True Discount

Introduction Suppose I have to pay you Rs 104 in a year's time.

But you want your money at once. Certainly you cannot demand and 1 cannot give full Rs 104. What to do then? Let us go to some good bank, say, the State Bank of India, Delhi and enquire the rate of interests allowed by it. Suppose the rate of interest is 4%. Clearly I can discharge my debt by paying you Rs 100 at once. You lose nothing. Why? Because if you deposit Rs 100 in the bank, it will in a year's time amount to Rs 104. Now you can easily understand that Rs 100 is the present value (or present worth) or Rs 104 due 1 >ear hence, and the portion deducted namely, Rs 4 is the discount.

The Present Value or Present Worth of a sum of Tioney due at the end of a given time is that sum which with ts interest for the given time at the given rate will amount to

Tie sum due. The sum due is called the amount. The True Discount is the difference between the sum

iue at the end of a given time and its present worth. I wn the definition it is clear that True Discount = Interest on Present Worth ind Amount = Present Worth + Discount

A = P.W.+T.D. Note:

1. True discount is also called Mathematical, Arithmeti-cal, Theoretical or Equitable Discount.

. In questions on Discount, True Discount, is gener-ally denoted by T.D. Present Worth is denoted by P.W. Amount or sum denoted by A. Rate is denoted by R. Time is denoted by T. Interest is denoted by I .

Thus T.D. = I on P.W. at given R and T = P.W.xRxT

100

P.W. lOQxT.D.

R x T

and A = P.W. + T.D. = P.W. +

R x T "

P.W.xRxT

= P.W. 1+- 100 P.W.

100

100+RxT

P.W = lOOxA

100 + R x T

100

0)

lOOxT.D. _ Again A = P.W. + T.D. = + T.D.

= T.D. 1 + 100 R x T

R x T

= T.D. RxT+100

R x T

T.D. = A x R x T

.(ii) 100 + R x T

Present Worth and True Discount can be obtained in a much simpler way with the help of formulae (i) and (ii).

Rule 1 To find Present Worth when Rate, Amount and Time are given.

100xA PW (Present Worth) =

100+RT Where A = Amount or sum; R = Rate per cent per annum;

T = Time in years

Illustrative Example

- J

Ex: Find the present worth of Rs 481.25 due 2 years

hence, reckoning simple interest at 4 p.c. per annum. . 1

Soln: Detail Method: InterestonRs lOOfor 2 yrsat4p.c.

= Rs 10

706 PRACTICE BOOK ON QUICKER MATHS

[v SI = 1 0 0 x - x 4

2 100

10]

present worth of Re 1

.-. amount of Rs 100 = Rs 100 + Rs 10 = Rs 110

.-. present worth of Rs 110 = Rs 100

100 110

.-. present worth of Rs 4 8 1 | = | ^ x 4 8 1 . 2 5

= Rs 437.50 Quicker Method: Applying the above formula, we get

100x481.25 48125 PW = Rs r - = Rs =Rs 473.50

100 + 4x 110

Exercise 1. Find the present worth (PW) and the true discount reck-

oning 6% per annum simple interest of Rs 176 due in 20 months time. a)Rsl60,Rsl6 b)Rsl30,Rs46 c) Rs 150, Rs 26 d) None of these

, 1 2. Find the present worth of Rs 9950 due 3 ~ years hence

at 7 per cent per annum simple interest. Also find the

discount. a) Rs 7000, Rs 2950 b) Rs 8000, Rs 1950 c) Rs 7950, Rs 2000 d) None of these

3. I want to sell may scooter. There are two offers, one at cash payment of Rs 8100 and another at a credit of Rs 8250 to be paid after 6 months. I f money being worth

A I 6 % per annum simple interest, which is the better

4 offer? a) Rs8100 in cash b) Rs 8250 due 6 months hence c) both are equally good d) Can't be said

4. The present worth fo Rs 1404 due in two equal half yearly instalments at 8% per annum simple interest is: a)Rsl325 b)Rsl300 c)Rsl350 d)Rsl500

5. A trader owes a merchant Rs 901 due 1 year's hence. However, the trader want to settle the account after 3 months. How much cash should he pay, i f rate of inter-est is 8% per annum: a)Rs870 b)Rs850 c)Rs 828.92 d)Rs 846.94

