Chapter 12

Compound Interest

Rule 1 lfPrincipal = RsP

Time = t years Rate = r per cent per annum and

Interest is compounded annually, then,

Amount = r

100

Illustrative Example Ex.: Rs 7500 is borrowed at CI at the rate of 4% per

annum. What will be the amount to be paid after 2 years?

Soln: Applying the above formula,

. n2

Amount = 7500 1 + -100

7500x26x26 25x25

Rs 8112.

Exercise I .

5.

Raviraj invested an amount of Rs 10,000 at compound interest rate of 10 per cent per annum for a period of three years. How much amount will Raviraj get after 3 years? |SBI Associates PO, 1999] a)Rs 12340 b)Rs 13210 c)Rs 13320 d)Rs 13310 Seema invested an amount of Rs 16000 for two years at compound interest and received an amount of Rs 17640 on maturity. What is the rate of interest?

|SBI Associates PO, 1999| a) 8 pcpa b) 5 pcpa c)4pcpa d) None of these Amit Kumar invested an amount of Rs 15000 at com-pound interest rate of 10 pcpa for a period of 2 years. What amount will he receive at the end of two years?

|GuwahatiPO,1999] a) Rs 18000 b)Rs 18500 c)Rs 17000 d)Rs 18150 Find the amount at compound interest of Rs 625 in 2 years at 4 per cent. a)Rs676 b)Rs756 c)Rs767 d)Rs675 Find the amount at compound interest of Rs 2700 in 3

years at 3 - per cent

a) Rs 2979.10 b)Rs2997.10 c) Rs 2797.10 d) None of these

6. What is the amount at compound interest for 2 years at 2 per cent? a) 2.0404 times the principal b) 1.0404 times the principal c) 1.404 times the principal d) Data inadequate

Answers

10 100

l.d;Hint: Amount = 10000 1 +

= lOOOOx j i x j ^ x j ^ =Rsl3310

2.b ;Hint:^= 1 + 4 or 17640 16000

111025 , r 5x7x3 or il H or

' A l 10000 100 ' 100

100 J

r "Too

105-100 or,

3.d 100 4. a

100 5.a

or,r = 5%

6.b

Rule 2 When interest is compounded half-yearly

l2l

Amount = P

r

i + i 100

i+-200

Illustrative Example Ex.: Rs 7500 is borrowed at CI at the rate of 4% per annum.

What will be the amount to be paid after 1 year, i f interest is compounded half-yearly?

294 PRACTICE BOOK ON QUICKER MATHS


Amount = 7 5 0 0 1 + -200 _

7500x102x102 Ton x 100

= Rs7803.

Exercise 1. Find the amount of Rs 1000 in 1 year at 5 per cnet com-

pound interest payable half-yearly, a) Rs 1050 (Approx) b) Rs 950 (Approx) c) Rs 1125 (Approx) d) Rs 1025 (Approx)

2..-Find the amount of Rs 6400 in 1 year 6 months at 5 per cent compound interest, interest being calculated half yearly. a) Rs 6882.10 b)Rs6892.10 c) Rs 6982.10 d) None of these

3. Find the compound interest on Rs 350 for 1 year at 4 per cent per annum, the interest being payable half yearly. a)Rs364.14 b)Rs365.15 c)Rs 14.14 d)Rs 15.15

Answers l .a 2.b

204 204 3. c; Hint: Amount = j 5 0 x ^ 0 " x ^ 0 " =Rs364.14

.-. Compound interest = Rs 364.14 - Rs 350 = Rs 14.14

Rule 3 ^ 0 When interest is compounded quarterly

Amount = P

r

100 = P 1 + -

400

Illustrative Example Ex.: Rs 7500 is borrowed at CI at the rate of 4% per annum.

What will be the amount to be paid after 6 months, i f interest is compounded quarterly?


Amount = 7500 4

400

= Rs 7650.75.

Exercise

7500x101x101

100x100

1 1. Find the compound interest on Rs 2000 for 1 years at

10 per cent per annum, the interest being payable quar-terly. a) Rs 2262.81 b)Rs 262.81 c) Rs 262.18 d) None of these

Find the compound interest on Rs 10000 in 9 months at 4 per cent interest payable quarterly. a) Rs 303 (Approx) b) Rs 313 (Approx) c) Rs 203 (Approx) d) None of these Find the compound interest on Rs 8000 in 3 months at 5 per cent interest payable quarterly. a)Rs250 b)Rs200 c)R 5 150 d)Rsl00

Answers

1. b; Hint: Amount - 200o| i +.d S S L -1. b; Hint: Amount - 200o| 400

= 2000| I 40

Rs 2262.81

2. a Compound interest = 2262.81 - 2000 = Rs 262.81 3.d

Rule 4 Let Principal = RsP

Time = t years Rate = r per cent per annum

CI (Compound Interest) = A - P = P 1 + ^ - 1 100x J

Where, n = 1, when the interest is compounded yearly n = 2, when the interest is compounded half-yearly n = 4, when the interest is compounded quarterly n = 12, when the interest is compounded monthly

When, interest is compounded yearly, n = 1

C I = P 1 + r Too

Note: Rule 1,2 and 3 are special cases of this general rule.

Illustrative Example Ex.: I f the interest is compounded annually, find the

compound interest on Rs 2000 for 3 years at 10% per annum.


CI= 2000 ( io V f i l l 3

i + - i = 2000 - 1 I IOOJ U o ,

= Rs662.

Exercise 1. The compound interest on any sum at the rate of 5% for

two years is Rs 512.50. Find the sum. [BSRB Hyderabad PO, 1999|

a)Rs5200 b)Rs4800 c)Rs5000 d)Rs5500 2. Find the amount on Rs 60,000, i f the interest is com-

Compound Interest

pounded half-yearly at 4 per cent per annum for 1

years. [NABARD, 1999] a) Rs 63672.48 b) Rs 62424.00 c)Rs 67491.84 d)Rs 64896.00

3. What will be the compound interest acquired on a sum of Rs 12000 for 3 years at the rate of 10% per annum?

[BSRBBhopalPO,2000] a)Rs2652 b)Rs3972 c)Rs3960 d)Rs3852

4. Find the compound interest on Rs 4000 for 3 years at 5 per cent per annum. a)Rs630 b)Rs 620.50 c) Rs 630.50 d) None of these

\d the compound interest on Rs 1600 for 3 years at

per cent per annum.

