32. TRUE DISCOUNT
IMPORTANT CONCEPTS
Suppose a man has to pay Rs. 156 after 4
years and the rate of interest is 14% per annum. Clearly, Rs. 100 at 14% will
amount to Rs. 156 in 4 years. So, the payment of Rs. 100 now will clear off the
debt of Rs. 156 due 4 years hence. We say that:
Sum due = Rs. 156 due 4 years hence;
Present Worth (P.W.) = Rs. 100;
True Discount (T.D.) = Rs. (156 - 100) = Rs. 56
(Sum due) - (P.W.).
We define : T.D. = Interest on
P.W.
Amount = (P.W.) + (T.D.).
Interest is reckoned on P.W. and
true discount is reckoned on the amount.
IMPORTANT FORMULAE
Let rate = R% per annum and Time = T years. Then,
1.
P.W.=[100 x Amount
/100 + (R x T)
=100 x T.D./
RxT
2. T.D.=[(P.W.) x R x T
/100]
= [ Amount x RxT/100
+ (R x T)]
3.(S.I.)*(T.D.) /(S.I.)-(T.D.)
4. (S.I.) - (T.D.) - S.I. on T.D.
5. When the sum is put at
compound interest, then
P.W. = Amount/[1 +R/100]^T
SOLVED EXAMPLES
Ex. 1. Find the present worth of Rs. 930
due 3 years hence at 8% per annum. Also find the discount.
Sol.
P.W=100 x Amount /[100 + (R x
T)]
=Rs.100 x 930/100+ (8x3)
= (100x930)/124
= Rs. 750,
T.D. = (Amount) - (P.W.) = Rs. (930 - 750) = Rs. 180.
Ex. 2. The true discount on a bill due 9
months hence at 12% per annum is Rs. Find the amount of the bill and its
present worth.
Sol. Let amount be Rs. x. Then,
x*R*T/100 + (R x T)
=T.D.
=>x * 12*3/ 4/[100+[12*3/4]]
=540
x= 540x109 = Rs.6540
Amount - Rs. 6540. P.W. = Rs. (6540 - 540) - Rs. 6000.
Ex. 3. The true discount on a certain
sum of money due 3 years hence is Rb. 250 and the simple interest on the same
sum for the same time and at the same rate is Rs. 375. Find the sum and the
rate percent.
Sol. T.D. = Rs. 250 and S.I. = Rs.
375.
Sum due =S.I. xT.D./ S.I. -T.D.
=375x250/375- 250
=Rs.750.
Rate=[100*375/750*3]%=16
2/3%
Ex. 4. The difference between the simple interest and true discount on
a certain sum
of money for 6 months at 12—% per annum is Rs. 25. Find the sum.
Sol. Let the sum be Rs. x. Then,
T.D. = (x*25/2*1/2)/(100+(25/2*1/2))=x*25/4*4/425=x/17
S.I=x*25/2*1/2*1/100=x/16
x/16-x/17=25
=>17x-16x=25*16*17
=>x=6800
Hence, sum due = Rs. 6800.
Ex. 5. A bill falls due in 1 year. The creditor agrees to accept
immediate payment of the half and to defer the payment of the other half for 2
years. By this arrangement
ins Rb. 40. What is the amount of the
bill, if the money be worth 12-z% ?
Sol. Let the sum be Rs. x. Then,
[x/2+(x/2*100)/100+(25/2*2)]-[(x*100)/(100+25/2*1]
=40
=>x/2+2x/5-8x/9=40
=>x=3600
Amount of the bill - Rs. 3600.