6. A man buys a watch for Rs 195 in cash and sells it for Rs

220 at a credit of 1 year. I f the rate of interest is 10%, the man: a) gains Rs 15 b) gains Rs 3 c) gains Rs 5 d) loses Rs 5

7. A man purchased a cow for Rs 300 and sold it the same day for Rs 360, allowing the buyer a credit of 9 years. If

the rate of interest be 7 % per annum, then the man

has a gain of:

1 a) 4 - %

2 c)6% d) 5%

8. A owes B, Rs 1120 payable 2 years hence and B owes A. Rs 1081.50 payable 6 months hence. I f they decide to settle their accounts forthwith by payment of read} money and the rate of interest be 6% per annum, then who should pay and how much: a)A,Rs50 b)B,Rs50 c)A,Rs70 d)B,Rs70

9. Find the present worth of Rs 264 due in 2 years reckon-ing simple interest at 5 per cent per annum. a)Rs240 b)Rs360 c)Rs540 d)Rs260

10. What is the present worth of Rs 272.61 due in 2 years 7?

_ 1 days at 7 per cent?

a)Rs334 b)Rs254 c)Rs234 d) None of these 11. Find the present value of Rs 1051.25 due a year hence at

a)Rsl200 b)Rsl000 c)Rsl500 d)Rsl050 12. What sum will discharge a debt of Rs 5300 due a year

and a half hence at 4% per annum? a)Rs5000 b)Rs4500 c)Rs4200 d)Rs5250

Answers

1. a; Hint: Applying the given rule we have

100x76 Present Worth = = Rs 160 100 + 6x 20

12 True Discount = Amount - Present Worth

= Rsl76-Rsl60 = Rsl6 [See Note of the Rule-2]

2. b 3. a; Hint: PW of Rs 8250 due 6 months hence = Rs

100x8250 (25 1

x 4 2

Rs8000 100 +

Rs 8100 in cash is a better offer.

True Discount

4. a; Hint: PW of Rs 702 due 6 months hence

= Rs i 100x702

100 + 8x-= Rs675

100x702 PW of Rs 702 due 1 year hence = R 1 JQO + (8X1)

= Rs650 .-. Total PW m Rs (675 + 650) = Rs 1325

5. b; Hint: PW of Rs 901 due 9 months hence at 8%

= Rs 100x901

100+ 8x

(100x901x1

106 Rs850.

6. c; Hint: PW of Rs 220 due 1 year hence '100x220

= , 100 + 10 Hence, the man gains Rs 5.

= Rs200

7. d; Hint: PW of Rs 360 due 2 years hence at 7 - % p . a .

100x360

100+j^yX2

100x360x7 800 Rs315

.-. SP = Rs315 15x100

Hence gain % = = 5%

8. b; Hint: PW of Rs 1120 due 2 years hence at 6%

' 100x1120 "

.100 + ( 6 x 2 ) J = R s 1 0 0 0

PW of Rs 1081.50 due 6 months hence at 6%

100x1081.50

100 + 6 x 1 2

100x1081.50 103 Rsl050

So, A owes B, Rs 1000 cash and B owes A Rs 1050 cash. .-. B must pay Rs 50 to A.

9. a 10. c; Hint: Required answer

100x272.61 27261 27261x2 33 15 73

100+ x 2 100 + 2 365 2

233 = Rs234.

707

11. b 12. a; Hint: Required answer

100x5300

100 + - x 4 2

100x5300 106

= Rs5000.

Rule 2 To find discount when, Amount or Sum, Rate and Time are given.

