True Discount(Ch16)

7/28/2019 True Discount(Ch16)

1/8

VISHAL JALAN (THE MATHS GURU) PURE TEACHING, NOTHING ELSECHAPTER: 16

TRUE DISCOUNT

16.1 DEFINITION

Suppose A has borrowed money from B. A sum of Rs 160 is due 4 years hence. But the borrower A wants to clear off th

debt right now, i.e. 4 years before the due date. Hence, A will have to pay definitely less than Rs 160. In fact, the monewhich amounts to Rs 160 after 4 years at a standard or agreed rate of interest, should be the amount paid by A to clear o

his debt right now.

Let the rate of interest in this case be 15% per annum. Then Rs 100 now becomes 160 after 4 years at the rate of 15% pe

annum. So, A pays Rs 100 right now, it is as good as to pay Rs 160 after 4 years from right now.

The sum due is called the Amount. The money to be paid right now is called the Present Worth (PW) of the sum due. Th

difference between the amount and the present worth is called the True Discount.

So, here true discount for A = 160 - 100 = Rs 60

[i.e. interest on Rs 100 for 4 years at 15% per annum)

So, True Discount = the interest on the present worth (PW) for the time which will elapse before the debtor bill is due t

be discharged.

100

.).(..

TRWPDT

=

16.2 IMPORTANT FORMULAE

(a) .).(100

.).(.. WPP

TRWPDT =

=

(b)

+

=

1001

..

RT

PWP

(c)

+

=

RT

PDT

1001

..

(d)100

..TRP

IS

=

(e))(

....

TDSI

DTISP

=

(f)100

........on..

TRDTDTISDTIS

==

(g)T

R

PWP

+

=

1001

.., when the sum is put at compound interest.

(h)1

2

2

1

100

100

)(

)(

RT

RT

PW

PW

++

=

Where PW1 = present worth of an amount due in T1 years

PW2 - present worth of the same amount due in T2 years . .

R - rate of interest per annum

JALAN TUTORIALS, 13/5, SWARN PATH, MANSAROWAR, JAIPUR (M: 9828266445)


2/8

VISHAL JALAN (THE MATHS GURU) PURE TEACHING, NOTHING ELSE(i)

)100(100..

RT

TRPTRPWPWPDT

+

=

==

(ii)( )

( ) 222

111

222

111

2

1

2

1

.).(

.).(

TRTDP

TRTDP

TRPW

TRPW

PWP

PWP

DT

DT

=

=

=

(k) Installments to pay-off debt

Suppose borrower is to pay a sum of Rs P due after T years hence, but he wants to pay through equal installments at anagreed interval (may be quarterly, monthly, half-yearly or yearly) to discharge off his debt in T years.

Hence, through equal installment payments, the debt of Rs P due in T years is cleared in T years.

Then,

2

)1(

100

+=

nn

y

raanP

Where P = amount due after T years hence,

n = number of installments to be actually paid to discharge the debt,

r = rate per cent per annum charged on simple interest,

y = number of installments per year,

= 1, if yearly installment is paid,= 4, if quarterly installment is paid,

= 2, if half yearly installment is paid,

= 12, if monthly installment is paid,

a = Amount of each installment in Rs

(l) Price)Sale(P100

100..

)%100(

100=

+

=

=RT

PPC

lorgPW

Where PW - Present worth of the amount P Rs given on credit for time T at R% per annum

C.P - Cost price of the article

% g or % l per cent gain or per cent loss.

Sale price S.P. = P = Amount of the bill due in T years hence (sale price of the article)

(m) Present Worth of a sum due in N equal installment.

( ) ( )

+

++

+= .......

1001

1001100

21 RTRTNPPW

T1,T2 - time for each installment to be paid.



3/8

VISHAL JALAN (THE MATHS GURU) PURE TEACHING, NOTHING ELSE

SOLVED EXAMPLES

E-1 Find the present worth (P.W.) and the true discount reckoning 6% per annum simple interest of Rs 176 due in 20 month

time.

S-1 Using the formula.

12100

2061

176

1001

..

+=

+

=RT

PWP

= Rs 160

T.D = P - P.W =176 - 160 =16.

E-2 Find the present worth of a bill of Rs 2420 due 2 years hence at 10% compound interest. Also find the true discount.

S-2 Here sum is put on compound interest, hence using the formula,

2000.Rs

100

101

2420

1001

..2=

+

=

+

=T

R

PWP

T.D = P.W. - P

True Discount = 2,420 - 2,000 = Rs 420.

E-3 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 presen

worth.

S-3 using the formula,

Present worth,3000Rs.

12

812

240100..100.. =

=

=

TR

DTWP

Present worth is Rs 3,000P = Amount of the bill = P.W. + T.D = 3,000 + 240 =3,240.

