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TRUE DISCOUNT

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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.

TRUE DISCOUNT -> IMPORTANT FORMULAE

Let rate = R% per annum and Time = T years. Then,
I. P.W. = 100 * Amount / 100 + (R*T) = 100 * T.D. / R * T
II. T.D. = (P.W.)* R * T / 100 = Amount * R * T / 100 + (R * T)
III. Sum = (S.I.) * (T.D.) / (S.I.) - (T.D.)
IV. (S.I.) - (T.D.) = S.I on T.D.
V. When the sum is put at compound interest, then P.W. = Amount / [1+R/100]T;

TRUE DISCOUNT -> SOLVED EXAMPLES

1. Find the present worth of Rs. 930 due 3 years hence at 8% per annum. Also find the discount.
  Sol. P.W. = 100 * Amount / 100 + (R*T)
= Rs.[100 * 930 / 100 + (8*3)] = Rs. [100 * 930 / 124] = Rs.750.
T.D. (Amount) - (P.W.) = Rs. (930 - 750) = Rs.180.

TRUE DISCOUNT -> EXERCISE

1. If the true discount on a sum due 3 years hence at 14% per annum be Rs. 168, the sum due is:
 
  • A. 698
  • B. 768
  • C. 1430
  • D. 1980
Ans: B.
Sol.
P.W. = 100 * T.D./R * T = 100 * 168 / 14 * 2 = 600.
∴ Sum = (P.W. + T.D.) = Rs. (600 + 168) = Rs. 768.
 
2. The true discount on Rs. 1760 due after a certain time at 12% per annum is Rs. 160. The time after which it is due is:
 
  • A. 4
  • B. 6
  • C. 8
  • D. 10
Ans: D.
Sol.
P.W. = Rs. (1760 - 160) = Rs. 1600
∴ S.I. on Rs.1600 at 12% is Rs. 160.
∴ Time = [100 * 160 / 1600 * 12] = 5/6 years = [5/6 * 12] months = 10 months.
 
3. The simple interest and the true discount on a certain sum for a given time and at a given rate are Rs. 85 and Rs. 80 respectively. The sum is:
 
  • A. 1360
  • B. 1450
  • C. 1600
  • D. 1800
Ans: A.
Sol.
Sum = S.I. * T.D. / (S.I)-(T.D.) = 85 * 80 / (85 - 80) = Rs.1360.
 
4. If Rs. 10 be allowed as true discount on bill of Rs. 110 due at the end of a certain time, then the discount allowedon the same sum due at the end of double the time is:
 
  • A. 16
  • B. 18.33
  • C. 20.21
  • D. 26
Ans: B.
Sol.
S.I. on Rs. (110 - 10) for a certian time = Rs. 10.
S.I. on Rs. 100 for double the time = Rs. 20.
T.D. on Rs. 120 = Rs. (120 - 100) = Rs.20
T.D. on Rs. 110 = Rs. [20 /120 * 110] = Rs. 18.33
 
5. 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. 10.40
  • B. 11
  • C. 14.80
  • D. 15.40
Ans: A.
Sol.
S.I. on Rs. (260 - 20) for a given time = Rs. 20.
S.I. on Rs. 240 for half the time = Rs. 10.
T.D. on Rs. 250 = Rs. 10.
∴ T.D. on Rs. 260 = Rs. [10/250 * 260] = Rs. 10.40