0

Ans: (b)

a. 1 b. 0 c. ½ d. None of these


Question 2: [1 − {1 − (1 − 𝑥2)−1}−1]−1 2⁄ is equal to

Solution: Consider[1 − {1 − (1 − 𝑥2)−1}−1]−1 2⁄ = [1 − {1 −1

1−𝑥2}−1

]−1 2⁄

Consider innermost brackets first, {1 −1

1−𝑥2}−1

= {1−𝑥2−1

1−𝑥2 }−1

∴ [1 − {−𝑥2

1−𝑥2}−1

]

−1 2⁄

= [1 − {−1

𝑥2 + 1}]−1 2⁄

= [1 +1

𝑥2 − 1]−1 2⁄

= [1

𝑥2]−1 2⁄

= [𝑥2]1 2⁄ = [𝑥]

Ans: (a)

Question 3: If 𝑥 = 31

3 + 3−1

3, then 3𝑥3 − 9𝑥 is

Solution: Consider 3𝑥3 -9𝑥 = 3 [31 3⁄ + 3−1 3⁄ ]3 - 9 [31 3⁄ + 3−1 3⁄ ]

Apply, (𝑎 + 𝑏)3 = 𝑎3 + 3𝑎2𝑏 +3a𝑏2 +𝑏3

= 3[(31 3⁄ )3

+ 3(31 3⁄ )2

3−1 3⁄ + 3 × 31 3⁄ × (3−1 3⁄ )2

+ (3−1 3⁄ )3

] – 9 [31 3⁄ + 3−1 3⁄ ]

=3[3 + 31+2

3−

1

3 + 31+1

3−

2

3 + 3−1] - 9[31 3⁄ + 3−1 3⁄ ]

=3[3 + 34 3⁄ + 32 3⁄ + 3−1] - [37 3⁄ + 35 3⁄ ]

=9 + 37 3⁄ + 35 3⁄ + 1 − 37 3⁄ − 35 3⁄

=10

Ans: (b)

Note: a𝑥3-3ax = 3a+1, if x = 𝑎1 3⁄ + 𝑎−1 3⁄

Example:

2𝑥3 – 6x = 6 +1 = 7

3𝑥3-9x = 9+1 = 10

4𝑥3- 12x = 12 +1 =13.

Question 4: If 𝑎𝑥 =b, 𝑏𝑦 = c, 𝑐𝑧 = a, then 𝑥𝑦𝑧 is

Solution: Consider 𝑎𝑥 = b & 𝑏𝑦 = c

(𝑎𝑥)𝑦 = C

⇨ 𝑎𝑥𝑦= C

& 𝑐𝑧 = a⇨ (𝑎𝑥𝑦)𝑧 = a

⇨ 𝑎𝑥𝑦𝑧= a

⇨ xyz = 1

Ans: a

a. X b. 1

𝑥

c. 1 d. None of these

a. 15 b. 10 c. 12 d. None of these



Question 5: If 2𝑥 = 3𝑦 = 6−𝑧, 1

𝑥+

1

𝑦+

1

𝑧 is

Solution:

Let 2𝑥 =3𝑦 = 6−𝑧 = k

⇨ 2𝑥 = k

∴2 = 𝑘1 𝑥⁄

3𝑦 = k

3 =𝑘1 𝑦⁄

6−𝑧 = k

6 = 𝑘−1 𝑧⁄

& (2× 3) = 𝑘−1 𝑧⁄

⇨ (𝑘1 𝑥⁄ × 𝑘1 𝑦⁄ ) = 𝑘−1 𝑧⁄

⇨𝑘1 𝑥⁄ +1/𝑦 = 𝑘−1 𝑧⁄

⇨1

𝑥 +

1

𝑦 = -

1

𝑧

⇨1

𝑥 +

1

𝑦 +

1

𝑧 = 0

Ans: b



7. Logarithms

Introduction

log𝑎 𝑛 = x is the transformation of 𝑎𝑥 = n , n > 0, a > 0 & a ≠ 1.

It brings the relationship between Indices and Logarithm

Points to Ponder:

1. 𝑎0 = 1 & log𝑎 1 = 0, log of 1 to any base is zero

2. 𝑎1 = a & log𝑎 𝑎 = 1, log of any quantity to same base in always 1

Laws

Product Rule 𝐥𝐨𝐠𝒂 𝒎𝒏 = 𝐥𝐨𝐠𝒂 𝒎 + 𝐥𝐨𝐠𝒂 𝒏 Base is not changed

Division Rule log𝑎𝑚

𝑛 = log𝑎 𝑚 - log𝑎 𝑛 Base is not changed

Power Rule log𝑎 𝑚𝑛 = nlog𝑎 𝑚 Base is not changed

Change of Base log𝑎 𝑚 = log𝑏 𝑚 × log𝑎 𝑏

log𝑏 𝑚 = log𝑎 𝑚

log𝑎 𝑏

change of base w.r.t. product

log𝑎 𝑚 = log𝑏 𝑚

log𝑏 𝑛 change of base w.r.t. division

Points to Ponder:

1. If number & base are interchanged then product of log values is 1, log𝑎 𝑎 = 1 =

log𝑏 𝑎 × log𝑎 𝑏

2. The log values of interchanged number & base is inversely proportional, log𝑏 𝑎 = log𝑎 𝑎

log𝑎 𝑏 =

1

log𝑎 𝑏

log2 100 =log10 100

log10 2 ➔ log2 100 =

2

0.3010=

6.445

log10 100 = log2 100 ×

log10 2

➔ 2 = 6.6445 × 0.3010

Simple Problems – Set 1

No Question Answer

1 Find the logarithm of 1728 to the base 2√3 log2√3 1728 = 𝑥

(2√3)𝑥

= 1728

2𝑥 . 3𝑥/2 = 1728 ➔ x= 6

2 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

log 𝑥 + log 𝑦 = log(𝑥 + 𝑦)

⇨log 𝑥𝑦 = log(𝑥 + 𝑦)

∴ xy = x +y

⇨ xy –y = x

y(x -1) = x


y = 𝑥

𝑥−1

3 log (1 + 2 + 3) is exactly equal to

(a) log 1 + log 2 + log 3 (b) log 1×2×3

(c) Both (d) None of these

log(1 + 2 + 3) = log(1 × 2 × 3)

Since, log 6 = log 6

log(1 × 2 × 3) = log 1 + log 2+ log 3

4 Given that log10 2 = x and log10 3 = y, the

value of log10 60 is expressed as

(a)x – y + 1 (b) x + y + 1 (c) x – y – 1 (d) none

of these

log10 2 = x & log10 3 = y

log10 60 = log10(10 × 6)

= log10 10 + log10 6

= 1 + log10(2 × 3)

= 1+log10 2 +log10 3

= 1+ x+ y

= x+y+1

5 The value of is 4log8

25 - 3log

16

125 - log 5 ia

(a ) 0 (b) 1 (c) 2 (d) -1

4log8

25 - 3log

16

125 - log 5

4log23

52 - 3log24

53 - log 5

4[3 log 2 − 2 log 5] - 3(4 log 2 −3 log 5) log 5

= 12log 2 - 8log 5 - 12log 2 + 9

log 5 - log 5

= -9log 5 + 9 log 5= 0

6 Change the base of log5 31 into the

common logarithmic base. Since log𝑎 𝑥 =

log𝑏 𝑥

log𝑏 𝑎

log5 31 = log10 31

log10 5

Practice Problems:

1. log 𝑎9 +log 𝑎 =10 if the value of a is given by

(a ) 0 (b) 10 (c) -1 (d) none

Solution:

log 𝑎9 +log 𝑎1 =10

→ 9log 𝑎 + log 𝑎 = 10

→ 10log 𝑎 = 10

→ log 𝑎 = 1

∴ a = 10 (since, log 10 = 1)

2. Given that log10 2 = x, log10 3 = y, then log10 1.2 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

Solution:


log10 1.2 = log1012

10

= log10 12 - log10 10

= log10 4 × 3 - 1

= log10 4 + log10 3 - 1

= log10

22 + log10 3 -1

= 2log10 2 + log10 3 -1

= 2 x + y -1

3. The value of (log𝑏 𝑎 × log𝑐 𝑏 × log𝑎 𝑐)3 is equal to

(a) 3 (b) 0 (c) 1 (d) none of these

Solution:

(log𝑏 𝑎 × log𝑐 𝑏 × log𝑎 𝑐)3

= (log𝑐 𝑎

log𝑐 𝑏× log𝑐 𝑏 × log𝑎 𝑐)

3

= log𝑐 𝑎 × log𝑎 𝑐 = log𝑎 𝑎 = 1

Or (Take all to base 10)

(𝑙𝑜𝑔 𝑎

𝑙𝑜𝑔 𝑏×

𝑙𝑜𝑔 𝑏

𝑙𝑜𝑔 𝑐 ×

𝑙𝑜𝑔 𝑐

𝑙𝑜𝑔 𝑎 )

