Chapter 13

Problems Based on Instalment

Introduction 1. Instalment Purchase Schemes: To increase their

sales, different business organisations offer different schemes to the buyers to enable them to buy costly articles even i f their income is very low. One such scheme is the usialment scheme. The cost of the article bought is paid in stalments.

2. In instalment scheme the buyer has to pay more aecause in addition to the selling price, the buyer has to pay

interest also on it. 3. Cash Price: The amount for which the article can

x purchased on full payment is called cash price. 4. Cash Down Payment: While purchasing an article

mier instalment scheme, some payment has to be made ini-mSh. It is called cash down payment. The remaining amount B paid in equal monthly, quarterly or annual instalements as av be decided at the time of purchase. Case-I: Based on Simple Interest

When the period over which the instalment scheme s operative is less than a year. The payment is made in a specified number of equal monthly instalments. Naturally we calculate the simple interest in such cases. T\e I: Monthly instalment being given, to find the rate of

interest.

Illustrative Examples Ei . 1: A coat is sold for Rs 60 or Rs 20 cash down payment

and Rs 8 per month for 6 months. Determine the rate of interest.

Soln: Cash price of the coat = Rs 60 Cash down payment = Rs 20 Amount paid in 6 instalments = Rs (8 x 6) = Rs 48 Total amount paid under instalment plan

= Rs20 + Rs48 = Rs68 Interest charged = Rs 68 - Rs 60 = Rs 8 Now we find the principals for each of the six months. Principal for the first month = Rs 60 - Rs 20 = Rs 40 Principal for the second month = Rs 40 - Rs 8 = Rs 32 Principal for the third month = Rs 32 - Rs 8 = Rs 24 Principal for the fourth month = Rs 24 - Rs 8 = Rs 16

Principal for the fifth month = Rs 16 - Rs 8 = Rs 8 Principal for the sixth month = Rs 8 - Rs 8 = Rs 0 .-. Total Principal = Rs 120

Therefore, interest on Rs 120 for 1 month or year 12 '

is Rs 8.

Rate % = 57x100 8x100

PxT 120x 12

8x100x12 = 80%

120x1 Hence the rate o f interest is SO%.

Ex2: A television is marked at Rs 3575 cash or Rs 1500 as cash down payment and Rs 420 a month for 5 months. Find the rate of interest for this instalment plan.

Soln: Cash price = Rs 3 575 Cash down payment = Rs 1500 Amount paid in 5 monthly instalments

= Rs(420x5) = Rs2100 .-. Total amount paid under instalment plan

= Rs 1500 + Rs2100 = Rs3600 .-. Interest charged = Rs 3600 - Rs 3575 = Rs 25 The principal for each month is as under: Principal for the lstmonth =Rs3575-Rs 1500

= Rs2075 Principal for the 2nd month = Rs 2075 - Rs 420

= Rsl655 Principal for the 3rd month = Rs 1655 - Rs 420

= Rs 1235 Principal for the 4th month = Rs 123 5 - Rs 420

= Rs815 Principal for the 5th month = Rs 815 - Rs 420

= Rs395 Total Principal = Rs 6175 The final (sixth) instalment of Rs 420 consists of the amount of Rs 395 and the interest of Rs 25 (Rs 395 + Rs25 = Rs420).

3 1 0 PRACTICE BOOK ON QUICKER MATHS

Thus, the interest on Rs 6175 for 1 month or year

is Rs 25.

PxTxR 1 =

Ex 4:

Soln: 100

7x100 R(Rate)= -j^r

25x100

6175x 1_ 12

25x100x12 1200 = 4.86%

6175x1 247 Hence the rate of interest is 4.86%.

Ex3: A house is sold for Rs 30000 cash or Rs 17500 as cash down payment and instalments of Rs 1600 per month for 8 month. Determine the rate of interest correct to one decimal place, under the instalment plan>

Soln: Cash price = Rs 30000 Cash down payment = Rs 17500 Total amount paid in 8 monthly instalments

= Rs(1600x8) = Rsl2800 Total amount paid under instalment plan

= Rs 17500 + Rs 12800 = Rs 30300 Interest charged = Rs 30300 - Rs 30000 = Rs 300 Principal for 1 st month = Rs 30000 - Rs 17500

