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 o f 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 inimSh. 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 E i . 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 PxT

8x100 120x

8x100x12

12

= 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).

PRACTICE BOOK ON QUICKER MATHS

310

Thus, the interest on Rs 6175 for 1 month or — year is Rs 25. 1=

PxTxR 100

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 o f interest charged. Soln: Cash price = Rs 950 Cash down payment = Rs 200 .-. Amount paid in 10 monthly instalments = Rs (80 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. Ex 4:

x

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 Rs12500 Principal for 2nd month Rs 12500-Rs 1600 Rs10900 Principal for 3rd month Rs10900-Rs 1600 Rs9300 Principal for 4th month Rs 9300-Rs 1600 Rs7700 Principal for 5th month Rs 7700-Rs 1600 Rs6100 Principal for 6th month Rs6100-Rsl600 Rs4500 Principal for 7th month Rs 4500-Rs 1600 Rs2900 Principal for 8th month Rs 2900-Rs 1600 Rsl300 Total Principal = Rs 55200 The last instalment of Rs 1600 includes Rs 1300 plus Rs 300 interest. 1 Time = 1 month = 12 — year, Interest = Rs 300 7=

PxTxR

R (Rate) •

100 300x100 12

PxT

300x100x12 _ 150 55200

55200x

7x100

23

6.52

Time = 1 month

l_ :

year, I = Rs 50

12 7=

PxTxR R(Rate)

100 Rate =

7x100 :

PxT

50x100

50x100x12

2oo

3900x—

3900

13

12

=

1 5

A 13

Hence the rate of interest is 15—% • 13 Type 2: Rate of interest being given, to find the monthh instalment.

Illustrative 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

= Rs

99x — x 2 5 12 100

Rs— = 7^8.25 4

Problems Based on Instalment

311 Cash down payment = Rs 105 .-. Balance to be paid in 3 equal monthly instalments = Rs485-Rsl05 = Rs380 Rate of interest = 16% p.a. Interest on Rs 380 for 3 months

PxTxR 100 .-. Amount due = Rs 99 + Rs 8.25 = Rs 107.25 ....(i) Let monthly instalment be x rupees .-. At the end of 4th month 1 st instalment of Rs x will amount to

*

Rs

3 ^ xx — x25 12 + 100

380x — x l 6 12 • = Rs 15.20 = Rs100

A', * + — ] = /& — 16 J 16

100

2nd instalment of Rs x will amount to ' Rs x +

f Rs *

2 25 25x < ( x\ 12 — = Rs Rs x + — = Rs 24 100 24

+

2 xx — x 16 — 1 2 — 100

Rs * +

2* 75

77*

= Rs

75

XZ:>

2nd instalment of Rs x will amount to

V 3rd instalment of Rs x will amount to

1

Rs xx — x25 12 = Rs * + Rs x + 100 48

Rs

49x

*

Rs\ +

25* 24

xx0xl6 :

100

„ f 7 7 * 76* ?| •• Rs\ + * = Rs I. 75 75 )

+

49* 48

+*

^

^75~

= Rsx

n

llx + 16x + 15x 75

_ 228 76 = Rs * = Rs—x 75 25 From (i) and (ii), we get

51* + 50* + 49x + 48* 48

_ 198* _ 33 = Rs = Rs—* 48 8

76*

. Total amount payable under instalment plan

Rs x

Total amount of 4 instalments at the end of 4th .17* month = Rs\—-+ ' 16

Rs

3rd instalment of Rs x will amount to

48

4th instalment of Rs x will amount to xx Ox 25 Rs * + 100

2* Rs * + 75

*x — x l 6 + 12 100

V

Rs

100

.-. Amount due = Rs 380 + Rs 15.20 = Rs 395.20... (i) Let the monthly instalment be Rs x. 1 st instalment of Rs x will amount to

PxTxR

A=P+I=P+

PxTxR

I

76 25 Cjil "~

* = 395.20

^

395.20x25

9880

76

76

* =•

130

Hence the monthly instalment = Rs 130.

W

From (i) and (ii), we get 33 107.25x28 858 ^ — * = 107.25 => x = = = 26 8 33 33 Hence the monthly instalment is Rs 26. Ex. 2: A ceiling fan is marked at Rs 485 cash or Rs 105 cash down payment followed by 3 equal monthly instalments. I f the rate of interest charged under this instalment plan is 16% per annum, find the monthly instalment. Soln: Cash price of the ceiling fan = Rs 485

Exercise 1.

2.

3.

An electric iron is sold for Rs 110 cash or for Rs 50 cash down payment followed by Rs 62 after a month. Find the rate of interest charged under instalment plan. A bicycle is sold for Rs 400 cash or Rs 160 cash down payment followed by 2 monthly instalments of Rs 130 each. Find the rate of interest. A pressure cooker is available on Rs 180 cash or for Rs 70 cash down payment followed by Rs 60 a month for 2 months. Find the rate of interest charged under instalment plan.

PRACTICE BOOK ON QUICKER MATHS

312 4.