3

= 1

Conceptual Problems

No Question Answer

1 If log2

𝑥 + log4

𝑥 + log16

𝑥

= 21

4,

then x is equal to

(a) 8 (b) 4

(c) 16 (d) none of these

log2

𝑥 + log4

𝑥 + log16

𝑥 = 21

4

log2

𝑥 + log2 𝑥

log2 4 +

log2 𝑥

log2 16 =

21

4

⇨log2

𝑥 + log2 𝑥

log2 22 + log2 𝑥

log2 24 = 21

4

⇨log2

𝑥 + log2 𝑥

2log2 2 +

log2 𝑥

4log2 2 =

21

4

⇨log2

𝑥 [1 +1

2+

1

4] =

21

4

⇨ log2 𝑥 [4+2+1

4] =

21

4

⇨ log2 𝑥 × 7

4 =

21

4

⇨ log2 𝑥 = 21

4 ×

4

7 =

3

⇨23 = x ⇨ x = 8

2 1

1+log𝑎 𝑏𝑐 +

1

1+log𝑏 𝑐𝑎 +

1

1 + log𝑐 𝑎𝑏 is equal to

(a)0 (b) 1 (c) 3 (d)-1

1

log𝑎 𝑎+log𝑎 𝑏𝑐 +

1

log𝑏 𝑏+log𝑏 𝑐𝑎 +

1

log𝑐 𝑐+log𝑐 𝑎𝑏 (since, log

𝑎𝑎 = 1)

= 1

log𝑎 𝑎𝑏𝑐 +

1

log𝑏 𝑎𝑏𝑐 +

1

log𝑐 𝑎𝑏𝑐

(product rule)

= log𝑎𝑏𝑐

𝑎 + log𝑎𝑏𝑐

𝑏

+ log𝑎𝑏𝑐

𝑐

= log𝑎𝑏𝑐

𝑎𝑏𝑐 = 1

3 Show that

𝑙𝑜𝑔3 √3√3√3 … … . . ∞ =

1

Let 𝑥 = √3√3√3 … … . . ∞

Then, 𝑥2 = 3√3√3 … … . ∞

𝑥2 = 3𝑥

∴ 𝑥 = 0𝑜𝑟 𝑥 − 3 =0

𝑥 ≠ 0, ℎ𝑒𝑛𝑐𝑒 𝑥 = 3

∴ 𝑙𝑜𝑔33 = 1


𝑥2 − 3𝑥 = 0

𝑥(𝑥 − 3) = 0

4 𝑎log 𝑏−log 𝑐 ×

𝑏log 𝑐−log 𝑎 × 𝑐log 𝑎−log 𝑏

has a value of

(a) 1 (b) 0 (c) -1 (d)

none

(Note: When log

appears in the power,

take log throughout)

Let x = 𝑎log 𝑏−log 𝑐 ×

𝑏log 𝑐−log 𝑎 × 𝑐log 𝑎−log 𝑏

= 𝑎log𝑏 𝑐⁄ × 𝑏log𝑐 𝑎⁄ × 𝑐log𝑎 𝑏⁄

∴ log 𝑥 =

log[𝑎log𝑏 𝑐⁄ . 𝑏log𝑐 𝑎⁄ . 𝑐log𝑎 𝑏⁄ ]

(Applying, log mn=log 𝑚 +

log 𝑛)

= log 𝑏 𝑐⁄ (log 𝑎)

+ log 𝑐 𝑎⁄ (log 𝑏) +

log 𝑎 𝑏⁄ (log 𝑐) (Applying,

log 𝑚𝑛 = nlog 𝑚)

= (log 𝑏 − log 𝑐)(log 𝑎) +

(𝑙𝑜𝑔 𝑐 − 𝑙𝑜𝑔 𝑎)(log 𝑏)

+ (log 𝑎 − log 𝑏)(log 𝑐)

(Applying, log𝑚

𝑛 = log 𝑚-

log 𝑛)

= log 𝑏 log 𝑎 – log 𝑐 log 𝑎 +

log 𝑐 log 𝑏 – log 𝑎 log 𝑏 +

log 𝑎 – log 𝑐 – log 𝑏 log 𝑐 = 0

∴ log 𝑥 = 0 ➔ 𝑥 = 1

𝑎log 𝑏−log 𝑐 ×

𝑏log 𝑐−log 𝑎 ×

𝑐log 𝑎−log 𝑏 has a

value of

(b) 1 (b) 0 (c) -1

(d) none

(Note: When log

appears in the

power, take log

throughout)

5 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

Consider the innermost

brackets 1 − 1

1−𝑥2

= [1 −1

1− 𝑥2]

−1

= [1 − 𝑥2 − 1

1 − 𝑥2]

−1

=[−𝑥2

1−𝑥2]

−1

= 1−𝑥2

−𝑥2

= - 1−𝑥2

𝑥2

∴ log[1 − {1 − (1 −

𝑥2)−1}−1]−1 2⁄

= log [1 −

(− 1−𝑥2

𝑥2)]

−1/2

=log [1 +1−𝑥2

𝑥2]

−1/2

= log [𝑥2+1−𝑥2

𝑥2]

−1/2

= log [1

𝑥2]

−1/2

= log 𝑥(−2)(−

1

2)

= log 𝑥1 = log 𝑥

6 If log 𝑎

𝑦−𝑧 =

log 𝑏

𝑧−𝑥 =

log 𝑐

𝑥−𝑦 , the

value of abc is

(a)0 (b) 1 (c) -1 (d) none

log 𝑎

𝑦−𝑧 =

log 𝑏

𝑧−𝑥 =

log 𝑐

𝑥−𝑦 = k

Consider, log 𝑎

𝑦−𝑧 = k

∴ log 𝑎 = ky – zk

Consider, log 𝑏

𝑧−𝑥 = k

∴ log 𝑏 = kz – kx


7 If log 𝑎 = 1

2 log 𝑏 =

1

5 log 𝑐

the value of 𝑎4𝑏3𝑐−2 is

(a)0 (b) 1 (c) -1 (d) none

Consider, log 𝑎 = 1

2 log 𝑏

∴ a =√𝑏

b = 𝑎2

Consider, log 𝑎 = 1

5log 𝑐

8 If 𝑎 = 𝑏2 = 𝑐3 = 𝑑4 the

value of log𝑎(𝑎𝑏𝑐𝑑) is

(a) 𝟏 +𝟏

𝟐+

𝟏

𝟑+

𝟏

𝟒 (b) 1 +

1

2! +

1

3! +

1

4!

(c) 1 + 2 + 3 + 4 (d) None

Here b = √𝑎 , 𝑐 = √𝑎3

, d=

√𝑎4

log𝑎(𝑎𝑏𝑐𝑑) =

𝑙𝑜𝑔𝑎

(𝑎1 × 𝑎𝟏

𝟐 𝑎 × 𝑎𝟏

𝟑 × 𝑎𝟏

𝟒)

= 𝑙𝑜𝑔𝑎 𝑎(1+

1

2+

1

3+

1

4)

= (1 +1

2+

1

3+

1

4) 𝑙𝑜𝑔

𝑎𝑎 =

1 +1

2+

1

3+

1

4

If 𝑎 = 𝑏2 = 𝑐3 =

𝑑4 the value of

log𝑎(𝑎𝑏𝑐𝑑) is

(a) 𝟏 +𝟏

𝟐+

𝟏

𝟑+

𝟏

𝟒

(b) 1 + 1

2! +

1

3! +

1

4!

(c) 1 + 2 + 3 + 4 (d)

None

Logarithm and Antilogarithm values

To find the value of log 4594 and log 0.4594

Name of the

parts

Name in the

log value

Method log 4594 =

3.6623

Log 0.4594 = -

1.6623

The

whole(integral)

part

Characteristic By

Inspection

3

One less

than the

number of

digits to

the lef

t of the

decimal

point

-1

One more than

the number of

zeros on the

right

immediately

after

the decimal

point

The Decimal

(Fractional)Part

Mantissa From Log

Tables

6623 6623

Points to Ponder:

1 If 𝟏𝟎𝒙 = 1, then x = 0 - i.e., log 1 = 0

If 𝟏𝟎𝒙 = 10, then x = 1 - i.e., log 10 = 1

➔ The log of a number lying between

(1, 10) lies in the range (0,1).

Example: log 2.5 =0.3979

It is the 0.3979 parts lying between (0,1)

If 𝟏𝟎𝒙 = 0.1 = 𝟏

𝟏𝟎, then x =-1, i.e., log 0.1 = -1

If 𝟏𝟎𝒙 = 0.01 = 𝟏

𝟏𝟎𝟎, then x = -2 - i.e., log 0.01

= -2

➔ The log of a number lying between

(0.01, 0.1) lies in the range (-2,-1).

For example, consider log 0.07 = -2.1549

It is the 0.1549 parts lying between (-1,0)

2 Negative Mantissa - convert into a positive mantissa before referring to a logarithm table.


Examples:

(i) Representation: – 3.6872 = – 4 + (4 – 3.6872) = -4 + 0.3128 = 4̅.3128

It may be noted that 4̅.3128 is different from – 4.3128 as – 4.3128 is a negative

number whereas, in 4̅.3128, 4 is negative while .3128 is positive.

(ii) Find the number whose logarithm is

“–2.4678”.