Principal for 2nd month

Principal for 3rd month

Principal for 4th month





Rs12500 Rs 12500-Rs 1600 Rs10900 Rs10900-Rs 1600 Rs9300 Rs 9300-Rs 1600 Rs7700 Rs 7700-Rs 1600 Rs6100 Rs6100-Rsl600 Rs4500 Rs 4500-Rs 1600 Rs2900 Rs 2900-Rs 1600 Rsl300

Hence the rate of interest is 6.5% A radio is available for Rs 950 cash or Rs 200 cash down payment and 10 monthly instalments of Rs 80 each. Find the rate of interest charged. Cash price = Rs 950 Cash down payment = Rs 200 .-. Amount paid in 10 monthly instalments = Rs (80 x 10) = Rs800 .-. Total amount paid under instalment plan =Rs800 + Rs 200 = Rs 1000 .. Interest charged = Rs 1000 - Rs 950 = Rs 50 Principal for 1 st month = Rs 950 - Rs 200 = Rs 750 Principal for 2nd month = Rs 750 - Rs 80 = Rs 670 Principal for 3rd month = Rs 670 - Rs 80 = Rs 590 Principal for 4th month = Rs 590 - Rs 80 = Rs 510 Principal for 5th month = Rs 510 - Rs 80 = Rs 430 Principal for 6th month = Rs430-Rs80 = Rs350 Principal for 7th month = Rs 350 - Rs 80 = Rs 270 Principal for 8th month = Rs 270 - Rs 80 = Rs 190 Principal for 9th month = Rs 190 - Rs 80 = Rs 110 Principal for 10th month = Rs 110 - Rs 80 = Rs 30

Total Principal for 1 month = Rs 3900 The last instalment of Rs 80 comprises Rs 30 plus Rs 50 interest.

Time = 1 month : l_ 12

year, I = Rs 50

7 = PxTxR

100 R(Rate) : 7x100 PxT

Rate = 50x100 50x100x12 2oo = 1 5 A 3900x 12

3900 13 13

Total Principal = Rs 55200 The last instalment of Rs 1600 includes Rs 1300 plus Rs 300 interest.

1 Time = 1 month = year, Interest = Rs 300 12

7 = PxTxR

100 R (Rate)

300x100 300x100x12

55200x 12

55200

7x100 PxT

_ 150 23

Hence the rate of interest is 15% 13

Type 2: Rate of interest being given, to find the monthh instalment.

I l lustrative Examples Ex. 1: A pocket transistor is sold for Rs 125 cash or for Ri

26 as cash down payment followed by 4 equal monthh instalments. I f the rate of interest charged is 25% per annum, determine the monthly instalment.

Soln: Cash price of the transistor = Rs 125 Cash down payment = Rs 26 Balance of price due = Rs 125 - Rs 26 = Rs 99 Rate of interest charged = 25% p.a. Interest on Rs 99 for 4 months

6.52 99x x25 = Rs 12

100 Rs = 7^8.25

4

14 PRACTICE BOOK ON QUICKER MATHS

225x100x12 600 = 54.55

4950x1 11 Hence the rate of interest is 54.55% pa

(ii) Cash price of the refrigerator = Rs 3580 Cash down payment = Rs 1500 .-. Balance to be paid = Rs 2080 Monthly instalment = Rs 440 .-. Amount paid in 5 equal monthly instalments = Rs (440x5) = Rs2200 .-. Interest charged = Rs 2200 - Rs 2080 = Rs 120 Principal for 1st month = Rs 2080 Principal for 2nd month = Rs 2080 - Rs 440 = Rs 1640 Principal for 3rd month = Rs 1640 - Rs 440 = Rs 1200 Principal for 4th month = Rs 1200 - Rs 440 = Rs 760 Principal for 5th month = Rs 760 - Rs 440 = Rs 320 Total principal for 1 month = Rs 6000 Thus, Rs 120 is the interest on Rs 6000 for 1 month or

^ y e a r

R = 7x100 120x100 120x100x12 PxT 6000 x

12 600x1

24

Hence the rate of interest is 24% pa (iii) Cash price of the typewriter = Rs 3600