A television set is priced at Rs 2400 cash or Rs 1200 cash down payment followed by 6 monthly instalments of Rs 225 each. What rate of interest will the dealer charge under instalment plan? 5. A mixi is marked at Rs 1000 cash or Rs 250 cash down payment followed by Rs 200 a month for 4 months. Find the rate of interest for this instalment plan. 6. A room cooler is marked at Rs 2000 cash or Rs 400 cash down payment followed by Rs 300 per month for 6 months. Determine the rate of interest charged under this instalment plan. 7. A watch is sold either for Rs 180 cash or for Rs 40 cash down payment followed by Rs 30 a month for 5 months. Determine the rate of interest. 8. Determine the interest rate charged under each of the following instalment plans. Article Cash Cash Each No. of price down instalment monthly payment instalments ©TV 2575 1000 300 6 (ii) Refrigerator 3580 1500 440 5 (iii) Typewriter 3600 1200 280 10 (iv) Tape recorder 1600 300 175 8 9. An article is sold for Rs 100 each or for Rs 10 as cash down payment followed by 5 equal monthly instalments. I f the rate of interest charged under instalment plan be 48% per annum, determine the monthly instalment.

Principal for 1 st month - Rs 240 Principal for 2nd month = Rs 240 - Rs 130 = Rs 110 Total Principal = Rs 350 Thus, Rs 20 is the interest on Rs 350 for 1 month or J_

year.

12 .-. 7 =

PxTxR 100

PxT

"

20x100

R=

7x100

R=

20x100x12 350

350 x

480 = 6 8 l 7 7

12 Hence the rate of interest is 68—%p

a

Cash price of the pressure cooker = Rs 180 Cash down payment = Rs 70 Balance to be paid = Rs 180 - Rs 70 = Rs 110 Monthly instalment = Rs 60 .-. Amount paid in 2 equal monthly instalments Rs (60x2) = Rsl20 .-. Interest charged = Rs 120 - Rs 110 = Rs 10 Principal for 1 st month = Rs 110 Principal for 2nd month = Rs 110 - Rs 60 = Rs 50 Total Principal for 1 month = Rs 160 Thus, Rs 10 is the interest on Rs 160 for 1 month or

Answers 1.

Cash price of the electric iron = Rs 110 Cash down payment = Rs 50 Balance to be paid after 1 month = R s l l O - R s 5 0 = Rs60 Monthly instalment = Rs 62 .-. Interest = Rs 62 - Rs 60 = Rs 2 Thus Rs 2 is charged as interest on Rs 60 for 1 month 1 or — year 12 3

PxTxR

R(Rate) =

7x100

100 2x100

2x100x12 60

60 x

PxT

= 40

12 Hence the rate of interest is 40%. Cash price of bicycle = Rs 400 Cash down payment = Rs 160 Balance to be paid = Rs 400 - Rs 160 = Rs 240 Monthly instalment = Rs 130 .-. Amount paid in 2 equal monthly instalments = Rs(130x2) = Rs260 .-. Interest charged = Rs 260 - Rs 240 = Rs 20

^

year. 7?:

7x100 PxT

10x100 160x

10x100x12 = 75 160

12 Hence the rate of interest is 75% per annum. Cash price of the television set = Rs 2400 Cash down payment = Rs 1200 Balance to be paid = Rs 2400 - Rs 1200 = Rs 1200 Monthly instalment = Rs 225 .-. Amount paid in 6 equal monthly instalments = Rs (225x6) = Rsl350 Interest charged = Rs 1350 - Rs 1200 = Rs 150 Principal for 1st month = Rs 1200 Principal for 2nd month = Rs 1200 - Rs 225 = Rs 975 Principal for 3rd month = Rs 975 - Rs 225 = Rs 750 Principal for 4th month = Rs 750 - Rs 225 = Rs 525 Principal for 5th month = Rs 525 - Rs 225 = Rs 300 Principal for 6th month = Rs 300 - Rs 225 = Rs 75 Total principal for 1 month = Rs 3825 Thus, Rs 150 is the interest on Rs 3 825 for 1 month or

12

year

Problems Based on Instalment R

7x100 PxT

Rs

313

150x100

7?:

7x100

200x100

PxT

5100x — 12

3825 x 12

150x100x12

800

3825

200x100x12 ,_ 800 _

:4717

5100

~ 17

4

?

J_ 17

Hence the rate of interest is 47 j y % pa

Hence, the rate of interest is 47—% p

Cash price of the mixi = Rs 1000 Cash down payment = Rs 250 Balance to be paid = Rs 1000 - Rs 250 = Rs 750 Monthly instalment = Rs 200 .-. Amount paid in 4 equal monthly instalments = Rs (200x4) = Rs800 .-. Interest charged = Rs 800 - Rs 750 = Rs 50 Principal for 1st month = Rs 750 Principal for 2nd month = Rs 750 - Rs 200 = Rs 550 Principal for 3rd month = Rs 550 - Rs 200 = Rs 350 Principal for 4th month = Rs 350 - Rs 200 = Rs 150 Total Principal for 1 month = Rs 1800 Thus Rs 50 is the interest on Rs 1800 for 1 month or