a) Rs 186.83 b)Rs 168.83 c)Rs 186.38 d)Rs 168.38 6. What principal will amount to Rs 13 52 in 2 years at 4 per

cent compound interest? a)Rsl520 b)Rsl260 c)Rsl250 d)Rsl220

7. On what principal will the compound interest for 3 years at 5 per cent amount to Rs 63.05? a)Rs400 b)Rs500 c)Rs450 d)Rs550

CS^A^ property decreases in value every year at the rate of

6 per cent on its value at the beginning of that year. I f

its value at the end of 3 years be Rs 21093.75, what was it worth at the beginning of these three years? a) Rs 25600 b)Rs 26500 c)Rs 25500 d)Rs 25800

9. One man offers Rs 80000 for an estate and another Rs 84270 to be paid in 3 year's time, allowing 6 per cent compound interest. Which is the better offer. a) first b) second c) both d) None of these

10. A merchant commences with a certain capital and gains annually at the rate of 25 per cent. At the end of 3 years he is worth Rs 10,000. What was his original capital. a)Rs5120 b)Rs5220 c)Rs5210 d)Rs5130

11. What sum put out for 1 years at 4 per cent compound

interest, payable half-yearly will amount to Rs 6632.55? a)Rs6250 b)Rs6520 c)Rs6350 d) None of these

12. Rs 800 at 5% per annum compound interest amount to Rs 882 in

[Clerical Grade 1991] a) 4 years b) 3 years c) 2 years d) 1 year

Answers 1. c; Hint: Let the sum be Rs x

512.50 = 100J 441-400

400

512.50x400 x = = Rs 5000

41

4 3' 2. a; Hint: Here r = - = 2% and t = - x 2 =3 years

Amount = 60000 1 + 100 60000x x x

50 50 50

= Rs 63672.48

3. b; Hint: Compound interest = 12000 1 + -v

10 Too

-12000

= 15972-12000 = Rs 3972 4. c 5. a 6. c; Hint: Let Rs P be the principal

.2

ThenP | l + I - H 5 2

25 25

p = 1352x25x25 26x26

.-. the required principal is Rs 1250 7. a; Hint: Let Rs P be the principal

ThenRsP jf1 + ^ j _ 1 f = R s 6 3 . 0 5

2 1 x 2 1 x 2 1 ^ _ 1 {20 20 20

6305 100

P = Rs400 the required principal is Rs 400

8. a; Hint: Here P 1 100

= 21093-

.-. P I T v l 6 y

84375

_ 84375x16x16x16 ' P = A k i< i< =Rs25600 4x15x15x15 .-. Original value of the property = Rs 25600

9. a; Hint: Rs 80000 after 3 years will become

106 106 106 80000 x ^ x ^ 0 " x J ^ = R s 95281.28, which is greater


than Rs 84270. Hence, first one is a better offer.

10000x100x100x100 10. a; Hint: Original capital = ~~z = 80

I ^ j X X l j

x64 = Rs5120

6 < J 2 3 . S S ~ 1 0 0 ~ l O O ^ l O O

11. a;Hint:Sum= =Rs6250 102x102x102

6632.55 = Sum

4

l + - 2 -100

12. c; Hint: Let time be t years

it , 882 = 800 1 +

00 882 800

21 20

t = 2 years

Rule 5 When rate of interest is rx %, r2 % and r3 % for 1st year, 2nd year, and 3rd year respectively, then Amount = P

X " i + 4 - " X 100 100 100

Illustrative Example Ex.: Rs 7500 is borrowed at CI at the rate of 2% for the first

year, 4% for the second year and 5% for the third year. Find the amount to be paid after 3 years.

Soln: Applying the above theorem,

Amount = 7 5 0 0 1 + -100 1 + -

100 1 + _5_

100

7500x102x104x105 100x100x100

= Rs 8353.8.

Exercise 1. Rs 50000 is borrowed at CI at the rate of 1 % for the first

year, 2% for the second year and 3% for the third year. Find the amount to be paid after 3 years, a) Rs 50355.3 b)Rs 53055.3 c)Rs 53505.3 d)Rs 53053.5 Rs 125000 is borrowed at CI at the rate of 2% for the first year, 3% for the second year and 4% for the third year. Find the amount to be paid after 3 years, a) Rs 135678 b)Rs 136587 c)Rs 163578 d)Rs 136578 Rs 25000 is borrowed at CI at the rate of 3% for the first year, 4% for the second year and 5% for the third year. Find the amount to be paid after 3 years.

2.

3.

a)Rs28119 b)Rs29118 c)Rs28129 d)Rs28117 4. What sum will amount to Rs 15916.59 in 3 years at com-

pound interest, the interest for 1st, 2nd and 3rd years being 3,2 and 1 per cent respectively. a) Rs 15900 b) Rs 15000 c) Rs 16000 d) None of these

5. What sum of money will amount to Rs 699.66 in 2 years, reckoning compound interest for 1 year at 4 per cent and

for the other at 3 per cent per annum.

a)Rs560 b)Rs650 c)Rs670 d)Rs580

Answers L b 2.d 3.a

' = P\4 b-Hinf 15916.59 = P\x ' UOO 100 100

P = 1591659x100x100 103x102x101

699.66x100x200

= Rs 15000

104x207

Rule 6

5. b; Hint: Sum =

Let, Amount = Rs A Principal = R s P Time = t years

Rate of interest (r) = n x 100

= Rs 650

% per annum.

Where, n = 1, when the interest is compounded yearly n = 2, when the interest is compounded half-yearly n=4, when the interest is compounded quarterly n = 12, when the interest is compounded monthly

If the interest is compounded yearly, then n = 1

. _ 100 % per annum.

Illustrative Examples Ex. 1: At what rate per cent per annum will Rs 1000 amount

to Rs 1331 in 3 years? The interest is compounded yearly.


r = 100 [1331] t1000J

1/3

100 i ir*-, 10

Ex.2: = 10% per annum

At what rate per cent compounded yearly will Rs 80,000 amount to Rs 88,200 in 2 years?

Compound Interest 297

Soln: Detail Method: We have 80,000 | 1 +

88200 441 or,

or,

1 + -100 80000 400

100

, , \

Rs88200

1+-21

00 20 r = 5%

Quicker Method: Applying the above formula,

100 f88200Y , = 100 2 1 1 - 1 = 100 1,80000 J _20~ .

= 5% per annum.