Cash down payment = Rs 1200 .-. Balance to be paid = Rs 3600 - Rs 1200 = Rs 2400 Monthly instalment = Rs 280 .-. Amount paid in 10 equal monthly instalments = Rs (280 x 10) = Rs2800 .-. Interest charged = Rs 2800 - Rs 2400 = Rs 400 Principal for 1 st month = Rs 2400 Principal for 2nd month = Rs 2400 - Rs 280 = Rs 2120 Principal for 3rd month = Rs 2120 - Rs 280 = Rs 1840 Principal for 4th month = Rs 1840 - Rs 280 = Rs 1560 Principal for 5th month = Rs 1560 - Rs 280 = Rs 1280 Principal for6th month = Rs 1280-Rs280 = Rs 1000 Principal for 7th month = Rs 1000 - Rs 280 = Rs 720 Principal for 8th month = Rs 720 - Rs 280 = Rs 440 Principal for 9th month = Rs 440 - Rs 280 = Rs 160 Principal for 10th month = Rs 160 - Rs 280=- Rs 120 Ignore the negative principal .-. Total principal for 1 month = Rs 11520 Thus, Rs 400 is the interest on Rs 11520 for 1 month

1 or year

12

7? = 7x100 400x100 PxT

400x100x12 11520x1

11520x

125 3

12

= 41.67

Hence, the rate of interest is 41.67% pa (iv) Cash price of the tape recorder = Rs 1600

Cash down payment = Rs 300 Balance to be paid = Rs 1600-Rs 300 = Rs 1300 Monthly instalment = Rs 175

Amount paid in 8 equal monthly instalments = Rs (175 x8) = Rs 1400 .-. Interest charged = Rs 1400-Rs 1300 = Rs 100 Principal for 1 st month = Rs 1300 Principal for 2nd month = Rs 1300-Rs 175 = Rs 1125 Principal for 3rd month = Rs 1125- Rs 175 = Rs950 Principal for 4th month = Rs 950 - Rs 175 = Rs 775 Principal for 5th month = Rs 775 - Rs 175 = Rs 600 Principal for 6th month = Rs 600 - Rs 175 = Rs 425 Principal for 7th month = Rs 425 - Rs 175 = Rs 250 Principal for 8th month = Rs250-Rsl75 = Rs75 Total principal for 1 month = Rs 5500 Thus, Rs 100 is the interest on Rs 5500 for 1 month or

12 year

7? = 7x100 100x100 PxT 5500x

12

100x100x12 240 = 21.81

5500x1 11 Hence, the rate of interest charged is 21.8% pa

9. Cash price of the article = Rs 100 Cash down payment = Rs 10 .-. Balance to be paid = Rs 100 - Rs 10 = Rs 90 Rate of interest charged = 48% pa

Interest on Rs 90 for 5 months or year is 12

90x x48 = Rs 12

100 ; R s 90x5x48 = R s l g

12x100 Amount due = Rs 90+ Rs 18 = Rs 108 ....(i) Let the monthly instalment be x rupees .-. At the end of 5th month: 1st instalment of Rs x wil l amount to

Rs x + xx x48

12 100

J A = P + I = P + P x T x R

100

= Rs 4x 29x

x + =Rs 25 J 25

Problems Based on Instalment 315

2nd instalment of Rs x wil l amount to

Rs x +

3 ^ x x x48

12 100

= Rs x + 3x 25 J

= Rs

J 3rd instalment of Rs x wil l amount to

Rs x +

2 ^ xx x48

12 100

Rs 2*"|

28* 25

27x x + =Rs . 2 5 j 25

) 4th instalment of Rs x wi l l amount to

Rs x +

1 * xx x48

12 100

= Rs x + 25

= Rs 26* 25

5th instalment of Rs * wil l amount to

Rs * + -* + 0x48 > l

100 Rsx

Total amount of 5 instalments at the end of 5 months

= Rs

Rs

29* 28* 27* 26* + + + +* 1, 25 25 25 25 29* + 28* + 27* + 28* + 25*

25

27* 135* = R S - 2 T = R S 5 ....(ii)

From (i) and (ii), we get

27* = 108

108x5 * = = 20

5 27 Each instalment = Rs 20

Type 3: To find the annual payment to discharge a debt if the rate per cent is given.

Theorem: The annual payment that will discharge a debt ofRs A due in t years at the rate of interest r% per annum is

\QQA

Illustrative Example Ex.: What annual payment wil l discharge a debt of Rs 770

due in 5 years, the rate of interest being 5% per an-num?

Soln: Detail Method: Let the annual payment be P rupees. The amount of Rs P in 4 years at 5%

100P + 4x5P 120P 100 100

The amount of Rs P in 3 years at 5% =

The amount of Rs P in 2 years at 5% :

The amount of Rs P in 1 year at 5% =

115P 100

nop 100

105 P 100

These four amounts together with the last annual payment of Rs P wil l discharge the debt of Rs 770.