A x R x T

True Discount (TD) = 1 0 0 + R T

Illustrative Example E K Find the true discount reckoning 4 per cent per an-

1

num simple interest of Rs 481.25 due in 2 years

time. Soln: Applying the above formula, we get

481.25x4x True Discount = :Rs43.75

100 + 4x

Note: True Discount can be found by first calculating present worth and then subtracting it from the sum or amount. [Since Present Worth = Amount + True Dis-count]

Exercise 1. I f the true discount on a sum due 2 years hence at 5%

per annum be Rs 75, then the sum due is: a)Rs750 b)Rs825 c)Rs875 d)Rs800

2. Find the true discount on a bill for Rs 1270 due 7 months hence at 10% per annum. a)Rs60 b)Rs75 c)Rs70 d) None of these Find the true discount reckoning 3 per cent per annum simple interest of Rs 1802 due in 2 years time. a)Rsl00 b)Rs98 c)Rsl02 d) None of these

Answers

3.

l .b : Hint: 75 = A x 2 x 5

100 + 2x5

110x75 or, A = = Rs 825

2. c; Hint: Required answer

1270x xlO , ~ n n n , n 12 _ 1270x7x10

100 + x l O 12

1270 = Rs 70.

3.c

708 PRACTICE BOOK ON QUICKER MATHS

Rule 3 To find (Present Worth) P. W, when True Discount (TD), Rate per cent (R), and Time (T) are given.

I O O X T D PW =

RxT

Illustrative Example Ex: The true discount on a bill due 9 months hence at

12% per annum is Rs 270. Find the present worth. Soln: Applying the above formula, we get

Present worth : 100x270

q =Rs3000. 12x

12

Exercise 1. The true discount on a bill due 8 months hence at 12%

per annum is Rs 240. Find the amount of the bill and its present worth. a)Rs3000,Rs3240 b) Rs 2000, Rs 2240 c) Rs 2100, Rs 2340 d) None of these

2. The true discount on a bill due 9 months hence at 6% per annum is Rs 180. Find the amount of the bill and its present worth. a)Rs3000,Rs3180 b) Rs 4000, Rs 4180 c) Rs 4500, Rs 4680 d) None of these

3. The interest on Rs 750 for 2 years is equal to the true disocunt on Rs 810 for the same time and at the same rate. The rate per cent is:

a) 4 j % b) 5i%

c)4% d)5% 4. Goods were bought for Rs 600 and sold and the same

day for Rs 650.25 at a credit of 9 months and still there was a gain of 2%. The rate per cent is:

1 a) 6-0/0 b) 8yO / o c) 8%

-,43 d ) 7 - o / o

Answers 1. a; Hint: Applying the given rule, we have,

IOOxTD 100x240 Present worth, PW = - = = Rs 3000

R x T 12x-12

.-. Present worth is Rs 3000

.-. A = Amount of bill = PW + TD = 3000 + 240 = 3240 2. b 3. c; Hint: Since TD is SI on PW, we have

Rs (810 - 750) or Rs 60 as SI on Rs 750 for 2 years.

Rate = 100x60 750x2 = 4%

4. b; Hint: SP = (102% of Rs 600) = Rs 102 100

x600 =R S612

.-. PW of Rs 650.25 due 9 months hence is Rs 612 or, Rs 38.25 is SI on Rs 612 for 9 months.

Rate =

f \

100x38.25

612x % = 8 - %

3 '

Rule 4 To find amount (A), when True Discount (TD), Rate per cent (R), and Time (T) are given.

100 Amount = TD 1+- RxT

Illustrative Example E K The true discount on a bill due 9 months hence at

12% per annum isRs270. Find the amount of the bill. Soln: Applying the above formula, we get

Amount = Rs 270 1 + -

100

12x 12

= Rs 270x109

= Rs30x 109 = Rs3270 Note: Amount = Present Worth + True Discount

[ I f you calculate present worth (PW), amount can t5e calculated by adding True Discount (TD) to PW] '

Exercise 1. The true discount on a bill due 10 months hence at 6%

per annum is Rs 26.25. Find the amount of the bill, a) Rs 551.25 b)Rs550 c)Rs551.50 c)Rs 550.25

\_ 2

per annum is Rs 265.10. Find the amount of the bill. a)Rs 16171.10 b)Rs 16711.10 c)Rs 16181.10 d) None of these

The true discount on a bill due 2 years hence at 5% per

annum is Rs 15. Find the amount of the bill. a)Rsl85 b)Rsl56 c)Rsl58 d)Rsl65

1 1 years hence at 3

per annum is Rs 24.50. Find the amount of the bill, a) Rs 224.50 b)Rs 124.50 c)Rs 324.50 d) None of these :