Cash price of the watch = Rs 180 Cash down payment = Rs 40 .-. Balance to be paid = Rs 180 - Rs 40 = Rs 140 Monthly instalment = Rs 30 .-. Amount paid in 5 equal monthly instalments = Rs(30x5) = Rsl50 .-. Interest charged = Rs 150-Rs 140 = Rs 10 Principal for 1 st month = Rs 140 Principal for 2nd month = Rs 140 - Rs 30 = Rs 110 Principal for 3rd month = Rs 110 - Rs 30 = Rs 80 Principal for 4th month = Rs 80 - Rs 30 = Rs 50 Principal for 5th month = Rs 50 - Rs 30 = Rs 20 Total principal for 1 month = Rs 400

^year

Thus Rs 10 is the interest for 1 month or — year on 12 Rs400

a

3

7? =

7x100 P

x

T

50x100 1800x — 12

7? =

50x100x12 1800

3

3

1 Hence the rate of interest is 33—% pa Cash price of the room cooler = Rs 2000 Cash down payment = Rs 400 Balance to be paid = Rs 2000 - Rs 400 = Rs 1600 Monthly instalment = Rs 300 .-. Amount paid in 6 equal monthly instalments = Rs (300x6)= 1800 .-. Interest charged = Rs 1800 - Rs 1600 = Rs 200 Principal for 1st month = Rs 1600 Principal for 2nd month = Rs 1600 - Rs 300 = Rs 1300 Principal for 3rd month = Rs 1300 - Rs 300 = Rs 1000 Principal for 4th month = Rs 1000 - Rs 300 = Rs 700 Principal for 5th month = Rs 700 - Rs 300 = Rs 400 Principal for 6th month = Rs 400 - Rs 300 = Rs 100 Total principal for 1 month = Rs 5100 Thus Rs 200 is the interest on Rs 5100 for 1 month or

8.(i)

year

PxT

10x100 400 x

j_

10x100x12 :30 400

12

Hence, the rate of interest is 30% pa Cash price ofTV = Rs 2575 Cash down payment = Rs 1000 .-. Balance to be paid = Rs 2575 - R s 1000 = Rs 1575 Monthly instalment = Rs 300 .-. Amount paid in 6 equal monthly instalments = R s ( 3 0 0 6 ) = Rsl800 .•. Interest charged = Rs 1800 - Rs 1575 = Rs 225 Principal for 1 st month = Rs 1575 Principal for 2nd month = Rsl575-Rs300 = Rsl275 Principal for 3rd month = Rs 1275 - Rs 300 = Rs 975 Principal for 4th month = Rs 975 - Rs 300 = Rs 675 Principal for 5th month = Rs 675 - Rs 300 = Rs 375 Principal for 6th month = Rs 375 - Rs 300 = Rs 75 Total principal for 1 month = Rs 4950 Thus Rs 225 is the interest on Rs 4950 for 1 month or x

12

year

7?: 12

7x100

7x100 PxT

225x100 4950 x — 12

PRACTICE BOOK ON QUICKER MATHS

14 225x100x12

600

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

= 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 ^year R =

7x100 PxT

120x100 6000 x

120x100x12 600x1

12 24

12

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 2 8 0 = - 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 PxT

7? =

7x100 PxT

9.

125 3

= 41.67

5500x — 12 240

5500x1

11

= 21.81

Hence, the rate of interest charged is 21.8% pa 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

= Rs

90x — x 4 8 12 100

; R s

90x5x48 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 will amount to

Rs

xx — x48 12 x+ 100

A =P+ I= P+

11520x

100x100

100x100x12

400x100 12

400x100x12 11520x1

year

J PxTxR 100

4x 29x = Rs x + — =Rs 25 J 25

=

R

s

l

g

Problems Based on Instalment

315

2nd instalment of Rs x will amount to 3 ^ x x — x48 3x 28* 12 Rs x + = Rs x + = Rs 100 25 J 25 J 3rd instalment of Rs x will amount to 2 ^ xx — x48 12 Rs x + 100

The amount of Rs P in 2 years at 5%

2*"| 27x Rs x + — =Rs . 25j 25

26* 25

5th instalment of Rs * will amount to * + 0x48 l Rs * + 100 >

Rsx

Total amount of 5 instalments at the end of 5 months 29*

Rs

28*

+

= Rs

27*

+

Annual payment •

26*

+

+*

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

-2T

27* = R S

5

....(ii)

1.

2.

From (i) and (ii), we get 3. 27* 5

= 108

108x5 „ „ * =— = 20 27 4.

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 tyears at the rate of interest r% per annum is \QQA

5.

6.

Illustrative Example

7.

Ex.:

What annual payment will discharge a debt of Rs 770 due in 5 years, the rate of interest being 5% per annum? 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

100 105 P 100

100x770 5x5(5-1) 100x5 +

fa 140

Exercise

25 135*

= R S

nop

These four amounts together with the last annual payment of Rs P will discharge the debt of Rs 770. 120P 115P HOP 105P -+P = 770 100 100 100 100 550 P 770 100 770x100 = 140 P= 550 Hence, annual payment = Rs 140 Quicker Method: Applying the above theorem, we have,

) 4th instalment of Rs x will amount to

= Rs

100

:

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

1 * xx — x48 12 Rs x + = Rs x + 100 25

115P

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

What annual instalment will discharge a debt of Rs 2210 due in 4 years at 7% simple interest? a)Rs450 b)Rs500 c)Rs550 d)Rs575 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 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 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 Find the annual instalment that will discharge a debt of Rs 5400 due in 5 years at 4% per annum simple interest. a)Rsl200 b)Rsl000 c)Rs800 d)Rsl050 What quarterly payment will 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) What annual payment will 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 7x4(4-1) 100x4 +