Solution: -2.4678 = - 3 + 3 - 2.4678 = - 3 +

.5322 = 3̅.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

(iii) Add -4.74628 and 3.42367

– 4 + .74628 + 3 + .42367 = –1 + 1.16995 = 0.16995

To find the value of Antilog 206 and Antilog 0.206

Number Method Antilog 206 Antilog 0.206

The number

together is

taken

From log

Tables

(placed after

the decimal

point)

2.1607

(“2”– One less than the

number of digits to the

left of the decimal point

)

-1. 1607

(“-1” - One more than the

number of zeros on the right

immediately after the decimal

point)

Simple Problems - Set 2

No Question Answer

1 log 6.25 to the base 2 is equal to

(a) 4 (b) 5

(c) 1 (d) none of these

log2 6.25 = 𝑥

log26.25

100 = x

log2 625 - log2 100

= log2 252 - log2 102

= 2log2 52 – 2log2 10

= 4log2 5 - 2[log2 5 + log2 2]

= 4log2 5 - 2log2 5 - 2log2 2

=2log2 5 – 2

2 Solve

1/2 log10 25 − 2 log10 3 +log10 18

log10 251/2 − log10 32 + log10 18

= log10 5 − log10 9 + log10 18

= log105 × 18

9=log10 10 = 1

3 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

log 6 = log 2 × 3

= log 2 + log 3

= 0.3010 + 0.4771

= 0.7781

SSA Business Mathematics A1.1

Annexure A1 - Number System

Set of real numbers in Venn Diagram

Number Definition Example

“N” – Natural Numbers Numbers from 1 onwards {1, 2, 3, 4, ……}


“Q” - Rational Numbers

(All Proper and

Improper fractions)

1. p/q – form

2. q = 0

3. p, q – Integers

All integers, Fractions and

Repeating Decimals (10

3 = 3.333)

“Q*” - Set of Positive

Rational Numbers

Only positive values 10

3, repeating decimals

Irrational Numbers Not real numbers Non-repeating decimals

“R” - Real Numbers Rational & Irrational

“C’ - Complex Numbers a + ib 3+ 2i

“Z” – All Integers Not decimals {.., -2, -1, 0, 1, 2, …}

𝒁+ - Positive Integers Only positive {1, 2, ……}

𝒁− - Negative Integers Only Negative {….., -2, -1}

Note: “0” is neither positive nor negatives

“P” - Prime Numbers The number divisible by

one and own only

2, 3, 5, 7, 11, ……

Even Numbers The number divisible by

two alone

2, 4, 6, 8, 10, ……

Odd Numbers The number not divisible

by two alone

1, 3, 5, 7, 9, ………..


Annexure A2 – Log and Antilog Tables


Annexure A3 – Time Value of Money Tables - Future Value of Interest Factor

1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16%

1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000 1.1100 1.1200 1.1300 1.1400 1.1500 1.1600

2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100 1.2321 1.2544 1.2769 1.2996 1.3225 1.3456

3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310 1.3676 1.4049 1.4429 1.4815 1.5209 1.5609

4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641 1.5181 1.5735 1.6305 1.6890 1.7490 1.8106

5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105 1.6851 1.7623 1.8424 1.9254 2.0114 2.1003

6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716 1.8704 1.9738 2.0820 2.1950 2.3131 2.4364

7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487 2.0762 2.2107 2.3526 2.5023 2.6600 2.8262

8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436 2.3045 2.4760 2.6584 2.8526 3.0590 3.2784

9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579 2.5580 2.7731 3.0040 3.2519 3.5179 3.8030

10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937 2.8394 3.1058 3.3946 3.7072 4.0456 4.4114

11 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531 3.1518 3.4785 3.8359 4.2262 4.6524 5.1173

12 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384 3.4985 3.8960 4.3345 4.8179 5.3503 5.9360

13 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523 3.8833 4.3635 4.8980 5.4924 6.1528 6.8858

14 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975 4.3104 4.8871 5.5348 6.2613 7.0757 7.9875

15 1.1610 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1772 4.7846 5.4736 6.2543 7.1379 8.1371 9.2655

16 1.1726 1.3728 1.6047 1.8730 2.1829 2.5404 2.9522 3.4259 3.9703 4.5950 5.3109 6.1304 7.0673 8.1372 9.3576 10.7480

17 1.1843 1.4002 1.6528 1.9479 2.2920 2.6928 3.1588 3.7000 4.3276 5.0545 5.8951 6.8660 7.9861 9.2765 10.7613 12.4677

18 1.1961 1.4282 1.7024 2.0258 2.4066 2.8543 3.3799 3.9960 4.7171 5.5599 6.5436 7.6900 9.0243 10.5752 12.3755 14.4625

19 1.2081 1.4568 1.7535 2.1068 2.5270 3.0256 3.6165 4.3157 5.1417 6.1159 7.2633 8.6128 10.1974 12.0557 14.2318 16.7765

20 1.2202 1.4859 1.8061 2.1911 2.6533 3.2071 3.8697 4.6610 5.6044 6.7275 8.0623 9.6463 11.5231 13.7435 16.3665 19.4608

21 1.2324 1.5157 1.8603 2.2788 2.7860 3.3996 4.1406 5.0338 6.1088 7.4002 8.9492 10.8038 13.0211 15.6676 18.8215 22.5745

22 1.2447 1.5460 1.9161 2.3699 2.9253 3.6035 4.4304 5.4365 6.6586 8.1403 9.9336 12.1003 14.7138 17.8610 21.6447 26.1864

23 1.2572 1.5769 1.9736 2.4647 3.0715 3.8197 4.7405 5.8715 7.2579 8.9543 11.0263 13.5523 16.6266 20.3616 24.8915 30.3762

24 1.2697 1.6084 2.0328 2.5633 3.2251 4.0489 5.0724 6.3412 7.9111 9.8497 12.2392 15.1786 18.7881 23.2122 28.6252 35.2364

25 1.2824 1.6406 2.0938 2.6658 3.3864 4.2919 5.4274 6.8485 8.6231 10.8347 13.5855 17.0001 21.2305 26.4619 32.9190 40.8742

26 1.2953 1.6734 2.1566 2.7725 3.5557 4.5494 5.8074 7.3964 9.3992 11.9182 15.0799 19.0401 23.9905 30.1666 37.8568 47.4141

27 1.3082 1.7069 2.2213 2.8834 3.7335 4.8223 6.2139 7.9881 10.2451 13.1100 16.7386 21.3249 27.1093 34.3899 43.5353 55.0004

28 1.3213 1.7410 2.2879 2.9987 3.9201 5.1117 6.6488 8.6271 11.1671 14.4210 18.5799 23.8839 30.6335 39.2045 50.0656 63.8004

29 1.3345 1.7758 2.3566 3.1187 4.1161 5.4184 7.1143 9.3173 12.1722 15.8631 20.6237 26.7499 34.6158 44.6931 57.5755 74.0085

30 1.3478 1.8114 2.4273 3.2434 4.3219 5.7435 7.6123 10.0627 13.2677 17.4494 22.8923 29.9599 39.1159 50.9502 66.2118 85.8499


Future Value of Interest Factor

17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%

1 1.1700 1.1800 1.1900 1.2000 1.2100 1.2200 1.2300 1.2400 1.2500 1.2600 1.2700 1.2800 1.2900 1.3000

2 1.3689 1.3924 1.4161 1.4400 1.4641 1.4884 1.5129 1.5376 1.5625 1.5876 1.6129 1.6384 1.6641 1.6900

3 1.6016 1.6430 1.6852 1.7280 1.7716 1.8158 1.8609 1.9066 1.9531 2.0004 2.0484 2.0972 2.1467 2.1970

4 1.8739 1.9388 2.0053 2.0736 2.1436 2.2153 2.2889 2.3642 2.4414 2.5205 2.6014 2.6844 2.7692 2.8561

5 2.1924 2.2878 2.3864 2.4883 2.5937 2.7027 2.8153 2.9316 3.0518 3.1758 3.3038 3.4360 3.5723 3.7129

6 2.5652 2.6996 2.8398 2.9860 3.1384 3.2973 3.4628 3.6352 3.8147 4.0015 4.1959 4.3980 4.6083 4.8268

7 3.0012 3.1855 3.3793 3.5832 3.7975 4.0227 4.2593 4.5077 4.7684 5.0419 5.3288 5.6295 5.9447 6.2749

8 3.5115 3.7589 4.0214 4.2998 4.5950 4.9077 5.2389 5.5895 5.9605 6.3528 6.7675 7.2058 7.6686 8.1573

9 4.1084 4.4355 4.7854 5.1598 5.5599 5.9874 6.4439 6.9310 7.4506 8.0045 8.5948 9.2234 9.8925 10.6045

10 4.8068 5.2338 5.6947 6.1917 6.7275 7.3046 7.9259 8.5944 9.3132 10.0857 10.9153 11.8059 12.7614 13.7858

11 5.6240 6.1759 6.7767 7.4301 8.1403 8.9117 9.7489 10.6571 11.6415 12.7080 13.8625 15.1116 16.4622 17.9216

12 6.5801 7.2876 8.0642 8.9161 9.8497 10.8722 11.9912 13.2148 14.5519 16.0120 17.6053 19.3428 21.2362 23.2981

13 7.6987 8.5994 9.5964 10.6993 11.9182 13.2641 14.7491 16.3863 18.1899 20.1752 22.3588 24.7588 27.3947 30.2875

14 9.0075 10.1472 11.4198 12.8392 14.4210 16.1822 18.1414 20.3191 22.7374 25.4207 28.3957 31.6913 35.3391 39.3738

15 10.5387 11.9737 13.5895 15.4070 17.4494 19.7423 22.3140 25.1956 28.4217 32.0301 36.0625 40.5648 45.5875 51.1859