The true discount on a bill due 8 years hence at 4%

The true discount on a bill due 3 years hence at 3 - %

True Discount 709

per annum is Rs 192.24. Find the amount of the bill, a) Rs 786.96 b)Rs 876.96 c)Rs 776.76 d)Rs 776.96

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

Rule 5 To find the difference between simple interest (SI) and True Discount (TD) when amount (A), time (T) and rate (R) are given

SI-TD = Ax(RT)

100(100 + RT)

Illustrative Example Ex: Find the difference between simple interest and true

discount on Rs 840 due 4 years hence at 5% per an-num simple interest. Applying the above formula, we get

840x(4x5) 2 840x20x20

Soln:

SI-TD = = Rs28. 100x(100 + 4x5) 100x120 Note: To find the amount or sum when difference in SI and TD, time (T) and rate (R) are given

(S1-TD)(100+RT)100 Amount (A) = (RT) 2

Ex: The difference between the simple interest and the true discount on a certain sum for 6 months at 4% is Rs 15. Find the sum.

Soln: Applying the above formula, we get

A =

100 + 4 x |>. 100 15x(102)xl00

4x 12

= Rs 38250.

Exercise 1. Find the difference between simple interest and true dis-

count on Rs 960 due 4 years hence at 5% per annum simple interest. a)Rs32 b)Rs52 c)Rs42 d) None of these

2. The difference between the simple interest and the true discount on a certain sum for 6 months at 4% is Rs 15. Find the sum. a) Rs 32850 b)Rs28250 c)Rs 38250 d)Rs 38350

3. The difference between the simple interest and the true discount on a certain sum of money for 6 months at 6% is Rs 27. Find the sum. a) Rs 30900 b)Rs 39000 c)Rs 20900 d)Rs 30600

4. The true discount on a certain sum of money due after

2 i years at 6% per annum is less than the simple inter-

est on the same sum for the same time by Rs 81. Find the sum. a)Rs4140 b)Rs4240 c)Rs4150 d)Rs4250

5. The difference between the simple interest and discount on a certain sum of money due 1 year 9 months hence at 4% is Rs 7.35. What is the sum? a)Rsl605 b)Rsl805 c)Rsl525 d)RsI625

6. I f the difference between the interest and discount on a certain sum of money for 6 months at 6% be Rs 2.25. Find the sum.

a)Rs7525 b)Rs2255 c)Rs2575 d)Rs2755

Answers 1. a; Hint: Applying the given rule, we have

S I - T D

A x ( R T ) 960 x (4x5)" 100000 + RT) 100x(100 + 4x5)

= Rs32 2. c; Hint: Applying the given rule, we have.

960x20x20

100+120

15 =

\

A x 4 x -2

I00j^l00 + 4x--A = Rs 38,250.

3.a

4. a; Hint: Amount =

81 100 + 6x^1100

6x 2)

8100x115 225

= Rs4140 Note: Here SI - TD = Rs 81 (given).

5.a 6.c

Rule 6 To find the time when TD, Amount (A) and Rate (R) are given.

IOOxTD 7 7 m * ( T ; = ( A _ T D ) x R

Illustrative Example Ex: The true discount on Rs 2040 due after a certain time

at 6% per annum is Rs 40. Find the time after which it is due.

Soln: Applying the above formula, we get

100x40 _ 100x40 _ 2

T i m e = (2040 - 40) x 6 ~ 2000x6 ~ 6 y e a r s

= 4 months.

710 PRACTICE BOOK ON QUICKER MATHS

Exercise 1. The true discount

5% is Rs 60. Find a) 6 months b) 5

2. The true discount 3% is Rs 25. Find a) 2 months b) 3

3. The true discount 8% is Rs 80. Find a) 4 months b) 6

Answers l .c

on Rs 1860 due after the time after which it months c) 8 months on Rs 2525 due after the time after which it months c) 4 months on Rs 4080 due after the time after which it months c) 3 months

a cerain time at is due. d) None of these a cerain time at is due. d) 6 months

a cerain time at is due. d)None of these

2.c 3.c

Rule 7 To find the rate (R) when TD, Amount (A) and Time (T) are given.