Exercise At what rate per cent compound interest, will Rs 400 amount to Rs 441 in 2 years? a) 4% b)5% c)6% d)3%

1 At what rate per cent compound interest will Rs 625 amount to Rs 676 in 2 years? a) 3% b)Rs2% c)4% d)5%

I At what rate per cent wil l the compound interest on Rs 2500 amount to Rs 477.54 in 3 years? a) 6% b)4% c)5% d) None of these

Answers l .b 2.c

3 a: Hint: rate % = 100 ( 2977.54 V

i 2500

V Here compound interest = Rs 477.54 (given) Amount (A) =Rs 2500+ Rs 477.54 = Rs 2977.54

< - 1 6., .50

Rule 7 Let Principal = P

Rate = r% per annum Amount = A and

Time is given in the form of fraction like 2 years.

\mount(A)= P 1 + -100

r

100

Illustrative Example L l : Find the compound interest on Rs 2000 at 5% per

, 1 t 1 annum, compounded yearly for - years.

Soln: We know that, compound interest = Amount - Princi-pal CI = A - P Now, using the above formula,

A = P 1 + -100 1 + -

200

"105" 2 X

"205" _100_ 200 _

= 2000

2000x105x105x205 Rs 2260.12

100x100x200 .-. CI = 2260.12-2000 = Rs260.12.

Exercise

1. Find the amount of Rs 800 at compound interest in 2-

years at 5 per cent. a) Rs 904.05 b)Rs 904.50 c)Rs904 d)Rs 905.04

On what sum wil l the compound interest for 2 years

at 10% amount to Rs 6352.50? a)Rs4900 b)Rs5500 c)Rs5000 d)Rs5800

[ ' 1 Find the amount of Rs 4000 for 2 years at 6% com-

b)Rs 4692.32 d) None of these

pound interest, a) Rs 4629.23 c)Rs 4639.32

Answers 1. a

6352.50x100x100x100 2. c;Hint: Sum = = Rs5000 110x110x105 3.a

Rule 8 Theorem: A sum of money, placed at compound interest, becomes n times in tyears andm times in xyears. We calcu-late the value of x from the equation given below

Illustrative Examples Ex. 1: A sum of money placed at compound interest doubles

itself in 4 years. In how many years will it amount to eight times itself?

Soln: Detail Method:

We have P 100 J

2P 1 + - = ? 100,


Cubing both sides, we get

1 + -100J

= 2 = 8

( V 2 = 8P Or P

u ' 1, 100 ^ Hence, the required time is 12 years. Quicker Method-1: x becomes 2x in 4 years 2x becomes 4x in next 4 years 4x becomes 8x in yet another 4 years Thus, x becomes 8x in 4 + 4 + 4=12 years Quicker Method - II Applying the above formula, we have

itself in 5 years. In how many years will it amount to 8 times itself? a) 18 years b) 15 years c) 16 years d) 12 years

2. A sum of money placed at compound interest doubles itself in 6 years. In how many years will it amount to 16 times itself? a) 24 years b) 26 years c) 22 years d) 20 years

3. A sum of money placed at compound interest thrice it-self in 4 years. In how many years will it amount to 27 times itself? a) 12 years b) 15 years c) 14 years d) 10 years

^ 4 ^ " l f a sum of money at compound interest amounts to thrice itself in 3 years, then in how many years will it be 9 times itself? | Bank PO Exam, 19811 a) 12 year b) 6 years c) 9 years d) 15 years

2 . / 4 = 8 l / , 2 I / 4 = 2 3 / * ^ I = 3

:.x = \2 years. Quicker Method - I II : Remember the following con-clusion: If a sum becomes n times in t years at CI then it will be (n)m times in mt years. Thus, i f a sum becomes 2 times in 4 years it will be (2) 3 times in 3 x 4 = 12 years.

Ex.2: A sum of money at compound interest amounts to thrice itself in three years. In how many years will it be 9 times itself?

Soln: Detail Method: Suppose the sum = Rs x Then, we have

>3

3x = x 1 + -100

Squaring both sides

or, 3 = 1 + -100

f i+1

Now multiply both sides by x; then 9x = x\ + 100 J

the sum x will be 9 times in 6 years. Quicker Method: Remember the following conclu-sion:

If a sum becomes n times in (years at CI then it will

be (n)m times in mt years.

Thus, i f a sum becomes 3 times in 3 years it will be

(3)2 times in 2 x 3 = 6 years. Exercise 1. A sum of money placed at compound interest doubles

Answers l .b 2. a 3.a 4.c

Rule 9 Theorem: If a certain sum becomes'm' times in 7' yean,

the rate of compound interest r is equal to 100 [(/w)17' - l i

Illustrative Example Ex.: At what rate per cent compound interest does a sum

of money become nine-fold in 2 years? Detail Method: Let the sum be Rs x and the rate of compound interest be r% per annum, then

9x = x(]+-^~) K lOOj

Soln:

or, 9 100

or, 3 1 + T o o ; o r ' i o o 200%

Quicker Method: Applying the above rule, we ha'

r = 1 Oo[(9)1/2 -1 ] = 100(3 -1) = 200% .

Exercise 1. At what rate per cent compound interest, does a sum

9 money become times itself in 2 years? a) 50% b)100% c)25% d)40%

2. At what rate per cent will the compound interest, does sum of money become four fold in 2 years?

IBank PO Exam, 19" a) 150% b)100% c)200% d)75%

3. At what rate per cent will the compound interest, does sum of money become 27 times in 3 years? a) 100% b) 150% c)75% ' d)200%

4. At what rate per cent wil l the compound interest, does sum of money become 16 times in 4 years? a) 100% b) 150% c)50% d)75%

Answers L a 2.b I d 4. a

Compound Interest \

Rule 10 Theorem: If the CI on a certain sum for t years atr%be Rs C, then the SI is given by the following formula, Simple

Interest (SI) =

100

(CI).