100x2210 442

= Rs500

PRACTICE BOOK ON QUICKER MATHS

316 2. d;

Hint: Here instalment is quarterly. Hence from the 16

.

question, we have, t = 4 and r = — - 4 /o Now applying the given rule, 100x2120 required answer • 4 x 4(4 - 1 ) 100x4 +

3. c;

100x2120 424

A sum of Rs 3 310 is to be paid back in 3 equal annual instalments. How much is each instalment if the interest 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

= Rs 500. Hint: Required answer A = P\l +

100x19350

100x19350 5x4x3 100x4 + 4. c;

Illustrative Example Ex:

0 /

430

= Rs4500. :Relx!K 11

e^ 11

R

100 J

Hint: Required answer P = Ix

100x12900 = Rs 3000 430

100x12900 5x4(4-1) 100x4 +

100

10

10

17

Similarly, the principal of 2nd instalment 5. b;

Hint: Required answer A 100x5400 4x5x(5-l) 100x5 +

6. d;

2280x100 912

Re

10

Relx

Re

!

100 121

1000 1331

Total of the three principals (10 = Rs ll

= Rs 250.

v

00 121

+

1000 +

1331

„ 1210 + 1100 + 1000 „ 3310 = Rs = Rs1331 1331

Hint: Required answer 100x580 8x5x(5-l) 100x5 +

2

Relx

The principal of 3rd instalment

Hint: Here, t = 8 and r = 4% [Since, payment is quarterly for 2 years] Now, applying the given rule, we have the required answer 100x2280 8x7x4 100x8 +

7. c;

100x5400 :Rsl000. 540

100x580 = Rsl00 500x580

Case - 2: Based on Compound Interest The problems o f money lending in which the payment 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

When the principal is Rs

3310 , . f

each instalment = Re 1 When the principal is Re 1 each instalment D i = Relx

1

3

3

1

3310 When the principal is Rs 3310 each instalment 1331-x3310 = R s i 3 3 i 3310 .-. Each instalment = Rs 1331 Second Method: Let each equal annual instalment be = Rs

Rs x and P ,P ,P , be respectively the principals for the three instalments. The first instalment is paid after a year l

2

i

.-. Principal (P ) of the first instalment l

by 100 IQ0 + R

100 100 + /?

2

/ + ...+

100

V100 + /?

xx-

10

\0x

11

11

Problems Based on Instalment

317 Principal:

Similarly p = 2

10

J

x

Forthe lstyear=

>

( loo V Uoo+/?,

in

For the 2nd year = RsX\ 3310

Now P + P + P x

2

3

2

lOx +x

i.e. 11

100 (ioo + /?

01;

ho] +X —

3310

OU

10^ , 10 fio 1+—+ — 11 \ 11

For the tth year =

X

( loo V 100 + /?

Interest Charged: = 3310 100 Interest in 1st instalment = ^* •*

TioY,

10 100

UiA

i i 121

x —

/

1+— +

100 + /?,

3310 Interest in 2nd instalment - Rs X 1 -

100 = 3310

Interest in nth instalment = ^

s

r

UUU21 11 121 x = 3310x — x 1331 10 331 Hence the required annual instalment = Rs 1331 Quicker Method: Applying the above theorem, we have the 3310 JQQ JQQQ"

required annual instalment = JQ

TT T 2 T 1331 +

+

_ 3310x1331 3310

/fr 1331-

T*pe II: Tofindthe 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 100

100

100 + /?

Uoo+/?

X

(

100 >

f

Ex:

A man borrowed some money and paid back in 3 equal annual instalments of Rs 2160 each. What sum did he borrow, if 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 instalment. 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

2

f

2160 20 1+ 100

= Rs 2 1 6 0 x - = Rs 1800 6 t' \ v A = 1+ —

H\

r

IOOJ

\ 20 \ 2160 = p 1 + I 100,

Uoo+/?j

100 "

100 + /?,

Illustrative Example

100

>

: 1. To find the total interest charged we use the following formula,

;

100 + /?,

1 0 Y 1 2 1 + 110 + 1001 = 3310 121 10Y33]

X

100

100 ^

[ l 0 0 + rt; * ^100 + /?; + ....+ 100 + /?J V

2. To calculate the principal and interest charged with each instalment following formula is used.

Principal for the 2nd year 25 = f a 2 1 6 0 x - 1 =/?s2160x — = /?il500 6) 36 Principal forthe 3rd year K

- Rs 2160 x = 2500 +1500 = 4000 .-. Cash price of the refrigerator ~- Rs 4000 Total sum paid = Rs (1500 + 102-J + 1003 + 990) = Rs4513 .-. Total interest charged = Rs (4513 - 4000) = Rs 513 Quicker Method: Applying the above theorem, we have

6.

7.