IOOxTD * " " W = ( A - T D ) x R

Illustrative Example Ex: The true discount on Rs 260 due 3 years hence is Rs

60. Find the rate per cent. Soln: Applying the above formula, we get

100x60 100x60 Rate = = 10% (260 - 60) x 3 200x3

.-. rate per cent is 10% per annum.

Exercise 1. The true discount on Rs 1,860 due 3 years hence is Rs

60. Find the rate per cent. a) 10% b)!2% c)5% d)15%

2. The true discount on Rs 2575 due 4 months hence is Rs 75. Find the rate per cent of interest. a) 8% b)6% c)9% d) None of these The true discount on Rs 340 due 5 years hence is Rs 40. Find the rate per cent.

3.

a) 3% b) 2-o/c c ) 3 ^ % d) 3 y %

4. The true discount on Rs 2080 due 2 years hence is Rs 80. Find the rate per cent. a) 4% b)8% c)2% d) None of these

, 1 If the true discount on Rs 161 due 2 years hence be

Rs 21, at what rate per cent is the interest calculated? a) 6% b)4% c)8% d)12% If the discount on Rs 2273.70 due at the end of a year and a half be Rs 128.70, what is the rate of interest?

a) 6% b)4% c)3% . 1

d) 4-o/o 2

7. What must be the rate of interest in order that the dis-count on Rs 774.76 payable at the end of 3 years may be Rs 83.01?

a) 3% b)2% c)4% d) None of these

Answers 1. a; Hint: Applying the above rule,

IOOxTD _ IOOxTD _ 100x60 R = PWxT " ( A - T D ) x T ~ 7800x3

= 10% [since PW = A - T D ] .-. The rate per cent is 10% per annum

2. c 3.b 4.c

5. a; Hint: Required answer = 100x21

= 6%. (161 -21)x2 -

2 6. b

7. c; Hint: Rate % = 100x83.1 8310

(774.76-83.0 l )x 3 691.75x3 4%

Rule 8 To find the sum or amount (A) when simple interest and true discount are given.

SIxTD Sum or Amount (A) =

o l 1 U

Illustrative Example The true discount on a certain sum of money due 6 years hence is Rs 200 and the simple interest on the same sum for the same time and at the same rate is Rs 300. Find the sum. Applying the above formula, we get

300x200 Sum(A) =

Ex:

Soln:

300-200 = Rs600.

Exercise 1. The true discount on a certain sum of money due 3 years

hence is Rs 100 and the simple interest on the same sum for the same time and at the same rate is Rs 120. Find the sum and the rate per cent.

a)Rs600, 6yO / o b)Rs400,10%

2. c) Rs 500,8% d) None of these The simple interest and the true discount on a certain sum for a given time and at a given rate are Rs 25 and Rs 20 respectively. The sum is: a)Rs500 b)Rs200 c)Rs250 d)Rsl00 The true discount on a certain sum of money due 5 years hence is Rs 34 and the simple interest on the same sum for the same time and at the same rate is Rs 51. Find the sum.

True Discount 711

a)Rsl02

Answers

b)Rsl20 c)Rsl05 d)Rs!10

1. a; Hint: Sum due ; SIxTD [120x100 S I - T D 20 Rs 600

100x120 ^2 Rate= . . . - = 6-o/ 0

600 x j 3 2.d 3. a

Rule 9 To find the rate (R) when simple interest (SI), time (T) and true discount (TD) are given.

Rate (R) 100 ' SI T TD

-1

Illustrative Example Ex:

Soln:

The true discount on a certain sum of money due 6 years hence is Rs 200 and the simple interest on the same sum for the same time and at the same rate is Rs 300. Find the rate per cent per annum. Applying the above formula, we get

R 100 ~6~

300 200

-1 2 5 - 8 ' / ~3~ 3 P e r a n n u m

100 SI

Vote: Time (T) = T D 1 years.