120P 115P HOP 105P 100

550 P 100

770

100 100 -+P = 770

100 770x100

P = 550

= 140

Hence, annual payment = Rs 140 Quicker Method: Applying the above theorem, we have,

Annual payment 100x5 +

100x770 5x5(5-1)

fa 140

Exercise 1. What annual instalment wil l discharge a debt of Rs 2210

due in 4 years at 7% simple interest? a)Rs450 b)Rs500 c)Rs550 d)Rs575

2. What quarterly payment will discharge a debt of Rs 2120 in one year at 16% per annum simple interest? a)Rsl000 b)Rs400 c)Rs850 d)Rs500

3. What annual payment will discharge a debt of Rs 193 5 0 due 4 years hence at the rate of 5% simple interest? a)Rs4600 b)Rs3500 c)Rs4500 d)Rs4550

4. Find the annual instalment that will discharge a debt of Rs 12900 due in 4 years at 5% per annum simple interest. a)Rs3500 b)Rs2500 c)Rs3000 d)Rs3200

5. Find the annual instalment that wil l discharge a debt of Rs 5400 due in 5 years at 4% per annum simple interest. a)Rsl200 b)Rsl000 c)Rs800 d)Rsl050

6. What quarterly payment wil l discharge a debt of Rs 2280 due in two years at 16% per annum simple interest? a)Rs500 b)Rs450 c)Rs550 d)Rs250

(Bank PO Exam 1989) 7. What annual payment wi l l discharge a debt of Rs 580

due in 5 years, the rate being 8% per annum? a)Rs 166.40 b)Rsl20 c)Rsl00 d)Rs65.60

Answers 1. b; Hint: Required annual payment

100x2210 100x2210

100x4 + 7x4(4-1) 442

= Rs500


2. d; Hint: Here instalment is quarterly. Hence from the 16 . 0 /

question, we have, t = 4 and r = - 4 /o

Now applying the given rule, 100x2120 100x2120

required answer 100x4 +

= Rs 500. 3. c; Hint: Required answer

100x19350

4 x 4(4 - 1 ) 424

100x19350

100x4 + 5x4x3 430

= Rs4500.

4. c; Hint: Required answer

100x12900 100x12900

100x4 + 5x4(4-1) 430

= Rs 3000

5. b; Hint: Required answer

100x5400 100x5400

100x5 + 4 x 5 x ( 5 - l ) 540

:Rsl000.

6. d; Hint: Here, t = 8 and r = 4% [Since, payment is quar-terly for 2 years] Now, applying the given rule, we have the required answer

100x2280 2280x100

100x8 + 8 x 7 x 4 912

= Rs 250.

7. c; Hint: Required answer

100x580 100x580

100x5 + 8 x 5 x ( 5 - l ) 500x580 = Rsl00

Case - 2: Based on Compound Interest

The problems of money lending in which the pay-ment is made in instalments and the range normally is in years. In such cases compound interest computations are used. Type I: To find each instalment when the instalments are

equal Theorem: A sum of Rs P is to be paid back in n equal annual instalments. If the interest is compounded annually atR% per annum, then the value of each instalment is given

by 100

IQ0 + R 100

100 + /?

2 / + . . .+

V

100 100 + /?

Illustrative Example Ex: A sum of Rs 3 310 is to be paid back in 3 equal annual

instalments. How much is each instalment i f the inter-est is compounded annually at 10% per annum.

Soln: First Method: Let each equal annual instalment be Re 1. .-. The 1st instalment is paid after a year .-. Principal of the 1st instalment

: R e l x ! K R e ^ 11 11

A = P\l + 100

100 J 10

P = Ix 10 17

Similarly, the principal of 2nd instalment

A 2 Relx Re

100 121

The principal of 3rd instalment

Relx 10

Re 1000

! 1331

Total of the three principals

= Rs (10 00 1000 v l l + 121 + 1331

1210 + 1100 + 1000 3310 = Rs = Rs-

1331

3310

1331

When the principal is Rs , f .

each instalment = Re 1 When the principal is Re 1 each instalment

D i 1 3 3 1 = Relx 3310

When the principal is Rs 3310 each instalment 1331 = Rs -x3310 = Rsi33i 3310

.-. Each instalment = Rs 1331 Second Method: Let each equal annual instalment be Rs x and Pl,P2,Pi, be respectively the principals for the three instalments. The first instalment is paid after a year

.-. Principal (Pl) of the first instalment

10 \0x xx- 11 11


Similarly p2 = x

i n Now Px + P2 + P3

i.e.