Note: Whent = 2

SI= -

rt

, 1 + - 1 v 100

2r

x Compound Interest

100 1 + r + 1 2 '00 100

2rxC/xlQQ

r 2 +200r

[ 200 S I = r + 200

200r r(r+ 200)

-xCI

xCI

xCI

Illustrative Example Ex.: If the CI on a certain sum for 2 years at 3% be Rs

101.50, what would be the SI? Soln: Detail Method:

CI on 1 rupee = | 1 + 3 \ / i m A 2

1 = 103

- 1

Re

100 J

609 10000

S I o n l r u p e e ^ R e ^ R e ^

6 10000 200 -x-

s i " C I 100 609 203

200 200 t M . S I = ^ o f C I = 2 0 3 - X l 0 1 - 5 = R s 1 0 0 Quicker Method I: Applying the above formula, we have,

mple Interest:

3x2

rt

100 1 + -100

x Compound Interest

;oo I IOOJ

X101.50:

100

10000

103 Y _ 100 J

xl01.50

100 609 xl01.50 =Rsl00

Quicker Method II: Here, t = 2

SI= 101.50 = Rsl00 (SeeNotel

Exercise I f the CI on a certain sum for 2 years at 5% be Rs 410. what would be the SI? a)Rs200 b)Rs300 c)Rs350 d)Rs400 I f the CI on a certain sum for 2 years at 4% be Rs 510, what would be the SI? a)Rs500 b)Rs505 c)Rs400 d)475 I f the CI on a certain sum for 2 years at 6% be Rs 25.75, what would be the SI? a)Rs25 b)Rs24 c)Rs20 d)Rsl5 I f the compound interest on a certain sum for 3 years at 5 per cent be Rs 50.44, what would be the simple inter-est? a)Rs49 b)Rs48 c)Rs44 d)Rs40 The compound interest on a sum of money for 3 years at 5 percent is Rs 1324.05. What is the simple interest? a)Rsl260 b)Rsl560 c)Rsl l60 d)Rsl360 The compound interest on a certain sum of money for 2 years at 10% per annum is Rs 420. The simple interest on the same sum at the same rate and same time will be:

|Clerical Grade Exam, 1991] a)Rs350 b)Rs375 c)Rs380 d)Rs400

Answers l . d 2. a 3.a 4.b 5.a 6.d

2.

3.

4.

5.

6.

Rule 11 ' T Y " Theorem: IftheSIon a certain sum for 2 years atr% beRs 'S' then the CI is given by the following formula, CI =

r + 200 200

-xS

Illustrative Example Ex.: The simple interest on a certain sum of money for 2

years at 10% per annum is Rs 400. Find the compound interest at the same rate and for the same time.

Soln: Using the above formula, 210

CI= ^ x 4 0 0 =Rs420.

Exercise 1. The simple interest on a certain sum of money for 2 years

at 5% per annum is Rs 100. Find the compound interest at the same rate and for the same time. a) Rs 102.50 b)Rsl03 c)Rs 103.50 d)Rs 102.25

2. The simple interest on a certain sum of money for 2 years at 6% per annum is Rs 300. Find the compound interest at the same rate and for the same time. a)Rs310 b)Rs308 c)Rs307 d)Rs309

3. The simple interest on a certain sum of money for 2 years


at 4% per annum is Rs 350. Find the compound interest at the same rate and for the same time. a)Rs387 b)Rs367 c)Rs357 d) None of these

4. The simple interest on a certain sum of money for 2 years at 20% per annum is Rs 200. Find the compound interest at the same rate and for the same time. a)Rs320 b)Rs220 c)Rs210 d)Rs225

5. SI on a sum at 4% per annum for 2 years is Rs 80. The CI on the same sum for the same period is:

[Asstt. Grade 1987] a) Rs 1081.60 b)Rs 81.60 c)Rsl60 d) None of these

Answers l .a 2.d 3.c 4.b 5.b

Rule 12 Theorem: If the compound interest on a certain sum for 2 years isRs'C and simple interest is Rs 'S', then the rate of

" 2 x ( C - S ) -xlOO 0/

/ 0 per or interest per annum is

2xDiff. xlOO

SI \

Illustrative Example Ex.: The compound interest on a certain sum for 2 years is

Rs 40.80 and simple interest is Rs 40.00. Find the rate of interest per annum and the sum.

Soln: Detail Method: A little reflection will show that the difference between the simple and compound inter-ests for 2 years is the interest on the first year's inter-est.

40 First year's SI = Rs = Rs 20

CI - SI = Rs 40.8 - Rs 40 = Re 0.80 Interest on Rs 20 for 1 year = Re 0.80

.'. Interest on Rs 100 for 1 year = Rs 80x100 100x20

= Rs4 .-. rate = 4% Now, principal P is given by

100x1 100x40 P = = . . =Rs500 tr 2x4 Quicker Method: Applying the above rule, we have

2x0.8 the rate =

40 -xl00 = 4%

40x100 and sum = -;- = Rs 500.

4 x 2

Exercise 1. The compound interest on a certain sum for 2 years is Rs

50.50 and simple interest is Rs 50. Find the rate of inter-est per annum and the sum. a) 4%, Rs 1000 b) 2%, Rs 1150 c) 2%, Rs 1250 d) 4%, Rs 1250

2. The compound interest on a certain sum for 2 years is Rs 105 and simple interest is Rs 100. Find the rate of interest per annum and the sum. a) 10%, Rs 500 b) 10%, Rs 1000 c) 20%, Rs 1000 d) None of these

3. The compound interest on a certain sum for 2 years is Rs 60.60 and simple interest is Rs 60. Find the rate of inter-est per annum and the sum. a) 2%, Rs 1600 b) 2%, Rs 1400 c) 3%, Rs 1500 d) 2%, Rs 1500

Answers l .c 2.a 3.d

Rule 13 Theorem: If on a certain sum of money, the simple interest for 2 years at the rate of r% per annum is Rs X, then the difference in compound interest and simple interest is given

(Xr

^ * s U o o ,

Note: This formula is applicable only for 2 years.

Illustrative Example Ex.: On a certain sum of money, the simple interest for 2

years is Rs 50 at the rate of 5% per annum. Find the difference in CI and SI.

Soln: Applying the above formula, we have 5 0 x 5

difference in CI and SI = = Rs 1.25.

Exercise 1. On a certain sum of money, the simple interest for 2 \a

is Rs 140 at the rate of 4% per annum. Find the differer.a in CI and SI. a)Rs3 b)Rsl.5 c)Rs2.8 d)Rsl.8

2. On a certain sum of money, the simple interest for 21 is Rs 160 at the rate of 5% per annum. Find the differeia in CI and SI. a)Rs4 b)Rs5 c)Rs6 d)Rs8

3. On a certain sum of money, the simple interest for 2 \a is Rs 150 at the rate of 3% per annum. Find the different in CI and SI. a)Rs5 b)Rs4.5 c)Rs2.5 d)Rs2.25

4. On a certain sum of money, the simple interest for 2 y is Rs 200 at the rate of 7% per annum. Find the differn^ in CI and SI.

a)Rs7 b)Rs6 c)Rs3.5 d) None of the!