Exercise 1. The true discount on a certain sum of money due 5 years

hence is Rs 75 and the simple interest on the same sum for the same time and at the same rate is Rs 150. Find the rate per cent per annum. a) 10% b)8% c)18% d)20%

2. The true discount on a certain sum of money due 8 years hence is Rs 150 and the simple interest on the same sum for the same time and at the same rate is Rs 450. Find the rate per cent per annum. a) 20% b) 15% c)25% d)30%

2 3. The true discount on a certain sum of money due 2

years hence is Rs 150 and the simple interest on the same sum for the same time and at the same rate is Rs 200. Find the rate per cent per annum.

a) I2j%

c)10%

Answers l . d

b) 12%

d) 8 - %

Rule 10 Application of the formula

(TD)2

PWX x g, x 7j PW2xR2xT2

(A-TD^xR, xTx

(A-TD2)xR2xT2

Illustrative Example Ex:

Soln:

I f Rs 60 be the true discount on Rs 240 for a certain time, what is the discount on the same sum for double the time, the rate being the same in both the cases. Applying the above formula, we have

60 T D ,

240

( 2 4 0 - 6 0 ) x R x T " ( 2 4 0 - T D 2 ) x R x 2 T

T D , T D ,

180 2x60 T D , =Rs96

The true discount on the same sum for double the time is Rs 96.

Exercise

2.c 3. a

1. I f Rs 21 be the true discount on Rs 371 for a certain time, what is the discount on the same sum for double the time, the rate being the same in both the cases. a)Rs40 b)Rs 39.75 c)Rs 40.25 d) None of these

2. I f Rs 120 be the true discount on Rs 480 for a certain time, what is the discount on the same sum for double the time, the rate being the same in both the cases. a)Rsl92 b)Rs96 c)Rs48 d) None of these

3. I f Rs 42 be the true discount on Rs 742 for a certain time, what is the discount on the same sum for double the time, the rate being the same in both the cases. a)Rs80 b)Rs 39.75 c)Rs79.5 d)Rs 69.25

Answers l .b 2. a 3.c

Rule 11 Application of the formula,

PW! _ 100 + R x T 2 PW 2 ~ 100+RxT,

Where PW, = Present worth of an amount due in T, years

PW 2 = Present worth of the same amount due in T 2

years

R = Rate of interest per annum.

Illustrative Example Ex: The present worth of a bill due 6 months hence is Rs

1500 and i f the bill were due at the end of 2 years, its present worth would be Rs 1200. Find the rate per cent per annum.

712 PRACTICE BOOK ON QUICKER MATHS

Soln: Applying the above formula, we have

1500 _ 100 + R x 2

" ^ l O O + Rx-L

[Since T, = 6 months = years, T 2 = 2 year]

5R HR > 500 + =400 + 8R= - = 100

2 2

.-. R= 18%

Exercise 1. The present worth of a bill due 7 months hence is Rs

1 1200, and if the bill were due at the end of 2 years, its

present worth would be Rs 1016. Find the rate per cent and the sum of the bill. a) 10%, Rs 1270 b) 8%, Rs 1720 c) 16%, Rs 1570 d) 18%, Rs 1560

2. The present worth of a bill due 1 year hence is Rs 3000 and i f the bill were due at the end of 4 years, its present worth would be Rs 2400. Find the rate per cent per an-num.

200 a ) - %

100 0 ) - %

300 0 - %

1

400

3. The present worth of a bill due 1 years hence is Rs 6

2400 and i f the bill were due at the end of 5 years, its present worth would be Rs 2032. Find the rate per cent per annum.

a) 10% b)20% c)5% d) 12%

Answers 1. a; Hint: Detail Method: Sum due = PW + TD = PW + SI on

PW Now,sumdue = (Rs 1200 + SIon Rs 1200 for 7 months)

Also, sum due = (Rs 1016 + SI on Rs 1016 for years)

{Rs 1200 +SI on Rs ^ 2 0 0 > < ^

= {Rs 1016 +SI on Rs

for 1 year}

for 1 year} ' 5^ 1016x-

2) or, {Rs 1200 + SI on Rs 700 for 1 year}

= {Rs 1016 + SI on Rs 2540 for 1 year} or, SI on Rs (2540 - 700) for 1 year

= Rs(1200-1016) or, SI on Rs 1840 for 1 year = Rs 184

Rate: 100x184 1840x1 = 10%.