10

3310

lOx 2 ho]

+ x + X 11 01; O U

10^ 11

, 10 fio 1 + +

11 \

3310

= 3310

TioY, 10 100 x 1 + + U i A i i 121

3310

/ 10Y121 + 110 + 1001

10Y33] U U U 2 1

121

= 3310

= 3310

11 121 x = 3310x x 10 331

1331

Hence the required annual instalment = Rs 1331 Quicker Method: Applying the above theorem, we have the

3310 required annual instalment = JQ JQQ JQQQ"

TT + T 2 T + 1331

_ 3310x1331 3310

/fr 1331-

T*pe II : To find the Principal when each instalment is given. TWorem: A man borrows some money on compound inter-

znd returns it in t years in n equal instalments. If the rate Wmterest is R% and the yearly instalment is Rs X, then the

nt borrowed is given by

X 100

100 + /? 100

Uoo+/? 100

Uoo+/?j

: 1. To find the total interest charged we use the follow-ing formula,

; X ( 100 >

* f 100 " 2 f 100 ^

[ l 0 0 + rt; * ^ 100 + /?; + ....+ V100 + /?J 2. To calculate the principal and interest charged with

each instalment following formula is used.

Principal: J 100 >

Forthe lstyear= (ioo + /?

For the 2nd year = RsX\

For the tth year = X

Interest Charged:

( loo V Uoo+/?,

( loo V 100 + /?

Interest in 1st instalment = ^* * 100

Interest in 2nd instalment - Rs X 1 -

100 + /?,

100 100 + /?,

Interest in nth instalment = ^s r 100

100 + /?,

Illustrative Example Ex: A man borrowed some money and paid back in 3 equal

annual instalments of Rs 2160 each. What sum did he borrow, i f the rate of interest charged by the money lender was 20% per annum compounded annually? Find also the total interest charged. Also calculate the principal and interest charged with each instal-ment.

Soln: Detail Method Amount of each annual instalment = Rs 2160 Rate of interest = 20% p.a. Number of instalments = 3

Principal for the 1st year = Rs 2160

1 + 20 100

= Rs 2160x- = Rs 1800 6

t ' r \ v A = H 1+ \ IOOJ

\ 2 0 > \ 2160 = p 1 +

I 100, Principal for the 2nd year

=fa2160x 25

- 1 =/?s2160x = /?il500 K6) 36

Principal forthe 3rd year

/ & 2 1 6 0 x =/?5l250 216

- Rs 2160


Amount borrowed = Sum of the principals for all the three years = Rs (1800 + 1500 + 1250) = Rs 4550 Total interest charged

= Total amount of the three instalments - Amount borrowed

= Rs (2160 * 3 - 4550) = Rs (6480 - 4550) = Rs 1930 Interest in 1st instalment

= 1st instalment - Principal for 1st instalment = Rs (2160-1800) = Rs 360

Interest in 2nd instalment = 2nd instalment - Principal for 2nd instalment = Rs (2160-1500) = Rs 660

Interest in 3rd instalment = 3rd instalment - Principal for 3rd instalment = Rs(2160-1250) = Rs910

Quicker Method: Applying the above theorem, we have (i) Amount borrowed

2160

2160

100 100 ( 100 100+20 UOO+20) V100 + 20

5 25 125 + + 6 36 216

= 1800+ 1500+ 1250 = Rs4550

(ii) Total interest charged = 2160

= Rs (6480- 4550) = Rs 1930

, .5 25 125 3 H - + +

6 36 216

(iii) Interests - in

1st instalment = 2160| 1 - - = Rs 360

2nd instalment - 2 1 6 0

3rd instalment = 2 1 6 0

(iv) Principal for the

I -

1 -

25

36.

125 216

Rs 660

= Rs9lO

lstyear= Rs 2160x- = Rs 1800 6

25 2nd year = Rs 2160x = Rs 1500

36

125 3rd year = Rs 2\60x~ = Rs 1250

Type III: To find each instalment when the instalments are not equal.