8.

pa compounded annually, what should be the annount of each instalment? A man borrows some money on compound interest and returns it in two years in two equal instalments. I f the rate of interest is 5% and yearly instalment is Rs 441: find the amount borrowed. A sum of money is to be paid back in 3 annua! instalments ofRs 2800, Rs 2700 and Rs 2600 payable at the end of 1 st year, 2nd year and 3rd year respectively. I f rate of interest be 4% pa, calculate the principal and the interest charged. One can purchase a flat from a house building socien for Rs 55000 cash or on the terms that he should pay Rs 4275 as cash down payment and the rest in three equai yearly instalments. The society charges interest at the rate of 165 per annum compounded half yearly. If the flat is purchased under instalment plan, find the value of each instalment. A sum o f Rs 5600 is paid back in yearly instalments How much is each instalment, i f the interest is compounded annually on the balance at 8% per annum anc is to be included in each instalment? A sum of Rs 6000 is paid back in 3 annual instalments How much is each instalment i f the interest is compounded annually on the balance at 10% per annum anc is to be included in each instalment? A sum o f Rs 8400 is to be returned in three annua instalments. What is the annual instalment, if the rate c: interest is 9— % per annum compounded annually on

Problems Based on Instalment 9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

the balance and is to be included in each instalment? The price of a tape-recorder is Rs 1561. A customer purchased it by paying a cash of Rs 300 and balance with due interest in 3 half yearly equal instalments. I f the dealer charges interest at the rate of 10% per annum compounded half yearly, find the value of each instalment. A loan ofRs 2550 is to be paid back in two equal half yearly instalments. How much is each instalment, if the interest is compounded half yearly at 8% per annum? A sum ofRs 2600 is to be paid back in 2 equal annual instalments. What is the annual instalment, if the rate of interest is 8% per annum compounded annually? A man borrows Rs 816 and agrees to return it in two equal annual instalments. What is the annual instalment, if the rate of interest is 12.5% per annum compounded annually? Govind borrowed money from a money lender and agreed to pay back in 3 equal annual instalments ofRs 665.50 each. What sum did he borrow, i f the rate of interest charged by the money lender was 10% per annum compounded annually? A man takes loan on compound interest and returns it in two equal annual instalments. I f the rate of interest is 16% per annum and the yearly instalment is Rs 1682, find the principal and the interest charged with each instalment. A man borrowed some money and paid back in 3 equal annual instalments o f Rs 2160 each. What sum did he borrow if the rate of interest charged by the money lender was 20% per annum compounded annually? Find also the total interest charged. Kusum borrowed money and returned it in 3 equal quarterly instalments ofRs 1630.50 each. What sum did she borrow if the rate of interest charged by the money lender was 20% per annum compounded quarterly? Find also the total interest charged. Naresh took loan from a bank and the payment was made in 3 annual instalments ofRs 2600, Rs 2400 and Rs 2200 payable at the end of first, second and third year respectively. Interest was charged at 10% per annum. Calculate the amount of loan taken and the interest paid by him. A person borrows Rs 5407.50 and agrees to pay the loan back with compound interest at the rate of 13 — % per

annum in 3 equal half yearly instalments. Find the amount of each instalment, i f the interest is compounded half yearly. 19. Sanjay bought a gas stove on instalment basis. He has to pay Rs 500 cash down payment and Rs 810 at the end of first year, Rs 520 at the end of second year and Rs 460 at the end of third year. Interest is charged at the rate of

321 15% per annum. Calculate the total cash price of the gas stove. 20. A dealer advertises that a cassette recorder is sold at Rs 450 cash down followed by two yearly instalments ofRs 680 and Rs 590 at the end of first year and second year respectively. If the interest charged is 18% per annum compounded annually, find the cash price of the cassette recorder. 21. A colour TV set purchased under instalment purchase system. Cash down payment is Rs 2000 and 3 annual instalments ofRs 1800, Rs 1560 and Rs 1430 are payable at the end o f first year, second year and third year respectively. I f the rate of interest is 10% per annum respectively, find the cash price of the TV set and the total interest charged under instalment plan. 22. A sewing machine is available at Rs 240 cash down payment followed by 3 annual instalments of Rs 380, Rs 240 and Rs 200 payable at the end of first year, second year and third year respectively. If the rate of interest is 25% per annum compound interest, find the cash price and total interest being charged under the instalment plan.

Answers 1.

Price of the tape recorder = Rs 1500 Paid cash = Rs 300 Balance to be paid = Rs (1500 - 300) = Rs 1200 Let each half-yearly instalment = Rs x r = 5% pa = (5/2)% half-yearly, Amount = Rs 1200 ;•. Using the formula A= P 1+100

x=P 1+

5/2

1200J

100

U0

Principal (P) included in the first instalment 40 40 xx — = —x 41 41 Similarly, Principal included in the second instalment '40

x 2

41 '40^ Principal included in the third instalment

40 41

(40} x+

40 41'

V

2

41;

+

f '40 x = 1200 x+ \, 4 1 , N

40 r40] +

41

\4\J

1200

:

.41

PRACTICE BOOK ON QUICKER MATHS

322 40 41'

40 1+ 41

1600^ 1681

The man paid Rs 441 as the amount at the end of first year and another Rs 441, as the amount at the end of second year. .-. Principal of the first year

= 1200

40 (1681 + 1640 +1600) _s —x\0 B 41 I, 1681 J

,„

n f t

, + = 441 + 1

4921 41 => x = 1200x — = 1230 ^ 1681 40

,,,1 = 441 + 2

1681 => x = 1230x — — 20.20 4921 Cash price = Rs 3000 Cash down payment = Rs 1000 Balance = Rs (3000 -1000) = Rs 2000 Let the amount of each instalment = Rs x

1

1 =441 „„, + 100 J 100 5

1 0 5

AA, =441x-= 2

0

R

s

4

2

0

= R s

' 5 f Principal for the second year = 441 + 1 + 100.