E-4 Find the difference between simple interest and true discount on Rs 960due A years hence at 5% per annum simp

interest.

S-4 Using the formula,



4/8


32.Rs.120100

2020960

)54100(100

)54(960

)100(100

)(....

2

2

=

=

+

+

=RT

RTPDTIS

E-5 The difference between the simple interest and the true discount on a certain sum for 6 months at 4% is Rs 15. Find th

sum.

S-5)100(100

)(....

2

RT

RTPDTIS

+

=

Rs.38250.=

+

PP

2

14100100

2

14

15

2

E-6 The true discount on Rs 2040 due after a certain time at 6% per annum is 40. Find the time after which it is due.


62000

40100

.).(

..100..100

=

=

= RDTPDT

RPW

DT

T (Since PW = P T.D.)

=2/6 year = 4 months

The bill is due 4 months hence.

E-7 The true discount on Rs 1,860 due 3 years hence is Rs 60. Find the rate per cent.


T.D.)PPW(Since%10

31800

60100

)(

..100

..100

==

=

=

=

TTDP

DT

TPW

DTR

The rate per cent is 10% per annum.

E-8 The true discount on a bill due 5 months hence at 12 % per annum is Rs 75. Find the amount of the bill.

S-8 using the formula,

+=

+=

12

512

100175

1001..Amount

RTDTP

Rs.15752175 ==Amount of the bill is Rs 1,575.

E-9 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 th

same time and at the same rate is Rs 300. Find the sum and the rate per cent.S-9 Using the formula,

%3

216%

3600

300100

..100)(

600.600.100

200300

.).(.).(

....

=

=

=

=

=

==

TP

ISRRate

RsissumTheRsDTIS

DTISPSum

The rate % is 163

2% per annum.



5/8

VISHAL JALAN (THE MATHS GURU) PURE TEACHING, NOTHING ELSEE-10 If 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, th

rate being the same in both the cases?


( )( )

96Rs.54802

3

602

1802402)240(

)60240(60

)(

)(

222

2

22

222

111

2

1

===

=

=

=

TDTDTD

TD

TRTD

TR

TD

TRTDP

TRTDP

TD

TD

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

E-11 The present worth of a bill due 6 months hence is Rs 1,500 and if the bill were due at the end of 2 years, its present wor

would be Rs 1,200. Find the rate per cent and the sum of the bill.


%11

218%

11

200,100

2

11

84002

5500

year2T,year2

1

month6TSince

2

1100

2100

1200

1500

100

100

)(

)(

21

1

2

2

1

===

+=+

===+

+=

++

=

RR

RR

R

R

RT

RT

PW

PW

Using the formula,

1636.30Rs.11

121500

15001211

2

1

10011

2001

1500

1001

=

=

=

+

=

+=

P

PP

RT

PPW

The rate per cent is 1811

2% per annum and the sum of the bill is Rs 1,636.30.

E-12 If 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 rat

find the rate % per annum.

S-12 Since S.I. on Rs. 600 = T.D. on Rs. 720

annumper4%5

20120100

)100(

720

100

600

===+

+

=

RRT

RT

TRTR

E-13 A man borrows a sum of money at 16% per annum simple interest, promising to pay Rs 8,480 after a year from the date o

borrowing. If he wants to discharge the debt by paying four equal quarterly installments, how much should be eac

installment?

S-13 Using the formula, (y=4)



6/8


Rs.2000106

84840258480

25

106

2

34

4100

1648480

2

)1(

100

=

==

+=

+=

aa

aa

nn

y

raanP

EXERCISE

1. Find the present worth and true discount of Rs 9,950 due 34

1years hence at 7

2

1% per annum simple interest.

[Rs 8,000, Rs 1950]

2. Find the present worth and true discount of Rs 617.50 due 22

1years hence at 7

2

1% per annum simple interest.

[Rs 520, Rs 97.50]

3. Find the present worth of a bill of Rs 2,160 due 3 years hence at 20% compound interest. Find the true discount also.

[Rs 1,250, Rs 910]

4. The true discount on Rs 1,400 due after a certain time at 10% is Rs 200. Find the time after which it is due.

years3

5

5. The true discount on a bill 2 years hence at 5% per annum is Rs 120. Find the amount of the bill and its present worth.[Rs 1,200, Rs 1,320]

6. The true discount on Rs 5,250 due 2 years hence is Rs 250. Find the rate per cent of interest.

annumper%2

12

7. The true discount on bill due 6 months hence at 12% per annum is Rs 36. Find the amount of the bill.

[Rs 636]

8. If T.D and S.I on certain bills are Rs 60 and Rs 62 respectively for a time of 8 months, find the amount of the bill and th

rate per cent.[Rs 1,860, 5%]

9. If Rs 20 is the true discount on Rs 260 due after a certain time, what will be the true discount on the same sum after ha

of the former time, the rate of interest being the same.