Theorem: A sum of Rs P is to be paid back in n annual instalments. If the interest is compounded annually on the

balance at R% and is to be included in each instalment, then the value of each instalment is given by the following,

Instalment at the end of 1 st year = Rs

Instalment at the end of 2nd year = Rs

Instalment at the end of 3rd year = Rs

Instalment at the end of 4th year = Rs

1 +

1 +

1 +

Rn 100

100

R(n-2) 100

R(n-3)~ 100

Instalment at the end of nth year = Rs 1 + -100

Note: Number of instalments = no. of years. Now, we can alternatively write the above theorem as follows. Value of instalment at the end of required year

Sum which is to be paid back No. of instalments

1 +

(Rate per cent) (no. of instalments - one less the year after which instalment is payable)

100

Illustrative Examples Ex.1: A sum of Rs 7500 is to be paid back in 3 annua,

instalments. How much is each instalment, i f the in-terest is compounded annually on the balance at 4 t and is to be included in each instalment.

Soln: The loan is to be paid in 3 annual instalments. .-. Each instalment will be of Rs (7500- 3) or Rs 2500 together with the interest on the balance for 1 year Amount payable at the end of 1 st year

= Rs 2500 + 4% of Rs 7500

= Rs2500 + Rs - i - x 7 5 0 0 100

= Rs 2500 + Rs 300 = Rs 2800 Balance at the end of first year

= Rs 7500 - Rs 2500 = Rs 5000 .-. Amount payable at the end of 2nd year

= Rs2500 + 4%ofRs5000

= Rs2500 + Rs *5000

= Rs 2500 + Rs 200 = Rs 2700


Balance at the end of 2nd year = Rs 5000 - Rs 2500 = Rs2500 .-. Amount payable at the end of 3rd year

= Rs 2500 + 4% of Rs 2500

4 = Rs 2500 + Rs x 2500

= Rs 2500 + Rs 100 = Rs 2600 Hence the three instalments are Rs 2800, Rs 2700 and Rs2600 Quicker Method: Applying the above theorem, we have Amount payable at the end of 1 st year

7500 1 + 4 x ( 3 - 0 ) 100 [see note]

= Rs2800

Amount payable at the end of 2nd year

75001~. 4 x ( 3 - l ) 1 + -100 Rs2700

Amount payable at the end of 3rd year

7500 1 + 4(3-2) 25

= 2500 + 100 = Rs 2600 Ex 2: A loan of Rs 12000 is to be paid back in 6 annual

instalments. How much is each instalment i f the interest is compounded annually on the balance at 5% and is included in each instalment?

Soln: Detail Method: The loan is to be paid in 6 annual instalments. .-. Each instalment wil l be ofRs 12000 + 6 or Rs 2000 together with interest on the balance for 1 year. Amount payable at the end of 1st year

= Rs 2000 + 5% of Rs 12000

= Rs2000 + Rs 12000

= Rs 2000 + Rs 600 = Rs 2600 Balance at the end of first year

= Rs 12000 - Rs 2000 = Rs 10000 Amount payable at the end of 2nd year

= Rs 2000 +5% ofRs 10000 5

= Rs2000 + Rs -x10000 100 = Rs 2000+ Rs 500 = Rs 2500

Balance at the end of 2nd year = Rs 10000 - Rs 2000 = Rs 8000

Amount payable at the end of 3rd year = Rs 2000+ 5% ofRs 8000

= Rs2000 + Rs 7 ^ * 8 0 0 0

= Rs 2000 + Rs 400 = Rs 2400 Balance at the end of 3rd year

= Rs 8000 - Rs 2000 = Rs 6000 Amount payable at the end of 4th year

= Rs 2000 + 5% ofRs 6000

= Rs2000 + Rs ] 0 ^ x 6 0 0 0 = Rs 2000 + Rs 300 = Rs 2300

Balance at the end of 4th year = Rs 6000 - Rs 2000 = Rs 4000

Amount payable at the end of 5th year = Rs 2000+ 5% ofRs 4000

= Rs2000 + Rs j ^ x 4 0 0 0 = Rs 2000 + Rs 200 = Rs 2200

Balance at the end of 5th year = Rs 4000 - Rs 2000 = Rs 2000

Amount payable at the end of 6th year = Rs 2000+ 5% ofRs 2000

= Rs2000 + Rs T 5 o " x 2 0 0 0 = Rs2000 + Rsl00 = Rs2100

Hence the six instalments are Rs 2600, Rs 2500, Rs 2400, Rs 2300, Rs 2200, Rs 2100. Quicker Method: Applying the given rule we have, Amount payable at the end of 1 st year

12000 1 + 5x (6 -0 ) 100

= 2000 + 2 0 0 0 X 3 = Rs 2600 10

Amount payable at the end of 2nd year

12000 1 + 5 x ( 6 - l ) 100

= 2000 +500 = Rs 2500 Similarly we can find the remaining instalments as Rs 2400, Rs 2300, Rs 2200 and Rs 2100.