Rate = 12 i % pa = (25/2)% pa

. . . 20 20 = 441x — x — =Rs400 21 21 .-. Total principal = Rs 420 + Rs 400 = Rs 820 Let the principals for the three annual instalments be

Principal included in the first instalment is given by

P P ,P .

25/2 1+ 100

x'-P

A=P

2

Then

3

UooJ

•:A = P\ +

" 5

25

/>, =Rs2800x

2800= P\ 1 + 100

loo J

P = xx

U

26

100

8 2

2

100

Similarly, principal included in the 2nd instalment =

2600=^3

' H i

2b

P =Rs2700x

2700= P \ +

2

P, =Rs2600x

1+100

6.

(2£ 26.

Total sum borrowed = P + P + P x

Principal included in the 3rd instalment = * | —

2

3

x

= Rs 2800 26) x x — + x\ 9 {?)

x\ | = 20ou

W l + l ^ t ^ 9 I 9 81

81

2

2000

8 217* „ „ „ „ -x = 2000 9 81 2000x81x9 8x217

= Rs 839.86

Amount of each instalment = Rs 839.86

2

5

25"

x — + 26 x — 26 26 26

2500f^ 675 625 28 + + — 26 I 26 26 B

=r = 2000

25

5 Rs^xlOO

s

R S

8 (81 + 72 + 64

{26)

I '25 + 2600 1 26

2500 ( 728 + 675 + 625 ;Rs

26 V

26

= Rs

2500 26

2028 x26

= Rs2500x3=Rs7500 Total money paid = Rs (2800 + 2700 + 2600) = Rs 8100 .-. Total interest charged = Rs 8100 - Rs 7500 = Rs 600 Cash price of the flat = Rs 55000 In the instalment plan, cash down payment = Rs 4275 .•. Present value of the price to be paid in instalments = Rs 55000 - Rs 4275 = Rs 50725

Problems Based on Instalment

323

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 +

Rs 1400 + Rs V x= P 1+

100 J

100

J5_ 100

(25)

'25^

= Rs 27

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

x

|'25>

2

x+\

\ 27J

X

V

It should be equal to Rs 50725

27'

f25> 1 1 + — +{llj 27 2

2~ = 50725

5

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

25 ("729 + 675 + 625 27 %

729

n

c

= 50725

25 2029 => —xx = 50725 • 27 729 M

W

50725x27x729 => x =

0

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

5600X

x2800

100

U00

xl400

= 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 -x6000 = Rs 2000 + Rs .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 10 ^ = Rs2000 + Rs — x 4 0 0 0 100

, ^ = 19683 0

_8_

= 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

3

x4200

100

8

Total principal for the three instalments 25 '25> Rs —x + 27 ^27;

_8_

= 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

Similarly, Principal forthe 2nd instalment

25

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

75 " 0

= Rs 1400+ Rs 448 = Rs 1848

= Rs 2000 + Rs 400 = Rs 2400 Balance at the end of 2nd year = Rs 4000 - Rs 2000 = Rs 2000 .-. Amount payable at the end o f 3rd year = Rs 2000+10% ofRs 2000 10 = Rs 2000 + Rs

100

x2000

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

PRACTICE BOOK ON QUICKER MATHS

324 8.

The sum is to be returned in 3 annual instalments. .-. Each instalment will be ofRs (8400 + 3), ie Rs 2800 together with interest on the balance for one year. .-. Amount payable at the end of 1 st year

Similarly, principals for the next two instalments are

P =Rs(f]*andP =Rs(f)* 2

19 = Rs 2800 + — % ofRs 8400 2

3

PP l+

i

P =1261

+

3

'20Y

f20^ 19 1 1 = Rs2800 + Rs — x x8400 2 100 J = Rs 2800+ Rs 798 = Rs 3598 Balance at the end of 1st year = Rs 8400 - Rs 2800 = Rs 5600 .-. Amount payable at the end o f 2nd year

Ui;

.21J

i1 + — + 20 21 21J

20

2

21 20 —> 21

19 = Rs 2800 + — % ofRs 5600 2

f'20V x + \ — x = 1261 \

x+\— .21)

1261

0

, 20 400 , , 1+— + =1261 21 441

20 r441 + 420 + 400 , , , , , —x\ = 1261 21 I, 441 s

19 1 Rs2800 + Rs — x 5 6 0 0 .200

J

20 1261 1261x21x441 —xx = 1261 => x = 21 441 20x1261

= Rs 2800 + Rs 532 = Rs 3332 Balance at the end of 2nd year = Rs 5600 - Rs 2800 = Rs 2800 .-. Amount payable at the end of 3rd year = Rs 2800 + ^ % ofRs 2800 2 19 = Rs2800 + Rs v

200

x2800

=Rs 2800 + Rs 266 = Rs 3066 Hence, the three instalments are: Rs 3598, Rs 3332 and Rs 3066 Cash price of the tape recorder = Rs 1561 Cash down payment = Rs 300 .-. Price to be paid in instalment has its present value = Rs 1561 -Rs300 = Rs 1261 Let each instalment be Rs x Rate = 10% pa = 5% half yearly .-. Principal (P) included in the 1 st instalment

rz> x =

10.