16 12.3303 14.1290 16.1715 18.4884 21.1138 24.0856 27.4462 31.2426 35.5271 40.3579 45.7994 51.9230 58.8079 66.5417

17 14.4265 16.6722 19.2441 22.1861 25.5477 29.3844 33.7588 38.7408 44.4089 50.8510 58.1652 66.4614 75.8621 86.5042

18 16.8790 19.6733 22.9005 26.6233 30.9127 35.8490 41.5233 48.0386 55.5112 64.0722 73.8698 85.0706 97.8622 112.4554

19 19.7484 23.2144 27.2516 31.9480 37.4043 43.7358 51.0737 59.5679 69.3889 80.7310 93.8147 108.8904 126.2422 146.1920

20 23.1056 27.3930 32.4294 38.3376 45.2593 53.3576 62.8206 73.8641 86.7362 101.7211 119.1446 139.3797 162.8524 190.0496

21 27.0336 32.3238 38.5910 46.0051 54.7637 65.0963 77.2694 91.5915 108.4202 128.1685 151.3137 178.4060 210.0796 247.0645

22 31.6293 38.1421 45.9233 55.2061 66.2641 79.4175 95.0413 113.5735 135.5253 161.4924 192.1683 228.3596 271.0027 321.1839

23 37.0062 45.0076 54.6487 66.2474 80.1795 96.8894 116.9008 140.8312 169.4066 203.4804 244.0538 292.3003 349.5935 417.5391

24 43.2973 53.1090 65.0320 79.4968 97.0172 118.2050 143.7880 174.6306 211.7582 256.3853 309.9483 374.1444 450.9756 542.8008

25 50.6578 62.6686 77.3881 95.3962 117.3909 144.2101 176.8593 216.5420 264.6978 323.0454 393.6344 478.9049 581.7585 705.6410

26 59.2697 73.9490 92.0918 114.4755 142.0429 175.9364 217.5369 268.5121 330.8722 407.0373 499.9157 612.9982 750.4685 917.3333

27 69.3455 87.2598 109.5893 137.3706 171.8719 214.6424 267.5704 332.9550 413.5903 512.8670 634.8929 784.6377 968.1044 1192.5333

28 81.1342 102.9666 130.4112 164.8447 207.9651 261.8637 329.1115 412.8642 516.9879 646.2124 806.3140 1004.3363 1248.8546 1550.2933

29 94.9271 121.5005 155.1893 197.8136 251.6377 319.4737 404.8072 511.9516 646.2349 814.2276 1024.0187 1285.5504 1611.0225 2015.3813

30 111.0647 143.3706 184.6753 237.3763 304.4816 389.7579 497.9129 634.8199 807.7936 1025.9267 1300.5038 1645.5046 2078.2190 2619.9956


Present Value of Interest Factor

1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 0.9009 0.8929 0.8850 0.8772 0.8696

2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264 0.8116 0.7972 0.7831 0.7695 0.7561

3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513 0.7312 0.7118 0.6931 0.6750 0.6575

4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830 0.6587 0.6355 0.6133 0.5921 0.5718

5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209 0.5935 0.5674 0.5428 0.5194 0.4972

6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645 0.5346 0.5066 0.4803 0.4556 0.4323

7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132 0.4817 0.4523 0.4251 0.3996 0.3759

8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665 0.4339 0.4039 0.3762 0.3506 0.3269

9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241 0.3909 0.3606 0.3329 0.3075 0.2843

10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855 0.3522 0.3220 0.2946 0.2697 0.2472

11 0.8963 0.8043 0.7224 0.6496 0.5847 0.5268 0.4751 0.4289 0.3875 0.3505 0.3173 0.2875 0.2607 0.2366 0.2149

12 0.8874 0.7885 0.7014 0.6246 0.5568 0.4970 0.4440 0.3971 0.3555 0.3186 0.2858 0.2567 0.2307 0.2076 0.1869

13 0.8787 0.7730 0.6810 0.6006 0.5303 0.4688 0.4150 0.3677 0.3262 0.2897 0.2575 0.2292 0.2042 0.1821 0.1625

14 0.8700 0.7579 0.6611 0.5775 0.5051 0.4423 0.3878 0.3405 0.2992 0.2633 0.2320 0.2046 0.1807 0.1597 0.1413

15 0.8613 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394 0.2090 0.1827 0.1599 0.1401 0.1229

16 0.8528 0.7284 0.6232 0.5339 0.4581 0.3936 0.3387 0.2919 0.2519 0.2176 0.1883 0.1631 0.1415 0.1229 0.1069

17 0.8444 0.7142 0.6050 0.5134 0.4363 0.3714 0.3166 0.2703 0.2311 0.1978 0.1696 0.1456 0.1252 0.1078 0.0929

18 0.8360 0.7002 0.5874 0.4936 0.4155 0.3503 0.2959 0.2502 0.2120 0.1799 0.1528 0.1300 0.1108 0.0946 0.0808

19 0.8277 0.6864 0.5703 0.4746 0.3957 0.3305 0.2765 0.2317 0.1945 0.1635 0.1377 0.1161 0.0981 0.0829 0.0703

20 0.8195 0.6730 0.5537 0.4564 0.3769 0.3118 0.2584 0.2145 0.1784 0.1486 0.1240 0.1037 0.0868 0.0728 0.0611

21 0.8114 0.6598 0.5375 0.4388 0.3589 0.2942 0.2415 0.1987 0.1637 0.1351 0.1117 0.0926 0.0768 0.0638 0.0531

22 0.8034 0.6468 0.5219 0.4220 0.3418 0.2775 0.2257 0.1839 0.1502 0.1228 0.1007 0.0826 0.0680 0.0560 0.0462

23 0.7954 0.6342 0.5067 0.4057 0.3256 0.2618 0.2109 0.1703 0.1378 0.1117 0.0907 0.0738 0.0601 0.0491 0.0402

24 0.7876 0.6217 0.4919 0.3901 0.3101 0.2470 0.1971 0.1577 0.1264 0.1015 0.0817 0.0659 0.0532 0.0431 0.0349

25 0.7798 0.6095 0.4776 0.3751 0.2953 0.2330 0.1842 0.1460 0.1160 0.0923 0.0736 0.0588 0.0471 0.0378 0.0304

26 0.7720 0.5976 0.4637 0.3607 0.2812 0.2198 0.1722 0.1352 0.1064 0.0839 0.0663 0.0525 0.0417 0.0331 0.0264

27 0.7644 0.5859 0.4502 0.3468 0.2678 0.2074 0.1609 0.1252 0.0976 0.0763 0.0597 0.0469 0.0369 0.0291 0.0230

28 0.7568 0.5744 0.4371 0.3335 0.2551 0.1956 0.1504 0.1159 0.0895 0.0693 0.0538 0.0419 0.0326 0.0255 0.0200

29 0.7493 0.5631 0.4243 0.3207 0.2429 0.1846 0.1406 0.1073 0.0822 0.0630 0.0485 0.0374 0.0289 0.0224 0.0174

30 0.7419 0.5521 0.4120 0.3083 0.2314 0.1741 0.1314 0.0994 0.0754 0.0573 0.0437 0.0334 0.0256 0.0196 0.0151


Present Value of Interest Factor

16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%

1 0.8621 0.8547 0.8475 0.8403 0.8333 0.8264 0.8197 0.8130 0.8065 0.8000 0.7937 0.7874 0.7813 0.7752 0.7692

2 0.7432 0.7305 0.7182 0.7062 0.6944 0.6830 0.6719 0.6610 0.6504 0.6400 0.6299 0.6200 0.6104 0.6009 0.5917

3 0.6407 0.6244 0.6086 0.5934 0.5787 0.5645 0.5507 0.5374 0.5245 0.5120 0.4999 0.4882 0.4768 0.4658 0.4552

4 0.5523 0.5337 0.5158 0.4987 0.4823 0.4665 0.4514 0.4369 0.4230 0.4096 0.3968 0.3844 0.3725 0.3611 0.3501

5 0.4761 0.4561 0.4371 0.4190 0.4019 0.3855 0.3700 0.3552 0.3411 0.3277 0.3149 0.3027 0.2910 0.2799 0.2693

6 0.4104 0.3898 0.3704 0.3521 0.3349 0.3186 0.3033 0.2888 0.2751 0.2621 0.2499 0.2383 0.2274 0.2170 0.2072

7 0.3538 0.3332 0.3139 0.2959 0.2791 0.2633 0.2486 0.2348 0.2218 0.2097 0.1983 0.1877 0.1776 0.1682 0.1594

8 0.3050 0.2848 0.2660 0.2487 0.2326 0.2176 0.2038 0.1909 0.1789 0.1678 0.1574 0.1478 0.1388 0.1304 0.1226

9 0.2630 0.2434 0.2255 0.2090 0.1938 0.1799 0.1670 0.1552 0.1443 0.1342 0.1249 0.1164 0.1084 0.1011 0.0943

10 0.2267 0.2080 0.1911 0.1756 0.1615 0.1486 0.1369 0.1262 0.1164 0.1074 0.0992 0.0916 0.0847 0.0784 0.0725

11 0.1954 0.1778 0.1619 0.1476 0.1346 0.1228 0.1122 0.1026 0.0938 0.0859 0.0787 0.0721 0.0662 0.0607 0.0558

12 0.1685 0.1520 0.1372 0.1240 0.1122 0.1015 0.0920 0.0834 0.0757 0.0687 0.0625 0.0568 0.0517 0.0471 0.0429