Answers l .c 2. a 3.d 4. a


Rule 14 Theorem: When difference between the compound interest and simple interest on a certain sum of money for 2 years at r% rate is Rs x, then the sum is given by

s 2

Sum = Difference x 100x100 x{\00)2

Rate x Rate r2 = x 100

Illustrative Example Ex.: The difference between the compound interest and

the simple interest on a certain sum of money at 5% per annum for 2 years is Rs 1.50. Find the sum.

Soln: Using the above theorem:

Sum = 1.5 100 5 = 1.5 x 400 = Rs 600.

Exercise The difference between the compound interest and the simple interest on a certain sum of money at 4% per annum for 2 years is Rs 2. Find the sum. a)Rsl260 b)Rsl225 c)Rsl250 d)Rsl230

1 The difference between the compound interest and the simple interest on a certain sum of money at 5% per annum for 2 years is Rs 3. Find the sum. a)Rs600 b)Rsl200 c)Rsl400 d) Data inadequate

: The difference between the compound interest and the simple interest on a certain sum of money at 8% per annum for 2 years is Rs 4. Find the sum. a)Rs625 b)Rsl260 c)Rs312.5 d) None of these

I The difference between the compound interest and the simple interest on a certain sum of money at 10% per annum for 2 years is Rs 2.50. Find the sum. a)Rs350 b)Rs275 c)Rs250 d)Rs325

: The difference between the compound interest and the simple interest on a certain sum of money at 4% per annum for 2 years is Rs 1.40. Find the sum. a)Rs875 b)Rs857 c)Rs785 d) None of these The difference between the compound interest and simple interest on a certain sum at 5% for 2 years is Rs 1.50. The sum is [Bank PO 1987] a)Rs600 b)Rs500 c)Rs400 d)Rs300

Rule 15 keorem: On a certain sum of money, the difference be-

mKen compound interest and simple interest for 2 years at

Answers JLc 2.b

n

3.a 4.c 5.a 6. a

rate is given by sum 100

Illustrative Example Ex: Find the difference between the compound interest

and the simple interest for the sum Rs 1500 at 10% per annum for 2 years. Using the above theorem, we have Soln:

Difference = Sum r Too 1500 I_

100 Rs 15.

Exercise 1. Find the difference between the compound interest and

the simple interest for the sum Rs 1250 at 4% per annum for 2 years. a)Rs2 b)Rs2.5 c)Rsl.5 d)Rel

2. Find the difference between the compound interest and the simple interest for the sum Rs 1500 at 5% per annum for 2 years. a)Rs3.25 b)Rs7.5 c)Rs3.75 d)Noneofthese

3. Find the difference between the compound interest and the simple interest for the sum Rs 625 at 8% per annum for 2 years. a)Rs3 b)Rs4 c)Rs4.5 d)Rsl.5

4. Find the difference between the compound interest and the simple interest for the sum Rs 2500 at 6% per annum for 2 years.

a)Rs9 b)Rs8 c)Rs7.5 d)Rs6

Answers l .a 2,c 3.b 4.a

Rule 16 Theorem: If the difference between CI and SI on a certain sum for 3 years at r% is Rs x, the sum will be

Difference x(lQO)3

/-2(300 + r ) Illustrative Example Ex.: I f the difference between CI and SI on a certain sum

of money for 3 years at 5% per annum is Rs 122, find the sum.

Soln: By the above theorem:

S u m . 1 2 2 x 1 0 0 x 1 0 0 x 1 0 0 =Rs 16,000. 52(300 + 5)

Exercise 1. On what sum will the difference between the simple and

compound interests for 3 years at 5 per cent per annum amount to Rs 12.20? a)Rsl600 b)Rs800 c)Rs 1200 d)Rs 1500

2. On what sum will the difference between the simple and compound interests for 3 years at 4 per cent per annum amount to Rs3.04? a)Rsl250 b)Rs625 c)Rs650 d)Rs675

3. On what sum will the difference between the simple and


compound interests for 3 years at 6 per cent per annum amount to Rs 13.77? a)Rsl250 b)Rsl320 c)Rsl520 d)Rsl l50

4. On what sum will the difference between the simple and compound interests for 3 years at 3 per cent per annum amount to Rs 27.27?

a)Rs5000 b)Rs 10000 c)Rs8000 d) None of these

Answers l.a 2.b 3.a 4.b

Rule 17 Theorem: On a certain sum of money, the difference be-tween compound interest and simple interest for 3 years at

Sum xr2 (300 + r)

3.b 4. a

r% per annum is given by difference: (100)3

Illustrative Example Ex.: Find the difference between CI and SI on Rs 8000 for

3 years at 2.5% pa. Soln: Using the above theorem,

Difference = Sumxr 2(300 + r )

(100)3

8000x2.5x2.5(300 + 2.5) 100x100x100

8x25x25x3025 121 100x100x100

= Rs 15.125.

Exercise 1. Find the difference between the simple and compound

interest on Rs 500 for 3 years at 4 per cent. a) Rs 2.432 b)Rs 3.432 c)Rs 2.342 d)Rs 2.423

2. What is the difference between the simple and compound interest for 3 years at 5 per cent? a) 0.0007625 times the principal b) 0.07625 times the principal c) 0.007625 times the principal d) Data inadequate

3. Find the difference between the simple and compound interest on Rs 10000 for 3 years at 3 per cent. a)Rs27.8 b)Rs 27.27 c)Rs 37.27 d)Rs37.8

4. Find the difference between the simple and compound interest on Rs 8000 for 3 years at 5 per cent. a)Rs6l b>Rs60 c)Rs51 d)Rs59

Answers 1. a 2. c; Hint: Let P be the principal

Difference between CI and SI Px5x5x(300 + 5)

Rule 18 Theorem: If an amount of money grows upto Rs A, in n years and upto RsAin(n +1) years on compound interest,

then the rate per cent is given by {a2-a] )100

or Difference of amount after n years and (n + \) years x 100

Amount after n years

Illustrative Examples Ex. 1: An amount of money grows upto Rs 4840 in 2 years

and upto Rs 5324 in 3 years on compound interes: Find the rate per cent.

Soln: Detail Method: We have, P + CI of 3 years = Rs 5324 .... (1) P + CI of 2 years = Rs 4840 ... (2) Subtracting (2) from (1), we get CI of 3rd year = 5324 - 4840 = Rs 484 Thus, the CI calculated in the third year which is 484 is basically the amount of interest on the amo generated after 2 years which is Rs 4840.