= 463.05 20 .-. Each instalment = Rs 463.05 Let each instalment be Rs x Rate = 8% pa = 4% half yearly .-. Principal (P ) included in the 1st instalment x

26 Rs

= Rs

x

+

25

Similarly, principal (P ) for the next instalment 2

25: \ Rs

26y

P, + P = 2550 f25^ '25] ie { 2 6 ;x + ^26) x = 2550

1+100

25 A= P 1+

21

1 +

26

25

26 + 25

26

26

i. ^ 26

2

Rs

25

26A

\x = P 1 + 100 100

105 100 = Rs x + = Rs x x 100J 105 (20

26.

1+ — V 100.

2

Rs

25^ Rs

x

:

x

ll 26

=

= 2550

= 2550

2550

Problems Based on Instalment

325

=> x = 2550x — x — = 1352 25 51 11.

= 665.50+ 1 +

.-. Each instalment = Rs 1352 Let each instalment be Rs x Rate = 8% pa

10 100 J

A = P 1+100

665.50 =P\+

.*. Principal (P,) included in the 1st instalment 108 = Rs

= Rs

*

1+100

10 V — 100 J

= 665.50 + — I = 665.50 x — =R 605 S

100 Principal (P ) for the second year 2

100 = Rs

25 665.50+ 1 +

108

Similarly, principal (P ) included in the 2nd instalment

10'

= 665.50 +

100

10

2

*^" *« io io ;

25 =

R

S

|

= 665.50x — x —

n 2

2T

Principal (P ) for the third year 3

.-. P, + P = 2600

10 V = 665.50 + 1 1 + 100

2

(25^ ie

12.

(25)

{21 j

x+

x = 2600

127 J

On solving this equation, you will obtain x = 1458 .-. Each annual instalment = Rs 1458 Let each annual instalment be Rs x Rate =12.5% pa .-. Principal (p,) included in the 1st instalment

= Rs x + 1+

12.5}

112.5 J

.-. Total principal = P +P l

14.

1000 xx-

1125

+P

2

i

= Rs (605 + 550 +500) = Rs 1655 The yearly instalment paid at the end of 1 st year and 2nd year = Rs 1682

= 1682 +

Rs

10)

r t c cr. 10 10 10 = 665.50 x — x — x — =Rs500

100

100, 100

Rs xx-

=665.50x

.-. Principal (P,) for the first year

112.5 = Rs

=Rs550

= Rs

i i 1 +100

8 9)

y A = P\l +

1682 = P.\ +

100 )

16 100J

Similarly principal (P ) included in the 2nd instal2

= Rs ! 6 8 2 + 100 r

ment = Rs P +P x

2

1 1 6

\9j =816

= Rs

1682x

f 25^

100 116

= Rsl450 Principal (P ) for the 2nd year 2

13.

On solving this equations you will get x = 486 .•. Each annual instalment = Rs 486 Govind paid Rs 665.50 as the amount at the end of 1st year, 2nd year and 3rd year. .-. Principal (P, )for the first year

= 1682 +

JL6_

\

1 + 100J

=

R

s

1682 x

25 25 = Rs 1682x — x — =Rsl250

25 29.

PRACTICE BOOK ON QUICKER MATHS

326 .-. Total principal = P + P x

2

= Rs 2600 +

= 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 2600 x

= Rs 2700x — =Rs432 100

15. 16.

HQ

Too 100 110

= Rs 2 6 0 0 x ^

=

R

11)

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

10 10 Rs 2 4 0 0 x - x = T

1+100

4630.50 x

=

4

4

1

Rs

0

200000 121 26000

Similarly, principal (P ) for the 2nd quarter

:. Total principal = Rs

2

240000

+

11 f20^ = Rs 4630.50

;Rs

20 20 = Rs 4630.50 x — x — 21 21 Principal (P ) for the 3rd quarter

Rs4200

= Rs

3

'20^

3

18.

121

20 20 20 = Rs 4630.50 x - - x — x — =Rs4000 21 21 21 Total principal =

17.

P +P +P i

2

R

s

2600 + 1 +

200000^1

121

121 J

286000 + 240000 + 200000 121 7260000

40

Rate= 1 3 - % pa

% pa

o/ 20 — % = — % half-yearly 0

.-. Principal {P ) for amount x at the end of 1 st six x

20/3'' months = Rs * "" 1 + 100 5

P = /I + 100

+

= Rs 6000 121 .-. The amount of loan taken = Rs 6000 Total amount paid = Rs (2600 + 2400 + 2200) = Rs 7200 Total interest paid = Rs (7200 - 6000) = Rs 1200 Let each instalment be ofRs x

4

i

= 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 o f 1st year = Rs 2600 .-. Principal of 1 st instalment =

s

10 10 10 = Rs 2200 x — x — x — 11 11 11

105

= Rs 4630.50

240000 R

Rs 2200 x= 1000 + 450= 1450

s

529

PRACTICE BOOK ON QUICKER MATHS

328

21.