13 0.1452 0.1299 0.1163 0.1042 0.0935 0.0839 0.0754 0.0678 0.0610 0.0550 0.0496 0.0447 0.0404 0.0365 0.0330

14 0.1252 0.1110 0.0985 0.0876 0.0779 0.0693 0.0618 0.0551 0.0492 0.0440 0.0393 0.0352 0.0316 0.0283 0.0254

15 0.1079 0.0949 0.0835 0.0736 0.0649 0.0573 0.0507 0.0448 0.0397 0.0352 0.0312 0.0277 0.0247 0.0219 0.0195

16 0.0930 0.0811 0.0708 0.0618 0.0541 0.0474 0.0415 0.0364 0.0320 0.0281 0.0248 0.0218 0.0193 0.0170 0.0150

17 0.0802 0.0693 0.0600 0.0520 0.0451 0.0391 0.0340 0.0296 0.0258 0.0225 0.0197 0.0172 0.0150 0.0132 0.0116

18 0.0691 0.0592 0.0508 0.0437 0.0376 0.0323 0.0279 0.0241 0.0208 0.0180 0.0156 0.0135 0.0118 0.0102 0.0089

19 0.0596 0.0506 0.0431 0.0367 0.0313 0.0267 0.0229 0.0196 0.0168 0.0144 0.0124 0.0107 0.0092 0.0079 0.0068

20 0.0514 0.0433 0.0365 0.0308 0.0261 0.0221 0.0187 0.0159 0.0135 0.0115 0.0098 0.0084 0.0072 0.0061 0.0053

21 0.0443 0.0370 0.0309 0.0259 0.0217 0.0183 0.0154 0.0129 0.0109 0.0092 0.0078 0.0066 0.0056 0.0048 0.0040

22 0.0382 0.0316 0.0262 0.0218 0.0181 0.0151 0.0126 0.0105 0.0088 0.0074 0.0062 0.0052 0.0044 0.0037 0.0031

23 0.0329 0.0270 0.0222 0.0183 0.0151 0.0125 0.0103 0.0086 0.0071 0.0059 0.0049 0.0041 0.0034 0.0029 0.0024

24 0.0284 0.0231 0.0188 0.0154 0.0126 0.0103 0.0085 0.0070 0.0057 0.0047 0.0039 0.0032 0.0027 0.0022 0.0018

25 0.0245 0.0197 0.0160 0.0129 0.0105 0.0085 0.0069 0.0057 0.0046 0.0038 0.0031 0.0025 0.0021 0.0017 0.0014

26 0.0211 0.0169 0.0135 0.0109 0.0087 0.0070 0.0057 0.0046 0.0037 0.0030 0.0025 0.0020 0.0016 0.0013 0.0011

27 0.0182 0.0144 0.0115 0.0091 0.0073 0.0058 0.0047 0.0037 0.0030 0.0024 0.0019 0.0016 0.0013 0.0010 0.0008

28 0.0157 0.0123 0.0097 0.0077 0.0061 0.0048 0.0038 0.0030 0.0024 0.0019 0.0015 0.0012 0.0010 0.0008 0.0006

29 0.0135 0.0105 0.0082 0.0064 0.0051 0.0040 0.0031 0.0025 0.0020 0.0015 0.0012 0.0010 0.0008 0.0006 0.0005

30 0.0116 0.0090 0.0070 0.0054 0.0042 0.0033 0.0026 0.0020 0.0016 0.0012 0.0010 0.0008 0.0006 0.0005 0.0004


Recurring Deposit Factor (Future Value of Annuity)

1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 2.0100 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000 2.1100 2.1200 2.1300 2.1400 2.1500

3 3.0301 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100 3.3421 3.3744 3.4069 3.4396 3.4725

4 4.0604 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410 4.7097 4.7793 4.8498 4.9211 4.9934

5 5.1010 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051 6.2278 6.3528 6.4803 6.6101 6.7424

6 6.1520 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156 7.9129 8.1152 8.3227 8.5355 8.7537

7 7.2135 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872 9.7833 10.0890 10.4047 10.7305 11.0668

8 8.2857 8.5830 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359 11.8594 12.2997 12.7573 13.2328 13.7268

9 9.3685 9.7546 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795 14.1640 14.7757 15.4157 16.0853 16.7858

10 10.4622 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374 16.7220 17.5487 18.4197 19.3373 20.3037

11 11.5668 12.1687 12.8078 13.4864 14.2068 14.9716 15.7836 16.6455 17.5603 18.5312 19.5614 20.6546 21.8143 23.0445 24.3493

12 12.6825 13.4121 14.1920 15.0258 15.9171 16.8699 17.8885 18.9771 20.1407 21.3843 22.7132 24.1331 25.6502 27.2707 29.0017

13 13.8093 14.6803 15.6178 16.6268 17.7130 18.8821 20.1406 21.4953 22.9534 24.5227 26.2116 28.0291 29.9847 32.0887 34.3519

14 14.9474 15.9739 17.0863 18.2919 19.5986 21.0151 22.5505 24.2149 26.0192 27.9750 30.0949 32.3926 34.8827 37.5811 40.5047

15 16.0969 17.2934 18.5989 20.0236 21.5786 23.2760 25.1290 27.1521 29.3609 31.7725 34.4054 37.2797 40.4175 43.8424 47.5804

16 17.2579 18.6393 20.1569 21.8245 23.6575 25.6725 27.8881 30.3243 33.0034 35.9497 39.1899 42.7533 46.6717 50.9804 55.7175

17 18.4304 20.0121 21.7616 23.6975 25.8404 28.2129 30.8402 33.7502 36.9737 40.5447 44.5008 48.8837 53.7391 59.1176 65.0751

18 19.6147 21.4123 23.4144 25.6454 28.1324 30.9057 33.9990 37.4502 41.3013 45.5992 50.3959 55.7497 61.7251 68.3941 75.8364

19 20.8109 22.8406 25.1169 27.6712 30.5390 33.7600 37.3790 41.4463 46.0185 51.1591 56.9395 63.4397 70.7494 78.9692 88.2118

20 22.0190 24.2974 26.8704 29.7781 33.0660 36.7856 40.9955 45.7620 51.1601 57.2750 64.2028 72.0524 80.9468 91.0249 102.4436

21 23.2392 25.7833 28.6765 31.9692 35.7193 39.9927 44.8652 50.4229 56.7645 64.0025 72.2651 81.6987 92.4699 104.7684 118.8101

22 24.4716 27.2990 30.5368 34.2480 38.5052 43.3923 49.0057 55.4568 62.8733 71.4027 81.2143 92.5026 105.4910 120.4360 137.6316

23 25.7163 28.8450 32.4529 36.6179 41.4305 46.9958 53.4361 60.8933 69.5319 79.5430 91.1479 104.6029 120.2048 138.2970 159.2764

24 26.9735 30.4219 34.4265 39.0826 44.5020 50.8156 58.1767 66.7648 76.7898 88.4973 102.1742 118.1552 136.8315 158.6586 184.1678

25 28.2432 32.0303 36.4593 41.6459 47.7271 54.8645 63.2490 73.1059 84.7009 98.3471 114.4133 133.3339 155.6196 181.8708 212.7930

26 29.5256 33.6709 38.5530 44.3117 51.1135 59.1564 68.6765 79.9544 93.3240 109.1818 127.9988 150.3339 176.8501 208.3327 245.7120

27 30.8209 35.3443 40.7096 47.0842 54.6691 63.7058 74.4838 87.3508 102.7231 121.0999 143.0786 169.3740 200.8406 238.4993 283.5688

28 32.1291 37.0512 42.9309 49.9676 58.4026 68.5281 80.6977 95.3388 112.9682 134.2099 159.8173 190.6989 227.9499 272.8892 327.1041

29 33.4504 38.7922 45.2189 52.9663 62.3227 73.6398 87.3465 103.9659 124.1354 148.6309 178.3972 214.5828 258.5834 312.0937 377.1697

30 34.7849 40.5681 47.5754 56.0849 66.4388 79.0582 94.4608 113.2832 136.3075 164.4940 199.0209 241.3327 293.1992 356.7868 434.7451


Recurring Deposit Factor (Future Value of Annuity)

16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.00 1.00 1.00 1.00 1.00

2 2.1600 2.1700 2.1800 2.1900 2.2000 2.2100 2.2200 2.2300 2.2400 2.2500 2.26 2.27 2.28 2.29 2.30

3 3.5056 3.5389 3.5724 3.6061 3.6400 3.6741 3.7084 3.7429 3.7776 3.8125 3.85 3.88 3.92 3.95 3.99

4 5.0665 5.1405 5.2154 5.2913 5.3680 5.4457 5.5242 5.6038 5.6842 5.7656 5.85 5.93 6.02 6.10 6.19

5 6.8771 7.0144 7.1542 7.2966 7.4416 7.5892 7.7396 7.8926 8.0484 8.2070 8.37 8.53 8.70 8.87 9.04

6 8.9775 9.2068 9.4420 9.6830 9.9299 10.1830 10.4423 10.7079 10.9801 11.2588 11.54 11.84 12.14 12.44 12.76

7 11.4139 11.7720 12.1415 12.5227 12.9159 13.3214 13.7396 14.1708 14.6153 15.0735 15.55 16.03 16.53 17.05 17.58