484x100 4840x1

Quicker Method (Direct Formula):

Difference of amount after n years and (n +1) years Rate = Amount after n years

In this case, n = 2

Difference of amount after 2 years and 3 years i Rate = Amount after 2 years

Note:

Ex.2:

Soln:

100x100x100 -0.007625 P

= (5324-4840) x [ 0 0 = 484x100 4840 4840

The above generalised formula can be used for. positive value of n. See in the following example A certain amount of money at compound intc grows upto Rs 51168 in 15 years and upto Rs 51 16 years. Find the rate per cent per annum. Using the above formula:

(51701-51168)xlOQ _ 533x100

51168 51168

96 24 24 '

Exercise 1. A certain amount of money at compound interest i

upto Rs 6560 in 3 years and upto Rs 7216 in 4 years the rate per cent per annum, a) 10% b)5% c)8% d)6%

Rate


A certain amount of money at compound interest grows upto Rs 7520 in 15 years and upto Rs 7896 in 16 years. Find the rate per cent per annum. a) 10% b)8% c)5% d) None of these A certain amount of money at compound interest grows upto Rs 3840 in 4 years and upto Rs 3936 in 5 years. Find the rate per cent per annum. a) 2.05% b)2.5% c)2% d)3.5% A certain amount of money at compound interest grows upto Rs 4950 in 19 years and upto Rs 5049 in 20 years. Find the rate per cent per annum. a) 2% b) 2.5% c) 1 % d) 1.5% A certain amount of money at compound interest grows upto Rs 12960 in 2 years and upto Rs 13176 in 3 years. Find the rate per cent per annum.

a) % b) c) 2 | % d) - %

Answers l.a 2.c 5.b 4. a 5.d

Rule 19 Theorem: If an amount of money grows upto Rs At in n years and upto Rs A in (n + l)yearson compound interest,

tit en the sum is given by Rs

Illustrative Example Ex.: An amount of money grows upto Rs 4840 in 2 years

and up to Rs 5324 in 3 years on compound interest. Find the sum.

Soln: Applying the above theorem,

n i

KA2 J

Sum = 4840 x ( 4 8 4 0 V 1^5324 J :Rs4000.

Exercise 1. What sum of money at compound interest will amount

to Rs 650 at the end of the first year and Rs 676 at the end of the second year? a)Rs625 b)Rs630 c)Rs620 d) None of these

I What sum of money at compound interest will amount to Rs 480 at the end of the first year and Rs 576 at the end of the second year? a)Rs420 b)Rs450 c)Rs400 d)Rs375

1 An amount of money grows upto Rs 2750 in 2 years and upto Rs 3125 in 3 years on compound interest. Find the sum. a) Rs 2129.6 b)Rs 1229.6 c)Rs22I9.6 d) Data inadequate

| An amount of money grows upto Rs 1200 in 2 years and upto Rs 1440 in 3 years on compound interest. Find the sum.

a) Rs 2200

c)Rs

Answers l .a 2.c

2500

b)Rs800

d) None of these

3.a 4.c

Rule 20 Theorem: To find the ratio of Compound Interest (CI) to Simple Interest (SI) on a certain sum at r% for 2 years, we

C I _ r i

use the following formula - TTT + 1 .

Illustrative Example Ex.: Find the ratio of CI to SI on a certain sum at 4% per

annum for 2 years. Soln: Applying the above formula,

CI _ 4 - 204 _ 51 SI ~ 200 ~ 200 " 50

Exercise 1. Find the ratio of CI to SI on a certain sum at 5% per

annum for 2 years. a)41:40 b)42:41 c)43:40 d)41:35

2. Find the ratio of CI to SI on a certain sum at 8% per annum for 2 years. a)27:26 b)26:25 c)26:21 d)25:24

3. Find the ratio of CI to SI on a certain sum at 45% per annum for 2 years. a) 49:47 b)49:42 c)49:40 d) None of these

4. Find the ratio of CI to SI on a certain sum at 15% per annum for 2 years. a)53:40 b)53:50 c)43:40 d)50:43

5. Find the ratio of CI to SI on a certain sum at 10% per annum for 2 years.

a)7:5 b)21:20 c)8:5 d)20: 19

Answers l .a 2.b 3.c 4.c 5.b

Rule 21 Theorem: If a sum 'A' becomes 'B' in ttyears at compound

rate of interest, then after t,years the sum becomes ^y 2 / ' , - i

rupees.

Illustrative Example Ex.: Rs 4800 becomes Rs 6000 in 4 years at a certain rate of

compound interest. What will be the sum after 12 years?


Soln: Detail Method: We have:

4800 1 + -100

6000

Illustrative Examples Ex. 1: Find the compound interest on Rs 18,750 in 2 years,

the rate of interest being 4% for the first year and 8 | for the second year.

or, 6000 5

- T -100 J 4800 4

Now, 1 + v4x3

100 125 ~64~

or, 1 + -100 125x75 9375 64x75 4800

0 2

or, 4800 1 + 100 = 9375

The above equation shows that Rs 4800 becomes Rs 9375 after 12 years.

Quicker Method: Applying the above rule, we have

(6000)1 2/4

the required amount = ( 4 g 0 0 y 2 / 4 - i

= Rs9375

(6000)3

(4800)2

Exercise 1. Rs 2400 becomes Rs 3000 in 3 years at a certain rate of

compound interest. What will be the sum after 6 years? a)Rs4750 b)Rs3750 c) Rs 3570 d) None of these

2. Rs 1200 becomes Rs 1500 in 2 years at a certain rate of compound interest. What will be the sum after 6 years? a)Rs2433.25 b)Rs2334.75c)Rs2343.75 d)Rs2343.25

3. Rs 9600 becomes Rs 12000 in 6 years at a certain rate of compound interest. What will be the sum after 12 years? a) Rs 15000 b)Rs 14000 c)Rs 16000 d)Rs 18000

4. Rs 1600 becomes Rs 2000 in 2 years at a certain rate of compound interest. What wil l be the sum after 4 years? a)Rs2500 b)Rs2400 c)Rs2200 d) None of these

Answers l . b 2.c 3.a 4.a

Rule 22 Theorem: If the compound rate of interest for the first /,

years is rx %,for the next t2 years is r2 %,for the next t}

years is r3 %,... and the last t years is rn %, then com-

pound interest on Rs x for (/, +t2 +t3 + ...tn) years is

x 1 + -100 100 100 J -x

Soln: After first year the amount = 18750 M + 100,

1 8 7 5 0 f ^ U00;

After 2nd year the amount = 18750 104Y108 100 1100

= 18750 27 25

= Rs21060 .-. CI =21060-18750 = Rs 2310.