Hence, the cash price of the casette recorder is Rs 1450 Let the cash price of the colour TV be Rs x Cash down payment = Rs 2000 .-. Remaining amount = Rs (x - 2000) Rate = 10% pa 1 st instalment paid at the end of first year = Rs 1800

.-. Remaining amount = R s ( x - 240) 1 st instalment at the end of first year = Rs 380 25

,\l of 1 st instalment = 380 + 1 1 +

100

v

r V :.P = A + 1 + — 100 J IOOJ

A = P\ +

r

v

.-. Principal of first instalment =Rs 1800+ 1 +

r }

ft

A=P 1+

I

=

R s

i

100

. ( r Y" :P = A + \ + — n

100,

1

Rs

110;

=R

S

1800^

.;. Principal of 2nd instalment = Rs 1560

'io^

2

10 10 156000 = R s l 5 6 0 * — x — =Rs 121 3rd instalment paid at the end of third year = Rs 1430

fioV .-. Principal of 3rd instalment = Rs 14301 — „ . . . . 10 10 10 130000 = Rs 1430x — x — x — = R 11 11 11 121 .-. Total principal of the three instalments S

= Rs = Rs

22.

156000 +

21

130000^1 '

121

198000 + 156000 + 130000 121 484000

100

\125

u = Rs380

s

= Rs304

-„ 16 Rs 2 4 0 x — = r 25 3rd instalment at the end of 3rd year = Rs 200 .-. Principal of 3rd instalment

18000

11

100,

=Rs4000

x- 2000 = 4000 => x = 2000 + 4000 = 6000 .-. The cash price of the colour TV = Rs 6000 Amount paid in instalment plan = Rs (2000 + 1800 +1560 + 1430) = Rs 6790 .-. Total interest paid = Rs 6790 - Rs 6000 = Rs 790. Let the cash price of the sewing machine be Rs x Cash down payment = Rs 240

= Rs 200 _

768

n

= Rs 240

100

'18000

125"

2nd instalment at the end of 2nd year = Rs 240 .-. Principal of 2nd instalment

11 2nd instalment paid at the end of 2nd year = Rs 1560

= Rs

Rs 380

IOOJ _

[l800+i!° 100

= Rs 1800x

Rs 380 +

s

5

4 4 4 = Rs 200x — x-^x — 5 5 5

512

The total principal of the three instalments ( . . . 768 512 = Rs 304 + — + _ '1520 + 768 + 512^ - Rs V 5 j 2800 = Rs — r — =Rs560 .-. x-240 = 560 =>x =560 + 240 = 800 .-. Cash price of the sewing machine = Rs 800 Total amount paid = Rs (240 + 380+240 + 200) = Rsl060 .-. Total interest paid = Rs 1060 - Rs 800 = Rs 260 Case 3: Instalments to pay off debt Suppose borrower is to pay a sum ofRs P due after T years, but he wants to pay through equal instalments at an agreed interval (may be quarterly, monthly, half-yearly or yearly) to discharge off his debt in T years. Hence, through equal instalment payments, the debt ofRs P due in T years is cleared in T years. Then, n(n-\) A = nx + xr lOOw

3:;

Problems Based on Instalment Where A = amount due after T years n = number of instalments to be actually paid to discharge the debt, r = rate per cent per annum charged on simple interest m = number of instalments per year, = 1, if yearly instalment is paid. = 4, if quarterly instalment is paid. = 2, if half yearly instalment is paid, = 12, i f monthly instalment is paid, x = amount of each instalment in Rs. Note: Also see Type 3 of Case I.

3.

100x4 106

c)16j%

%

d)33i%

A sum ofRs 10 is lent out to be returned in 11 monthly instalments of Re 1 each, interest being simple. The rate of interest is . b) 11%

c)9^-%

d) 2 1 - ^ %

Answers 1. a;

Ex:

8480 = Ax +

2 2

a) 10%

Illustrative Example A man borrows a sum of money at 16% per annum simple interest, promising to pay Rs 8480 after a year from the date of borrowing. I f he wants to discharge the debt by paying four equal quarterly instalments, how much should be each instalment. Soln: Applying the above formula, we have

b) f

a)26|%

Hint: Applying the given rule, we have A = Rs2,x = R e l , n = 3,m= 12 or, 2 = 3x1 +

fxl

x

100x12 2. a;

2

or, r = 400% *. the rate per cent per annum = 400% Hint: Applying the given rule, we have, lxr 9 = 10x1 + 100x12

x [ : m = 4 (quarterly^ 2

3x2

10x(l0-l)

15r or, " T T T = - 1 [we omit the -ve sign] 400

8480 => x = /?s 2000

25 ' 400

Exercise 1.

2.

A sum ofRs 2 is lent to be paid back in 3 equal monthly instalments of Re 1 each. Find the rate per cent. a) 400% b)140% c)340% d)40% A money lender lends out Rs 9 on the condition that the loan is payable in 10 months in 10 equal instalments of Re 1 each. Find the rate per cent per annum.

80 2 = — = 26—% 15 3 3

. r= 3d-

IA

,,

I

l

X

r

Hint: 10 = 11x1 + x mm. 100x12

o

r

>

,

240 „ , 9 „, _ = 21 %.

=

n

Hx(ll-l)

i 2

'-