8 14.2401 14.7733 15.3270 15.9020 16.4991 17.1189 17.7623 18.4300 19.1229 19.8419 20.59 21.36 22.16 23.00 23.86

9 17.5185 18.2847 19.0859 19.9234 20.7989 21.7139 22.6700 23.6690 24.7125 25.8023 26.94 28.13 29.37 30.66 32.01

10 21.3215 22.3931 23.5213 24.7089 25.9587 27.2738 28.6574 30.1128 31.6434 33.2529 34.94 36.72 38.59 40.56 42.62

11 25.7329 27.1999 28.7551 30.4035 32.1504 34.0013 35.9620 38.0388 40.2379 42.5661 45.03 47.64 50.40 53.32 56.41

12 30.8502 32.8239 34.9311 37.1802 39.5805 42.1416 44.8737 47.7877 50.8950 54.2077 57.74 61.50 65.51 69.78 74.33

13 36.7862 39.4040 42.2187 45.2445 48.4966 51.9913 55.7459 59.7788 64.1097 68.7596 73.75 79.11 84.85 91.02 97.63

14 43.6720 47.1027 50.8180 54.8409 59.1959 63.9095 69.0100 74.5280 80.4961 86.9495 93.93 101.47 109.61 118.41 127.91

15 51.6595 56.1101 60.9653 66.2607 72.0351 78.3305 85.1922 92.6694 100.8151 109.6868 119.35 129.86 141.30 153.75 167.29

16 60.9250 66.6488 72.9390 79.8502 87.4421 95.7799 104.9345 114.9834 126.0108 138.1085 151.38 165.92 181.87 199.34 218.47

17 71.6730 78.9792 87.0680 96.0218 105.9306 116.8937 129.0201 142.4295 157.2534 173.6357 191.73 211.72 233.79 258.15 285.01

18 84.1407 93.4056 103.7403 115.2659 128.1167 142.4413 158.4045 176.1883 195.9942 218.0446 242.59 269.89 300.25 334.01 371.52

19 98.6032 110.2846 123.4135 138.1664 154.7400 173.3540 194.2535 217.7116 244.0328 273.5558 306.66 343.76 385.32 431.87 483.97

20 115.3797 130.0329 146.6280 165.4180 186.6880 210.7584 237.9893 268.7853 303.6006 342.9447 387.39 437.57 494.21 558.11 630.17

21 134.8405 153.1385 174.0210 197.8474 225.0256 256.0176 291.3469 331.6059 377.4648 429.6809 489.11 556.72 633.59 720.96 820.22

22 157.4150 180.1721 206.3448 236.4385 271.0307 310.7813 356.4432 408.8753 469.0563 538.1011 617.28 708.03 812.00 931.04 1,067.28

23 183.6014 211.8013 244.4868 282.3618 326.2369 377.0454 435.8607 503.9166 582.6298 673.6264 778.77 900.20 1,040.36 1,202.05 1,388.46

24 213.9776 248.8076 289.4945 337.0105 392.4842 457.2249 532.7501 620.8174 723.4610 843.0329 982.25 1,144.25 1,332.66 1,551.64 1,806.00

25 249.2140 292.1049 342.6035 402.0425 471.9811 554.2422 650.9551 764.6054 898.0916 1,054.7912 1,238.64 1,454.20 1,706.80 2,002.62 2,348.80

26 290.0883 342.7627 405.2721 479.4306 567.3773 671.6330 795.1653 941.4647 1,114.6336 1,319.4890 1,561.68 1,847.84 2,185.71 2,584.37 3,054.44

27 337.5024 402.0323 479.2211 571.5224 681.8528 813.6759 971.1016 1,159.0016 1,383.1457 1,650.3612 1,968.72 2,347.75 2,798.71 3,334.84 3,971.78

28 392.5028 471.3778 566.4809 681.1116 819.2233 985.5479 1,185.7440 1,426.5719 1,716.1007 2,063.9515 2,481.59 2,982.64 3,583.34 4,302.95 5,164.31

29 456.3032 552.5121 669.4475 811.5228 984.0680 1,193.5129 1,447.6077 1,755.6835 2,128.9648 2,580.9394 3,127.80 3,788.96 4,587.68 5,551.80 6,714.60

30 530.3117 647.4391 790.9480 966.7122 1,181.8816 1,445.1507 1,767.0813 2,160.4907 2,640.9164 3,227.1743 3,942.03 4,812.98 5,873.23 7,162.82 8,729.99


Annuity Factor (Present Value of Annuity)

1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 0.9009 0.8929 0.8850 0.8772 0.8696

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355 1.7125 1.6901 1.6681 1.6467 1.6257

3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869 2.4437 2.4018 2.3612 2.3216 2.2832

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699 3.1024 3.0373 2.9745 2.9137 2.8550

5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908 3.6959 3.6048 3.5172 3.4331 3.3522

6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553 4.2305 4.1114 3.9975 3.8887 3.7845

7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684 4.7122 4.5638 4.4226 4.2883 4.1604

8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349 5.1461 4.9676 4.7988 4.6389 4.4873

9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590 5.5370 5.3282 5.1317 4.9464 4.7716

10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446 5.8892 5.6502 5.4262 5.2161 5.0188

11 10.3676 9.7868 9.2526 8.7605 8.3064 7.8869 7.4987 7.1390 6.8052 6.4951 6.2065 5.9377 5.6869 5.4527 5.2337

12 11.2551 10.5753 9.9540 9.3851 8.8633 8.3838 7.9427 7.5361 7.1607 6.8137 6.4924 6.1944 5.9176 5.6603 5.4206

13 12.1337 11.3484 10.6350 9.9856 9.3936 8.8527 8.3577 7.9038 7.4869 7.1034 6.7499 6.4235 6.1218 5.8424 5.5831

14 13.0037 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667 6.9819 6.6282 6.3025 6.0021 5.7245

15 13.8651 12.8493 11.9379 11.1184 10.3797 9.7122 9.1079 8.5595 8.0607 7.6061 7.1909 6.8109 6.4624 6.1422 5.8474

16 14.7179 13.5777 12.5611 11.6523 10.8378 10.1059 9.4466 8.8514 8.3126 7.8237 7.3792 6.9740 6.6039 6.2651 5.9542

17 15.5623 14.2919 13.1661 12.1657 11.2741 10.4773 9.7632 9.1216 8.5436 8.0216 7.5488 7.1196 6.7291 6.3729 6.0472

18 16.3983 14.9920 13.7535 12.6593 11.6896 10.8276 10.0591 9.3719 8.7556 8.2014 7.7016 7.2497 6.8399 6.4674 6.1280

19 17.2260 15.6785 14.3238 13.1339 12.0853 11.1581 10.3356 9.6036 8.9501 8.3649 7.8393 7.3658 6.9380 6.5504 6.1982

20 18.0456 16.3514 14.8775 13.5903 12.4622 11.4699 10.5940 9.8181 9.1285 8.5136 7.9633 7.4694 7.0248 6.6231 6.2593

21 18.8570 17.0112 15.4150 14.0292 12.8212 11.7641 10.8355 10.0168 9.2922 8.6487 8.0751 7.5620 7.1016 6.6870 6.3125

22 19.6604 17.6580 15.9369 14.4511 13.1630 12.0416 11.0612 10.2007 9.4424 8.7715 8.1757 7.6446 7.1695 6.7429 6.3587

23 20.4558 18.2922 16.4436 14.8568 13.4886 12.3034 11.2722 10.3711 9.5802 8.8832 8.2664 7.7184 7.2297 6.7921 6.3988

24 21.2434 18.9139 16.9355 15.2470 13.7986 12.5504 11.4693 10.5288 9.7066 8.9847 8.3481 7.7843 7.2829 6.8351 6.4338

25 22.0232 19.5235 17.4131 15.6221 14.0939 12.7834 11.6536 10.6748 9.8226 9.0770 8.4217 7.8431 7.3300 6.8729 6.4641

26 22.7952 20.1210 17.8768 15.9828 14.3752 13.0032 11.8258 10.8100 9.9290 9.1609 8.4881 7.8957 7.3717 6.9061 6.4906

27 23.5596 20.7069 18.3270 16.3296 14.6430 13.2105 11.9867 10.9352 10.0266 9.2372 8.5478 7.9426 7.4086 6.9352 6.5135

28 24.3164 21.2813 18.7641 16.6631 14.8981 13.4062 12.1371 11.0511 10.1161 9.3066 8.6016 7.9844 7.4412 6.9607 6.5335

29 25.0658 21.8444 19.1885 16.9837 15.1411 13.5907 12.2777 11.1584 10.1983 9.3696 8.6501 8.0218 7.4701 6.9830 6.5509

30 25.8077 22.3965 19.6004 17.2920 15.3725 13.7648 12.4090 11.2578 10.2737 9.4269 8.6938 8.0552 7.4957 7.0027 6.5660


Annuity Factor (Present Value of Annuity)

16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%

1 0.8621 0.8547 0.8475 0.8403 0.8333 0.8264 0.8197 0.8130 0.8065 0.8000 0.7937 0.7874 0.7813 0.7752 0.7692

2 1.6052 1.5852 1.5656 1.5465 1.5278 1.5095 1.4915 1.4740 1.4568 1.4400 1.4235 1.4074 1.3916 1.3761 1.3609

3 2.2459 2.2096 2.1743 2.1399 2.1065 2.0739 2.0422 2.0114 1.9813 1.9520 1.9234 1.8956 1.8684 1.8420 1.8161