Also, sum due = Rs 1200 + (SI on Rs 1200 for 7 months at 10%)

= Rs 7 10

1200+ 1200x x 12 100 J = Rs 1270

Quicker Method: Applying the given rule, we have

1 2 Q 0 100 + R x |

1016 100+Rx

or,

12

I20U l O i i R x I = 1016fl00 + R x - A 12 J 1 2

or,3680R = 36800 .-. R=10% .-. Sum due = 1200 + (SI on Rs 1200 for 7 months at 10%)

Rs

2.b

1200 +

3.c

1200x x 12 100 = Rsl270.

Rule 12 Theorem: If the simple interest on Rs Pfor Tyears be equal to the true discount on Rs A for the same time and at the same rate, then the rate per cent per annum is given by

1 0 0 \A p1 \A-t\PxT

Illustrative Example Ex: I f the simple interest on Rs 600 for 5 years be equal to

the true discount on Rs 720 for the same time and at the same rate, find the rate per cent per annum.

Soln: Applying the above theorem, we have the required rate%

720-600 t M 120x100 -xl00 = =4%. 600x5 600x5

Exercise 1. I f the simple interest on Rs 400 for 8 years be equal to the

true discount on Rs 800 for the same time and at the same rate, find the rate per cent per annum.

a) 12-% b) 12% c ) 1 0 ^ % d ) 8 ^ %

2. I f the interest on Rs 50 at 4 % be equal to the discount

on Rs 59 for the same time and at the same rate when is the latter sum due? a) 2 years b) 4 years c) 6 years d) 3 years

3. If the interest on Rs 150.62 at 8% be equal to the dis-

True Discount 713

count on Rs 225.93 for the same time, and at the same rate, when is the latter sum due?

a ) 4 years b) 6 years c) 6 years d) 12 years

4. I f the discount on Rs 3050 be equal to the simple interest on Rs 3000 for the same time, find the time, the rate of interest being 5% per annum. a) 4 months b) 6 months c) 3 months d) None of these

Answers 1. a 2. b; Hint: Applying the given rule, we have

9 59-50 .-. T = 4years. -xl00

3. b

4. a; Hint: 5

2 50xT

3050-3000 ,s j . x 1 0 0 or, T = - years = 4 months.

Rule 13 To find present worth of A rupees due n years hence at r per cent compound Interest payable annually, We have

(i) A = Present Worth

(ii) Present Worth -

1 + 100J

1+-100 J

(Hi) True Discount - Amount - Present Worth

Illustrative Example Ex: Find the present worth and discount of Rs 1722.25

/ 3 due in 2 years reckoning compound interest at 3%.

Soln: Applying the above formula, we have

1722.25 1722.25x400x400 PW =

15/ 1 + -

100

415x415 = Rsl600

True Discount = A-PW = 1722.25 -1600 = Rs 122.25.

Exercise 1. Find the present worth of a bill of Rs 2420 due 2 years

hence at 10% compound interest. Also find the true dis-count. a)Rs2000,Rs420 b) Rs 2200, Rs 520 c) Rs 2100, Rs 460 d) None of these

2. Find the present worth of a bill of Rs 2916 due 2 years hence at 8% compound interest. Also calculate the true

discount. a)Rs2500,Rs416 b)Rs2400, Rs516 c) Rs 2600, Rs 316 d) None of these

3. Find the present worth of Rs 220.50 due in 2 years reck-oning compound interest at 5%. a)Rs200 b)Rs250 c)Rs210 d)Rs310

4. Find the present worth of Rs 169 due in 2 years reckon-ing compound interest at 4%. a) Rs 156.25 b)Rs 150.50 c) Rs 158.50 d) None of these

5. Find the true discount on Rs 39.69 due in 2 years reckon-ing compound interest at 5% a)Rs3.69 b)Rs5 c)Rs5.69 d)Rs4.69

6. Find the true discount on Rs 226.59 due in one year 9 months, reckoning compound interest at 5%. a)Rsl8.6 b)Rs 18.59 c)Rs 16.59 d)Rs 28.59

Answers 1. a; Hint: Here sum is put on compound interest, hence

applying the given rule, we have

PW = 2420

100 J 1 + 10}

100

: Rs 2000

TD = PW-P .-. True Discount = 2420-2000 = Rs 420.