Ex. 2: Find the compound interest on Rs 10000 for 3 year 1 the rate of interest is 4% for the first year, 5% for 1 second year and 6% for the third year.

Soln: The compound interest on Rs x in ' t ' years i f the rae of interest is r^/o for the first year, r 2% for the secca^ year ... and r t% for the tth year is given by

100 100

\ f r \ 1 +

/ [ 100J

Note: Here, t] = t2 In this case

= 10000 (II:

c i = 10000 1 + v 100

11 20

1 + -100

53 50

11575.20-10000 = Rs 1575.2.

6 100

-1000c

-10000

Exercise 1. Find the compound interest on Rs 9375 in 2 years.!

rate of interest being 2% for the first year and 4% fa second year. a)Rs570 b)Rsl l40 c)Rsll55 d)Rs67

2. Find the compound interest on Rs 8000 in 2 yean^ rate of interest being 5% for the first year and 1C the second year. a)Rsl340 b)Rsl420 c)Rsl240 d) Nor ; these

3. Find the compound interest on Rs 3200 in 2 year^J

rate of interest being 7 j % for the first year and

for the second year. a)Rs620 b)Rs670 c)Rs770 d)Rs76(

Compound Interest 2 1 5

4. Find the compound interest on Rs 50000 for 3 years i f the rate of interest is 5% for the first year, 6% for the second year and 10% for the third year. a) Rs 10632 b)Rs 16032 c)Rs 10362 d)Rs 13062

5. Find the compound interest on Rs 80000 for 3 years i f the rate of interest is 5% for the first year, 4% for the second year and 5% for the third year. a)Rs 17128 b)Rs 11728 c)Rs 11278 d)Rs 11738

Answers l.a 2.c 3.b 4.a 5. b

Rule 23 Theorem: Certain sum of money at compound interest will

mmount to Rs A in (tx + t 2 +13 +... +1) years. If the rate of

interest for the first /, years is %,for the next t2 years is

-_ %,for the next t3 years is r 3 %... and the last tn years is

r. %, then the sum is given by

100 Y 100 Y 100

100 + r, ){\00 + r2){\00 + r3) a here t, = / , = / , = . . . = f = 1

100

100+7, n J

111 ustrative Example Ex.: What sum of money at compound interest will amount

to Rs 2249.52 in 3 years, i f the rate of interest is 3% for the first year, 4% for the second year and 5% for the third year?

Soln: Detail Method: The general formula for such ques-tion is

A = P\ + 100

1+-100

1+-100

Where, A = Amount, P = Principal, and r,, r2, r3 are the rates of interest for different years. In the above case,

2249.52= P\ + T l + - 4 1 + I O O A I O O A 100

or, 2249.52 = P( 1.03) (1.04) (1.05)

2249.59 ' P = l . 03xl.04xl.05 = R S 2 0 0 -Quicker Method: Applying the above rule, we have

1 0 0 Y 1 0 0 Y 1 0 0 ' principal = 2249.52

103 A 104 A 105 = Rs2000.

I vercise What sum of money at compound interest will amount to Rs 5305.53 in 3 years, i f the rate of interest "is 1% for the first year, 2% for the second year and 3% for the third year?

a)Rs5000 b)Rs5200 c)Rs5130 d)Rs4910 What sum of money at compound interest will amount to Rs 1365.78 in 3 years, i f the rate of interest is 2% for the first year, 3% for the second year and 4% for the third year? a)Rsl360 b)Rsl250 c)Rsl l60 d)Rsl240 What sum of money at compound interest will amount to Rs 562.38 in 3 years, i f the rate of interest is 3% for the first year, 4% for the second year and 5% for the third year? a)Rs400 b)Rs450 c)Rs500 d)Rs520 What sum of money at compound interest will amount to Rs 2893.8 in 3 years, i f the rate of interest is 4% for the first year, 5% for the second year and 6% for the third year/ a)Rs2500

Answers l .a 2.b

b) Rs 2400 c) Rs 2200 d) None of these

3.c 4. a

Rule 24 Theorem: If a man borrows Rs P at r% compound interest and pays back Rs A at the end of each year, then at the end of the nth year he should pay

Rs 1 + -100 -A 1 + -

100 +1+- -T

100 J 100 J

Illustrative Example Ex.: A man borrows Rs 3000 at 10% compound rate of

interest. At the end of each year he pays back Rs 1000. How much amount should he pay at the end of the third year to clear all his dues? Using the above formula, the required answer

Soln:

= 3000 1 + L_ 100

1000 , 1 0 V C 1 0 1 + + 1 +

100 J I 100

3 0 0 0 f l i x l i x l l U o 10 10

1000 u 10

3993- 1000x + 1 0 0 0 x 1 1 100 10

= 3993-1210-1100 = Rsl683.

Exercise 1. A man borrows Rs 1500 at 5% compound rate of inter-

est. At the end of each year he pays back Rs 500. How much amount should he pay at the end of the third year to clear all his dues?


a) Rs 680 7 7 b)Rs600 7 7 16 16

3 C)RS660T7 d) None of these

16 2. A man borrows Rs 4000 at 20% compound rate of inter-

est. At the end of each year he pays back Rs 1500. How much amount should he pay at the end of the third year to clear all his dues? a)Rs2592 b)Rs2852 c)Rs2952 d)Rs2953

3. A man borrows Rs 3000 at 30% compound rate of inter-est. At the end of each year he pays back Rs 1000. How much amount should he pay at the end of the third year to clear all his dues?

a)Rs3602 b)Rs3601 c)Rs3603 d)Rs3604

Answers l .c 2.c 3.b

. Rule 25 Theorem: If a sum of money say Rs x is divided among it parts in such a manner that when placed at compound in-terest, amount obtained in each case remains equal while

the rate of interest on each part is r,, r2, r 3 , r n respec-

tively and time period for each part is t]t t2, tlt tn respectively, then the divided parts of the sum will in the ratio of

1 1 1 1

r 1 + A f : f 1 + i f : f , t i Y' :~y I + i . f . t 100 J t 100 J ^ 100J t 100 J

Illustrative Example Ex.: Divide Rs 3903 between A and B, so that A's share at

the end of 7 years may equal B's share at the end of 9 years, compound interest being at 4%.