4 2.7982 2.7432 2.6901 2.6386 2.5887 2.5404 2.4936 2.4483 2.4043 2.3616 2.3202 2.2800 2.2410 2.2031 2.1662

5 3.2743 3.1993 3.1272 3.0576 2.9906 2.9260 2.8636 2.8035 2.7454 2.6893 2.6351 2.5827 2.5320 2.4830 2.4356

6 3.6847 3.5892 3.4976 3.4098 3.3255 3.2446 3.1669 3.0923 3.0205 2.9514 2.8850 2.8210 2.7594 2.7000 2.6427

7 4.0386 3.9224 3.8115 3.7057 3.6046 3.5079 3.4155 3.3270 3.2423 3.1611 3.0833 3.0087 2.9370 2.8682 2.8021

8 4.3436 4.2072 4.0776 3.9544 3.8372 3.7256 3.6193 3.5179 3.4212 3.3289 3.2407 3.1564 3.0758 2.9986 2.9247

9 4.6065 4.4506 4.3030 4.1633 4.0310 3.9054 3.7863 3.6731 3.5655 3.4631 3.3657 3.2728 3.1842 3.0997 3.0190

10 4.8332 4.6586 4.4941 4.3389 4.1925 4.0541 3.9232 3.7993 3.6819 3.5705 3.4648 3.3644 3.2689 3.1781 3.0915

11 5.0286 4.8364 4.6560 4.4865 4.3271 4.1769 4.0354 3.9018 3.7757 3.6564 3.5435 3.4365 3.3351 3.2388 3.1473

12 5.1971 4.9884 4.7932 4.6105 4.4392 4.2784 4.1274 3.9852 3.8514 3.7251 3.6059 3.4933 3.3868 3.2859 3.1903

13 5.3423 5.1183 4.9095 4.7147 4.5327 4.3624 4.2028 4.0530 3.9124 3.7801 3.6555 3.5381 3.4272 3.3224 3.2233

14 5.4675 5.2293 5.0081 4.8023 4.6106 4.4317 4.2646 4.1082 3.9616 3.8241 3.6949 3.5733 3.4587 3.3507 3.2487

15 5.5755 5.3242 5.0916 4.8759 4.6755 4.4890 4.3152 4.1530 4.0013 3.8593 3.7261 3.6010 3.4834 3.3726 3.2682

16 5.6685 5.4053 5.1624 4.9377 4.7296 4.5364 4.3567 4.1894 4.0333 3.8874 3.7509 3.6228 3.5026 3.3896 3.2832

17 5.7487 5.4746 5.2223 4.9897 4.7746 4.5755 4.3908 4.2190 4.0591 3.9099 3.7705 3.6400 3.5177 3.4028 3.2948

18 5.8178 5.5339 5.2732 5.0333 4.8122 4.6079 4.4187 4.2431 4.0799 3.9279 3.7861 3.6536 3.5294 3.4130 3.3037

19 5.8775 5.5845 5.3162 5.0700 4.8435 4.6346 4.4415 4.2627 4.0967 3.9424 3.7985 3.6642 3.5386 3.4210 3.3105

20 5.9288 5.6278 5.3527 5.1009 4.8696 4.6567 4.4603 4.2786 4.1103 3.9539 3.8083 3.6726 3.5458 3.4271 3.3158

21 5.9731 5.6648 5.3837 5.1268 4.8913 4.6750 4.4756 4.2916 4.1212 3.9631 3.8161 3.6792 3.5514 3.4319 3.3198

22 6.0113 5.6964 5.4099 5.1486 4.9094 4.6900 4.4882 4.3021 4.1300 3.9705 3.8223 3.6844 3.5558 3.4356 3.3230

23 6.0442 5.7234 5.4321 5.1668 4.9245 4.7025 4.4985 4.3106 4.1371 3.9764 3.8273 3.6885 3.5592 3.4384 3.3254

24 6.0726 5.7465 5.4509 5.1822 4.9371 4.7128 4.5070 4.3176 4.1428 3.9811 3.8312 3.6918 3.5619 3.4406 3.3272

25 6.0971 5.7662 5.4669 5.1951 4.9476 4.7213 4.5139 4.3232 4.1474 3.9849 3.8342 3.6943 3.5640 3.4423 3.3286

26 6.1182 5.7831 5.4804 5.2060 4.9563 4.7284 4.5196 4.3278 4.1511 3.9879 3.8367 3.6963 3.5656 3.4437 3.3297

27 6.1364 5.7975 5.4919 5.2151 4.9636 4.7342 4.5243 4.3316 4.1542 3.9903 3.8387 3.6979 3.5669 3.4447 3.3305

28 6.1520 5.8099 5.5016 5.2228 4.9697 4.7390 4.5281 4.3346 4.1566 3.9923 3.8402 3.6991 3.5679 3.4455 3.3312

29 6.1656 5.8204 5.5098 5.2292 4.9747 4.7430 4.5312 4.3371 4.1585 3.9938 3.8414 3.7001 3.5687 3.4461 3.3317

30 6.1772 5.8294 5.5168 5.2347 4.9789 4.7463 4.5338 4.3391 4.1601 3.9950 3.8424 3.7009 3.5693 3.4466 3.3321


𝑒𝑥 = 2.71828𝑥

x 0 1 2 3 4 5 6 7 8 9

0.0 1.00000 1.01005 1.02020 1.03045 1.04081 1.05127 1.06184 1.07251 1.08329 1.09417

0.1 1.10517 1.11628 1.12750 1.13883 1.15027 1.16183 1.17351 1.18530 1.19722 1.20925

0.2 1.22140 1.23368 1.24608 1.25860 1.27125 1.28403 1.29693 1.30996 1.32313 1.33643

0.3 1.34986 1.36343 1.37713 1.39097 1.40495 1.41907 1.43333 1.44773 1.46228 1.47698

0.4 1.49182 1.50682 1.52196 1.53726 1.55271 1.56831 1.58407 1.59999 1.61607 1.63232

0.5 1.64872 1.66529 1.68203 1.69893 1.71601 1.73325 1.75067 1.76827 1.78604 1.80399

0.6 1.82212 1.84043 1.85893 1.87761 1.89648 1.91554 1.93479 1.95424 1.97388 1.99372

0.7 2.01375 2.03399 2.05443 2.07508 2.09594 2.11700 2.13828 2.15977 2.18147 2.20340

0.8 2.22554 2.24791 2.27050 2.29332 2.31637 2.33965 2.36316 2.38691 2.41090 2.43513

0.9 2.45960 2.48432 2.50929 2.53451 2.55998 2.58571 2.61170 2.63794 2.66446 2.69123

N 0 1 2 3 4 5 6 7 8 9

1.0 2.71828 2.74560 2.77319 2.80107 2.82922 2.85765 2.88637 2.91538 2.94468 2.97427

1.1 3.00417 3.03436 3.06485 3.09566 3.12677 3.15819 3.18993 3.22199 3.25437 3.28708

1.2 3.32012 3.35348 3.38719 3.42123 3.45561 3.49034 3.52542 3.56085 3.59664 3.63279

1.3 3.66930 3.70617 3.74342 3.78104 3.81904 3.85743 3.89619 3.93535 3.97490 4.01485

1.4 4.05520 4.09596 4.13712 4.17870 4.22070 4.26311 4.30596 4.34924 4.39295 4.43710

1.5 4.48169 4.52673 4.57223 4.61818 4.66459 4.71147 4.75882 4.80665 4.85496 4.90375

1.6 4.95303 5.00281 5.05309 5.10387 5.15517 5.20698 5.25931 5.31217 5.36556 5.41948

1.7 5.47395 5.52896 5.58453 5.64065 5.69734 5.75460 5.81244 5.87085 5.92986 5.98945

1.8 6.04965 6.11045 6.17186 6.23389 6.29654 6.35982 6.42374 6.48830 6.55350 6.61937

1.9 6.68589 6.75309 6.82096 6.88951 6.95875 7.02869 7.09933 7.17068 7.24274 7.31553

N 0 1 2 3 4 5 6 7 8 9

2.0 7.38906 7.46332 7.53832 7.61409 7.69061 7.76790 7.84597 7.92482 8.00447 8.08492

2.1 8.16617 8.24824 8.33114 8.41487 8.49944 8.58486 8.67114 8.75828 8.84631 8.93521

2.2 9.02501 9.11572 9.20733 9.29987 9.39333 9.48774 9.58309 9.67940 9.77668 9.87494

2.3 9.97418 10.0744 10.1757 10.2779 10.3812 10.4856 10.5910 10.6974 10.8049 10.9135

2.4 11.0232 11.1340 11.2459 11.3589 11.4730 11.5883 11.7048 11.8224 11.9413 12.0613

2.5 12.1825 12.3049 12.4286 12.5535 12.6797 12.8071 12.9358 13.0658 13.1971 13.3298

2.6 13.4637 13.5991 13.7357 13.8738 14.0132 14.1540 14.2963 14.4400 14.5851 14.7317

2.7 14.8797 15.0293 15.1803 15.3329 15.4870 15.6426 15.7998 15.9586 16.1190 16.2810

2.8 16.4446 16.6099 16.7769 16.9455 17.1158 17.2878 17.4615 17.6370 17.8143 17.9933

2.9 18.1741 18.3568 18.5413 18.7276 18.9158 19.1060 19.2980 19.4919 19.6878 19.8857