2. a 3. a 4. a 5. a; Hint: Present Worth

39.69 3969x100x100

100x105x105

I 100 J

True discount = Rs 39.69 - 36 = Rs 3.69

= Rs36

6.b

Rule 14 To find equal annual payments: Theorem: A sum of Rs A is borrowed to be paid back in n years in n equal annual payments, R per cent compound interest being allowed. Then the value of annual payment

A is given by Rs

100 R

100 100 + /?

Illustrative Example Ex: A sum of Rs 2550 is borrowed to be paid back in two

years by equal annual payments, 4 per cent, com-pound interest being allowed. What is the annual payment?

Soln: Detail Method: Let Rs x be the annual payment.

714 PRACTICE BOOK ON QUICKER MATHS

The present value of the 1st payment

x _ 25x

100 J

The present value of the 2nd payment

x 625x

100 J

676

25 625 N 0 W ' 2 6 X + 6 7 6 * = 2 5 5

1275* 676

= 2550 1352

.-. the annual payment = Rs 1352. Quicker Method: Applying the above theorem, we have the reqd annual payment

= Rsl352. 2550 _ 2550x676

100 25x51

4 1 UJ Exercise 1. A sum of Rs 820 is borrowed to be paid back in two

years by equal annual payments, 5 per cent, compound interest being allowed. What is the annual payment? a)Rs441 b)Rs410 c)Rs420 d) None of these

2. A sum of Rs 2600 is borrowed to be paid back in two years by equal annual payments, 8 per cent, compound interest being allowed. What is the annual payment? a)Rsl358 b)Rsl458 c)Rs!498 d) None of these

Answers l .a 2.b

Miscellaneous 1. I f Rs 10 be allowed as true discount on a bill of Rs 110

due at the end of a certain time, then the discount al-lowed on the same sum due at the end of double the time is: a)Rs20 b)Rs21.81 c)Rs22 d)Rs 18.33

2. Rs 20 is the true discount on Rs 260 due after a certain time. What will be the true discount on the same sum due after half of the former time, the rate of interest being the same: a)Rsl0 b)Rs 10.40 c)Rs 15.20 c)Rsl3

3. A has to pay Rs 220 to B after 1 year. B asks A to pay Rs 110 in cash and defer the payment of Rs 110 for 2 years. A agrees to it. Counting, the rate of interest at 10% per annum in this new mode of payment: a) there is no gain or loss to any one

b) A gains Rs 7.34 c) A loses Rs7.34 d) A gains Rs 11

1 4. A owes B Rs 456.75 payable 4 months hence and B

owes A Rs 455.51 payable 3 months hence. I f they agree to settle their account by a ready money payment, what sum should be paid over and to whom, reckoning the rate of true discount at 4 per cent per annum? a )Re l ,A b)Rs2,B c)Rs2,A d)Re l ,B

Answers 1. d; Hint: SI on Rs (110 -10) for a given time = Rs 10

SI on Rs 100 for double the time = Rs 20 Sum = Rs(100 + 20)Rsl20.

T D on Rs i 10 Rs | T ^ x l l Rs 18.33.

2. b; Hint: SI on Rs 240 for a given time = Rs 20 SI on Rs 240 for half the time = Rs 10 .-. Rs 10isTDonRs250.

So, TD on Rs 260 = Rs f ^ x 2 6 0 j = Rs 10.40.

3. b; Hint: A has to pay the PW of Rs 220 due 1 year hence, which is

= Rs 100x220

100 + (10xl) = Rs200

A actually pays = Rs [110 + PW of Rs 110 due 2 years hence]

= Rs 110 + 100x110

100+ (8x2)

.-. A gains = Rs [200 -192.66] = Rs 7.34.

= Rs 192.66

.-1 3 4. a; Hint: time = 4 months = yr, rate = 4 per cent

203 amount of Rs 100 = Rs

20"* PW = Rs 456.75 + ^-x 100 =Rs450

2

Again, time = 3 months = yr; rate = 4 per cent

PW = Rs 455 .51x =R S 451 104

Hence the required sum to be paid to A = Rs451-Rs450=Rel.