Soln: Applying the above theorem, A's share : B's share =

B's present share = Rs 3903 - Rs 2028 = Rs 1875.

Exercise 1. Divide Rs 2708 between A and B, so that A's share at the

end of 6 years may equal B's share at the end of 8 years, compound interest being at 8%. a) Rs 1458, Rs 1250 b) Rs 1448, Rs 1260 c)Rs 1438, Rs 1270 d)Rs 1468, Rs 1240

2. Divide Rs 1105 between A and B, so that A's share at the end of 5 years may equal B's share at the end of 7 years, compound interest being at 10%. a)Rs505,Rs600 b) Rs605, Rs 500 c)Rs705,Rs400 d)Rs625,Rs480

3. Divide Rs 6100 between A and B, so that A's share at the end of 3 years may equal B's share at the end of 5 years, compound interest being at 20%.' a)Rs3600,Rs2500 b)Rs 3500, Rs 2600 c)Rs 3400, Rs 2700 d) Rs 3450, Rs 2650

Answers l .a 2.b 3.a

Miscellaneous 1. The difference between the simple and the compour..:

interest compounded every six months at the rate of I I percent per annum at the end of two years is Rs 12415 What is the sum? [SBI PO Exain, 20

Compound Interest

9.

become more than double in amount i f put out at com-pound interest at the rate of 10% per annum, a) 6th year b) 7th year c) 8th year d) Data inadequate A sum of money put out at compound interest amounts in 2 years to Rs 578.40 and in 3 years toRs 614.55. Find the rate of interest.

a) 6% 1 1

10.

11.

12.

b ) 6 - % c) 6-0/0 d ) 6 - % 4 2 4

Divide Rs 3903 between A and B, so that A's share at the end of 7 years may equal to B's share at the end of 9 years, compound interest being at 4 per cent. a)Rs2028,Rsl875 b) Rs 2018, Rs 1885 c) Rs 2008, Rs 1895 d) Rs 2038, Rs 1865 Vijay obtains a loan of Rs 64,000 against his fixed depos-its. I f the rate of interest be 2.5 paise per rupee per an-num, calculate the compound interest payable after 3 years. a)Rs4921 b)Rs5020 c)Rs4821 d)Rs4920 A certain sum is interested at compound. The interest accrued in the first two years is Rs 272 and that in the first three years is Rs 434. Find the rate per cent.

a) 1 2 ^ % b) 7 - %

Answers 1. d; Let the sum be Rs x.

Then x 1 + -100

c) 1 7 1 %

x x l 0 x 2 100

d)25%

124.05

Solving the above eqn, we get x = Rs 8000

900x100 2. c; Certain sum for the person ;

.-. interest on Rs 5000 by CI ( c \

: Rs 5000

= 5000 1 + -100

-5000 =R S 955.08

.-. More interest = Rs (955.08 - 900) = Rs 55.08. 3. c; The amount of Rs 100 in one year at compound inter-

est at 5% per annum payable half-yearly

= Rsl00 1 +

5 > 2_

100 = Rs 100(l.025)2 =Rs 105.0625

Thus the nominal rate of 5% payable half-yearly has the same effect as the rate 5.0625 per cent would have if payable yearly. Hence 5.0625 per cent is calle the effective annual rate of 5% per annum payable half-yearly

4. a; Effective annual rate

5.b;

' 4 ' 4

= 100 1 + - * - -100 100

^ J

104.0604-100 = 4.0604%

V 1 + -

100 A (See Rule-1)

390625 l + I =456976 100,

1 + 4 V Arcane. (i/;A4

100

' 2 6 V 26 25

456976 390625

4

26

.25;

t = 4 25 J

the required time is 4 years

20 6.c; HereP| l + I > 2P

' 6 V or, > 2

6 6 6 6 By trial ^ X ^ " X " J X J > 2

.-. the required time is 4 years.

f 4 f

7.b; 62501 1+-^- 63:.:' 100

or,

or,

1 + -

' 5 0 2 T

V 50y

J

2 ) 21 663255 132651 100 y 625000 12500 Uo

or,2t = 3

t = years

10 8.c; HereP| l + | > 2P

or, > 2


l l x l l x l l x l l x l l x l l x l l x l l Bv trial > 2

* 10x10x10x10x10x10x10x10 Hence, the first year in which a sum of money will become more than double in amount is 8th year.

9. b; Clearly, the difference between Rs 578.40 and Rs 614.55 is the interest on Rs 578.40 for 1 year. .-. interest on Rs 578.40 for 1 year = Rs 614.55 - Rs 578.40 = Rs 36.15

interest on Rs 100 for 1 year = Rs 36.15 x 100

578.40

.-. the required rate is 6 per cent.

3615 100 _ A = R S x = Rs 6

57840 1 4 1

10. a; We have at once

(A's present share) 100 J

= (B's present share) 1 + -T 100 )

A's present share B's present share 100 J 1,25

676 625

Dividing Rs 3903 in the ratio 676 : 625

.-. A's present share: 676

of Rs 3903 676 + 625

= Rs2028 .-. B's present share = Rs 3903 - Rs 2028 = Rs 1875

11.a; P=Rs64000 r = 2.5 paise per rupee per annum (given)

= 0.025 rupee per rupee per annum. = 0.025 x 100 rupee per hundred rupee per annum. = 0.025 x 100 per cent per annum = 2.5 per cent per annum

t = 3 yrs

C I = 64000 1 + 0.025x100

100

6400o[(l + 0.025)2 -1 ] = Rs 492

.-. The compound interest payable is Rs 4921 Note: Remember that x paise per rupee per annum = Rs x per

cent per annum.

12. a; Amount A = P 1 + -100

CI = P 1+-100

Putting ' + 100 J = 1 m t n e a bo v e equation,

.-. fortwoyears,t = 2,then,P [ ^ 2 - l j = 2 7 2 (i)

For three years P [g-3 - l j =434 (ii)

Dividing (ii) by (i)

(q2 +q + \)(q-\) 434 (q + \)(q-\) 272

or, ql + q + l _ 217

=> t l

q + l 136 q + \6

or, 136?2 +81^ + 81 = 0

or, q =

r 9 1 1 .-. 1 + = - = > r = - x l 0 0 % = 1 2 - %

100 8 8 2 '

Documents

Chapter 12