Type IV: To find cash price when different instalments are given.

Theorem: A person buys an item on the terms that he is required to Rs P cash down payment followed by Rs x at the end offirst year, Rsy at the end of second year and Rs z at the end of third year. Interest is charged at the rate ofR% per annum, then the

(i) Cash price of the item is given by ,2"

Rs P + -100

00

x + y 100

+ 2 100

\Q0+R \100 + RJ {100+R, The total interest charged is given by

Rs[P + x + y + z- Cash Price]

and


Let each instalment be Rs x Rate = 16% pa - 8% half-yearly .-. Principal for the 1 st instalment at the end of 1 st

half yearly = Rs g 1 + -

100

v A = P] 1 + V

x = P 1 + 100 J

100 ( 2 5 )

Similarly, Principal forthe 2nd instalment

' 2 5 ^

J5_ 100

= Rs 27

25 Principal for the 3rd instalment = Rs | ~ I x

Total principal for the three instalments

Rs

It should be equal to Rs 50725

25 '25> 2 | '25> 3 x + x + \ X 27 ^27; \ V27J

1 2 5 1 + +

27

f25> 2~

{llj = 50725

25 27'

25 f, 25 625'. e n ^ c x 1 + + =50725 27 I 27 729,

25 ("729 + 675 + 625 27 % 729

= 50725

25 2029 M W => xx = 50725

27 729 50725x27x729 , ^ 0 0

=> x = = 19683 25x2029

.-. Each instalment = Rs 19683 The sum is to be paid back in 4 annual instalments. .-. Each instalment will be ofRs (5600 * 4) ie Rs 1400 together with interest on the balance for one year .-. Amount payable at the end of 1 st year

= Rs 1400+ 8% ofRs 5600

= Rs 1400 + Rs I 5 6 0 0 X 75 0 " = Rs 1400+ Rs 448 = Rs 1848

Balance at the end of 1st year = Rs (5600 -1400) = Rs 4200

Amount payable at the end of 2nd year = Rs 1400+ 8% ofRs 4200

Rs 1400 + Rs _8_ 100

x4200

= Rs 1400+ Rs 336 = Rs 1736 Balance at the end of 2nd year

= Rs (4200 -1400) = Rs 2800 Amount payable at the end of 3rd year

= Rs 1400+ 8% ofRs 2800

Rs 1400 + Rs _8_ 100

x2800

= Rs 1400 + Rs 224 = Rs 1624 Balance at the end of 3rd year

= Rs(2800-1400) = Rs 1400 .-. Amount payable at the end of 4th year

= Rs 1400+ 8% ofRs 1400

Rs 1400 + Rs 8

xl400 U00

= Rs 1400 + Rs 112 = Rs 1512 Hence the four instalments are Rs 1848, Rs 1736, Rs 1624 and Rs 1512 The sum is to be paid back in 3 annual instalments .-. Each instalment will be ofRs (6000 + 3), ie Rs 2000 together with interest on the balance for one year .-. Amount payable at the end of 1 st year

= Rs 2000 + 10% ofRs 6000 ' 10

= Rs 2000 + Rs -x6000 .100 ) = Rs 2000 + Rs 600 = Rs 2600

Balance at the end of 1 st year = Rs (6000 - 2000)=Rs 4000

.-. Amount payable at the end of 2nd year = Rs 2000 + 10% ofRs 4000

= Rs2000 + Rs 10 ^

x 4 0 0 0 100

= Rs 2000 + Rs 400 = Rs 2400 Balance at the end of 2nd year

= Rs 4000 - Rs 2000 = Rs 2000 .-. Amount payable at the end of 3rd year

= Rs 2000+10% ofRs 2000

= Rs 2000 + Rs 10

100 x2000

= Rs 2000 + Rs 200 = Rs 2200 Hence the three instalments are: Rs 2600, Rs 2400 and Rs2200.


15. 16.

17.