N 0 1 2 3 4 5 6 7 8 9

3.0 20.0855 20.2874 20.4913 20.6972 20.9052 21.1153 21.3276 21.5419 21.7584 21.9771

3.1 22.1980 22.4210 22.6464 22.8740 23.1039 23.3361 23.5706 23.8075 24.0468 24.2884

3.2 24.5325 24.7791 25.0281 25.2797 25.5337 25.7903 26.0495 26.3113 26.5758 26.8429


3.3 27.1126 27.3851 27.6604 27.9383 28.2191 28.5027 28.7892 29.0785 29.3708 29.6660

3.4 29.9641 30.2652 30.5694 30.8766 31.1870 31.5004 31.8170 32.1367 32.4597 32.7859

3.5 33.1155 33.4483 33.7844 34.1240 34.4669 34.8133 35.1632 35.5166 35.8735 36.2341

3.6 36.5982 36.9661 37.3376 37.7128 38.0918 38.4747 38.8613 39.2519 39.6464 40.0448

3.7 40.4473 40.8538 41.2644 41.6791 42.0980 42.5211 42.9484 43.3801 43.8160 44.2564

3.8 44.7012 45.1504 45.6042 46.0625 46.5255 46.9931 47.4654 47.9424 48.4242 48.9109

3.9 49.4024 49.8990 50.4004 50.9070 51.4186 51.9354 52.4573 52.9845 53.5170 54.0549

N 0 1 2 3 4 5 6 7 8 9

4.0 54.5982 55.1469 55.7011 56.2609 56.8263 57.3975 57.9743 58.5570 59.1455 59.7399

4.1 60.3403 60.9467 61.5592 62.1779 62.8028 63.4340 64.0715 64.7155 65.3659 66.0228

4.2 66.6863 67.3565 68.0335 68.7172 69.4079 70.1054 70.8100 71.5216 72.2404 72.9665

4.3 73.6998 74.4405 75.1886 75.9443 76.7075 77.4785 78.2571 79.0436 79.8380 80.6404

4.4 81.4509 82.2695 83.0963 83.9314 84.7749 85.6269 86.4875 87.3567 88.2347 89.1214

4.5 90.0171 90.9218 91.8356 92.7586 93.6908 94.6324 95.5835 96.5441 97.5144 98.4944

4.6 99.4843 100.484 101.494 102.514 103.544 104.585 105.636 106.698 107.770 108.853

4.7 109.947 111.052 112.168 113.296 114.434 115.584 116.746 117.919 119.104 120.301

4.8 121.510 122.732 123.965 125.211 126.469 127.740 129.024 130.321 131.631 132.954

4.9 134.290 135.639 137.003 138.380 139.770 141.175 142.594 144.027 145.474 146.936

N 0 1 2 3 4 5 6 7 8 9

5.0 148.413 149.905 151.411 152.933 154.470 156.022 157.591 159.174 160.774 162.390

5.1 164.022 165.670 167.335 169.017 170.716 172.431 174.164 175.915 177.683 179.469

5.2 181.272 183.094 184.934 186.793 188.670 190.566 192.481 194.416 196.370 198.343

5.3 200.337 202.350 204.384 206.438 208.513 210.608 212.725 214.863 217.022 219.203

5.4 221.406 223.632 225.879 228.149 230.442 232.758 235.097 237.460 239.847 242.257

5.5 244.692 247.151 249.635 252.144 254.678 257.238 259.823 262.434 265.072 267.736

5.6 270.426 273.144 275.889 278.662 281.463 284.291 287.149 290.035 292.949 295.894

5.7 298.867 301.871 304.905 307.969 311.064 314.191 317.348 320.538 323.759 327.013

5.8 330.300 333.619 336.972 340.359 343.779 347.234 350.724 354.249 357.809 361.405

5.9 365.037 368.706 372.412 376.155 379.935 383.753 387.610 391.506 395.440 399.415

N 0 1 2 3 4 5 6 7 8 9

6.0 403.429 407.483 411.579 415.715 419.893 424.113 428.375 432.681 437.029 441.421

6.1 445.858 450.339 454.865 459.436 464.054 468.717 473.428 478.186 482.992 487.846

6.2 492.749 497.701 502.703 507.755 512.859 518.013 523.219 528.477 533.789 539.153

6.3 544.572 550.045 555.573 561.157 566.796 572.493 578.246 584.058 589.928 595.857

6.4 601.845 607.894 614.003 620.174 626.407 632.702 639.061 645.484 651.971 658.523

6.5 665.142 671.826 678.578 685.398 692.287 699.244 706.272 713.370 720.539 727.781

6.6 735.095 742.483 749.945 757.482 765.095 772.784 780.551 788.396 796.319 804.322

6.7 812.406 820.571 828.818 837.147 845.561 854.059 862.642 871.312 880.069 888.914

6.8 897.847 906.871 915.985 925.191 934.489 943.881 953.367 962.949 972.626 982.401


6.9 992.275 1002.25 1012.32 1022.49 1032.77 1043.15 1053.63 1064.22 1074.92 1085.72

N 0 1 2 3 4 5 6 7 8 9

7.0 1096.63 1107.65 1118.79 1130.03 1141.39 1152.86 1164.45 1176.15 1187.97 1199.91

7.1 1211.97 1224.15 1236.45 1248.88 1261.43 1274.11 1286.91 1299.84 1312.91 1326.10

7.2 1339.43 1352.89 1366.49 1380.22 1394.09 1408.10 1422.26 1436.55 1450.99 1465.57

7.3 1480.30 1495.18 1510.20 1525.38 1540.71 1556.20 1571.84 1587.63 1603.59 1619.71

7.4 1635.98 1652.43 1669.03 1685.81 1702.75 1719.86 1737.15 1754.61 1772.24 1790.05

7.5 1808.04 1826.21 1844.57 1863.11 1881.83 1900.74 1919.85 1939.14 1958.63 1978.31

7.6 1998.20 2018.28 2038.56 2059.05 2079.74 2100.65 2121.76 2143.08 2164.62 2186.37

7.7 2208.35 2230.54 2252.96 2275.60 2298.47 2321.57 2344.90 2368.47 2392.27 2416.32

7.8 2440.60 2465.13 2489.91 2514.93 2540.20 2565.73 2591.52 2617.57 2643.87 2670.44

7.9 2697.28 2724.39 2751.77 2779.43 2807.36 2835.57 2864.07 2892.86 2921.93 2951.30

N 0 1 2 3 4 5 6 7 8 9

8.0 2980.96 3010.92 3041.18 3071.74 3102.61 3133.79 3165.29 3197.10 3229.23 3261.69

8.1 3294.47 3327.58 3361.02 3394.80 3428.92 3463.38 3498.19 3533.34 3568.85 3604.72

8.2 3640.95 3677.54 3714.50 3751.83 3789.54 3827.63 3866.09 3904.95 3944.19 3983.83

8.3 4023.87 4064.31 4105.16 4146.42 4188.09 4230.18 4272.69 4315.64 4359.01 4402.82

8.4 4447.07 4491.76 4536.90 4582.50 4628.55 4675.07 4722.06 4769.52 4817.45 4865.87

8.5 4914.77 4964.16 5014.05 5064.45 5115.34 5166.75 5218.68 5271.13 5324.11 5377.61

8.6 5431.66 5486.25 5541.39 5597.08 5653.33 5710.15 5767.53 5825.50 5884.05 5943.18

8.7 6002.91 6063.24 6124.18 6185.73 6247.90 6310.69 6374.11 6438.17 6502.88 6568.23

8.8 6634.24 6700.92 6768.26 6836.29 6904.99 6974.39 7044.48 7115.28 7186.79 7259.02

8.9 7331.97 7405.66 7480.09 7555.27 7631.20 7707.89 7785.36 7863.60 7942.63 8022.46

N 0 1 2 3 4 5 6 7 8 9

9.0 8103.08 8184.52 8266.78 8349.86 8433.78 8518.54 8604.15 8690.62 8777.97 8866.19

9.1 8955.29 9045.29 9136.20 9228.02 9320.77 9414.44 9509.06 9604.62 9701.15 9798.65

9.2 9897.13 9996.60 10097.1 10198.5 10301.0 10404.6 10509.1 10614.8 10721.4 10829.2

9.3 10938.0 11047.9 11159.0 11271.1 11384.4 11498.8 11614.4 11731.1 11849.0 11968.1

9.4 12088.4 12209.9 12332.6 12456.5 12581.7 12708.2 12835.9 12964.9 13095.2 13226.8

9.5 13359.7 13494.0 13629.6 13766.6 13904.9 14044.7 14185.8 14328.4 14472.4 14617.9

9.6 14764.8 14913.2 15063.0 15214.4 15367.3 15521.8 15677.8 15835.3 15994.5 16155.2

9.7 16317.6 16481.6 16647.2 16814.6 16983.5 17154.2 17326.6 17500.8 17676.7 17854.3

9.8 18033.7 18215.0 18398.1 18583.0 18769.7 18958.4 19148.9 19341.3 19535.7 19732.1

9.9 19930.4 20130.7 20333.0 20537.3 20743.7 20952.2 21162.8 21375.5 21590.3 21807.3

10.0 22026.5 22247.8 22471.4 22697.3 22925.4 23155.8 23388.5 23623.6 23861.0 24100.8

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