[Rs 10.40]10. The simple interest on true discount of a certain sum due 2 years hence at 4% per annum is Rs 16. Find the sum.

[Rs 2,700]

11. A man purchased a radio for Rs 3,000 and sold it for Rs 3,960 on credit for sometime and still gained 10%. If money b

worth 5%, find the time for which the credit was allowed.[4 years]

12. An HMT watch was bought for Rs 1,250 and sold at once for Rs 1278 due in 9 months hence. Find the gain or loss pe

cent if the rate of interest is 8% per annum. [3..54% gain]

13. Find the present worth of Rs 1,040 in two equal half yearly installments at 8% per annum simple interest.[Rs 981.48]

SPEED TEST - 16

Paper No. 1 [Time - 5 min.]

1. True discount is the interest calculated on

(a) Principal (b) Amount (c) Sum (d) Present worth

(e) None of these

2. If the sum due is Rs P and true discount is TD, then present worth (PW) is

(a) P + T.D (b) P - T.D. (c) T.D P (d)DT

P

.

2

(e) P T.D.

3. The difference between simple interest and the true discount is the simple interest on(a) Sum due (b) Present worth (c) true discount (d) None of these

4. The true discount on a sum, due T years hence at R% per annum is Rs T.D. Then the sum due (P) is given by



7/8

VISHAL JALAN (THE MATHS GURU) PURE TEACHING, NOTHING ELSE(a)

RT

RTTD

)100( (b)

100

.. TRDT (c)

TR

DT

..(d)

)100(

..

RT

RTDT

+

(e) Data insufficient

5. A borrower has to pay Rs P due T years hence at R% per annum, but if he desires to clear off his debt right now, he has t

pay

(a) Rs P (b)

+1001RT

P

(c)

)100( RT

P

+(d) P

TRP+

100

(e) Data insufficient

6. The difference between simple interest and true discount is equal to

(a) simple interest on true discount

(b)100

.. TRDT

(c) Always a positive quantity because simple interest is greater than true discount

(d) all of above three

(e) None of these

7. Simple interest is greater than true discount because

(a) Simple interest is on the sum due

(b) True discount is interest on present worth

(c) Present worth is less than the sum due(d) All the above three

(e) None of these


8. The present worth of Rs 1,404 due in two equal half yearly instalments at 8% per annum simple interest is

(a) Rs 1,404 (b) Rs 1,400 (c) Rs 1,275 (d) Rs 1,325

(e) Rs 1,483

9. If the simple interest on Rs 500 for 3 years equal to the true discount on Rs 600 for the same time and same rate, then th

rate % per annum is

(a) 4% (b) 5% (c) 10% (d) Data insufficient(c) 8%

10. A Timex watch was bought for Rs 600 and sold the same day for Rs 650.25 at a credit of 9 months and still there was

gain of 2%. The rate % per annum is

(a) 5% (b) 6% (c) 43

2% (d) 8

3

1% (e) 6

2

1%

11. A Leo toy was bought for Rs 195 in cash and sold for Rs 220 at a credit of 1 year. If the rate of interest is 10%, then theris a

(a) Loss of Rs 5 (b) Gain of Rs 5 (c) Gain of Rs 10 (d) Loss of Rs 20

(e) None of these


12. The simple interest and the true discount on a certain sum for a given time and rate are Rs 45 and Rs 40 respectively

Then the sum is

(a) Rs 360 (b) Rs 400 (c) Rs 600 (d) Data insufficient(e) Rs 540

13. The true discount on Rs 550 due after a certain time at 5% is Rs 50. The time after which it is due is(a) 10 months (b) 2 months (c) 2 years (d) 9 months

(e) None of these

14. The true discount on Rs 1,860 due 10 months hence, is Rs 60. The rate % per annum is

(a) 52

1% (b) 4 % (c) 5% (d) 8

3

1%

(e) 43

1%

15. If Rs 75 be the true discount on Rs 300 for a certain time, what is the discount on the same sum for double the time, th

rate % being same in both the cases?

(a) Rs 150 (b) Rs 120 (c) Rs 75 (d) Rs 225



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VISHAL JALAN (THE MATHS GURU) PURE TEACHING, NOTHING ELSE(e) Rs l80

16. The true discount of a sum due 2 years hence at 10% per annum is Rs 88. The sum due is

(a) Rs 528 (b) Rs 548 (c) Rs 488 (d) Rs 400

(e) Rs 508

17. The true discount of Rs 2,624, the sum due 8 months is Rs 224. Find the rate % per annum.

(a) 14% (b) 10% (c) 5% (d) 8%(e) 6%

*****End*****


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True Discount(Ch16)