.-. Total principal = Px + P2 = Rs 1450+ Rs 1250 = Rs 2700

Total amount paid = Rs (1682 * 2) = Rs 3364 .-. Total interest = Rs 3364 - Rs 2700 = Rs 664 Interest charged with first instalment

= Rs 2700x =Rs432 100

Interest charged with second instalment = Rs664-Rs432 = Rs232

See the solution of Q. No. 12. The quarterly instalment paid at the end of 1 st, 2nd and 3rd quarter = Rs 4630.50

.-. Principal (Px) for the first quarter

= 4630.50- 1+-

-Rs

100

4630.50 + V 100

: R s 4630.50 x 100 105

Rs 20

4630.50 x | = 4 4 1 0

Similarly, principal (P2) for the 2nd quarter

f 2 0 ^ = Rs 4630.50

20 20 = Rs 4630.50 x x

21 21 Rs4200

Principal (P3) for the 3rd quarter

= Rs 4630.50 ' 2 0 ^ 3 121

20 20 20 = Rs 4630.50 x - - x x =Rs4000

21 21 21

Total principal = Pi+P2+Pi = Rs (4410 +4200 + 4000) = Rs 12610

Total amount paid = Rs (4630.50 x 3) = Rs 13891.50 .-. Total interest = Rs 13891.50-Rs 12610

=Rs 1281.50 Instalment paid at the end of 1st year = Rs 2600 .-. Principal of 1 st instalment

= R s 2600 + 1 + 100

18.

= Rs 2600 +

= Rs 2600 x

H Q

Too

100 110

= Rs 2 6 0 0 x ^ = R s ^ 11) K S i i

2nd instalment paid at the end of 2nd year = Rs 2400

.-. Principal of 2nd instalment = Rs 2400 x 10 11

240000 10 10 Rs 2 4 0 0 x - x T T = R s

3rd instalment paid at the end of 3rd year = Rs 2200 .-. Principal of 3rd instalment

Rs 2200

Problems Based on Instalment 3 2 7

= Rs x + -320s] 300

300 15 = R s * | ^ J = R s ^ 1 6 ;

Similarly, the principal for amount x at the end of sec-

:\15 ond six months = Rs *| ~

Principal (P3)for amount x at the end of third six

1 5 " , 1 6 , months = Rs * V

/> + p2 + p, = 5407.50

ie 05" f f 1 5 l \ x \ J6, 06, i tiej

1 + + 16

15_N

,16.

16A256

(HI 2" U6y

15407.50

= 5407.50

256 + 240 + 225

721

256

= 5407.50

= 5407.50

.-. x = 5407.50 x x = 2048 15 721

.-. Each instalment = Rs 2048 19. Let the cash price of the gas stove be Rs x

Cash down payment = Rs 500 .-. Remaining amount = Rs (x - 500) 1 st instalment paid at the end of first year = Rs 810

( 1 5 V .-. Principal of 1 st instalment = Rs 810 + ^1 +

115 100

= R s 810 +

A = P\\100

:.P = A*\ +

= Rs 810x 100 115;

2nd instalment = Rs 520

= Rs 810 20^ v23 , Rs

100

16200 23

.-. Principal of 2nd instalment = Rs 520 (20s

C A 20 20 208000 = Rs 520x x = R s

23 23 S 529

3rd instalment = Rs 460 .-. Principal of 3rd instalment

\3

= Rs 460

160000

20Y A , n 20 20 20 =Rs 460x x x 23 23 23 23

= Rs 529

.-. Total principal = Rs 16200 208000 160000)

+ ; + -23 529 529

372600 + 208000 + 160000 740600 = Rs = Rs

JC -500 =

529

740600 x = 500 +

529

740600

x =

529 ^ >29

264500+740600 1005100 = 1900

529 529 .-. Cash price of the gas stove = Rs 1900

20. Let the cash price of the cassette recorder be Rs x Cash down payment = Rs 450 .-. Remaining amount = Rs(x-450) Rate = 18% pa 1st instalment paid at the end of first year = Rs 680

( 18 .-. Principal of first instalment =Rs 680 + I1 +

A = P 1 + 100 { IOOJ

= R s 6 80 + l i 8 - = R s 680 100

Rs 680 50^ v59y Rs

(100 v 118

34000 59

2nd instalment at the end of 2nd year = Rs 590

/ 5 0 f .59 J .-. Principal of second instalment = Rs 590

c n n 50 50 25000 Rs 590x x = R S

59 59 59

Total principal = Rs 34000 25000

59 59

59000 = Rs 5 9 =Rs 1000

x-450=1000 zz>x= 1000 + 450= 1450

Chapter 13

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