Aptitude True Discount Theory

Important Concepts & Formulas

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 R. 156 in 4 years. So, the payment of Rs. 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. = $\dfrac{100 \times Amount }{100 + (R \times T)}=\dfrac{100 \times T.D.}{R \times T}$

2. T.D. = $\dfrac{(P.W.) \times R \times T}{100} =\dfrac{Amount \times R \times T}{100 + (R \times T)}$

3. Sum = $\dfrac{(S.I.) \times (T.D.)}{(S.I.) - (T.D.)}$

4. (S.I.) - (T.D.) = S.I. on T.D.

5. When the sum is put at compound interest, then P.W. =$\dfrac{Amount}{\left(1+\dfrac{R}{100}\right)^{T}}$

Description :

True discount is a method of calculation and deduction of Interest in advance from the borrower at the time of disbursing the loan at a specific rate of Interest at the same time same rate of interest is considered for calculating interest on interest. The amount received plus the interest equals the amount to be paid at the maturity of the obligation.

Example:
If a customer is sanctioned a loan amount of Rs.10000 and let us suppose interest amount is Rs.1000 and the interest on the interest amount of Rs.1000 is Rs.100. The loan amount that will be disbursed to the customer will be (10000-1000+100 = 9100), i.e. the interest has been deducted at the time of disbursement of loan.

Let rate = R% per annum and Time = T years, Simple Interest = S.I, Amount = A, Present worth = P.W, True discount = T.D. Then,

1. Present Worth =$\dfrac{100 \times Amount}{100 + (R \times T)}=\dfrac{100 \times T.D.}{R \times T}$

2. True Discount =$\dfrac{(P.W.) \times R \times T}{100}=\dfrac{Amount \times R \times T}{100 + (R \times T)}$

3. Sum =$\dfrac{(S.I.) \times (T.D.)}{(S.I.) - (T.D.)}$

4. (Simple Interest) - (True Discount) = Simple Interest on True Discount

5. When the sum is put at compound interest, then Present Worth =$\dfrac{Amount}{\left(1+\dfrac{R}{100}\right)^{T}}$

Type 1:

The true discount on a certain sum of money due 3 years hence is Rs. 300 and the simple interest on the same sum for the same time and at the same rate is Rs. 400. Find the sum and the rate percent ?

Solution:
Given that,
True discount = Rs. 300
Simple interest = Rs. 400
Time = 3 years
Consider,
Sum due=Rs.$\dfrac{(S.I.) \times (T.D.)}{(S.I.) - (T.D.)}$
=>Sum due=Rs.$\dfrac{(400) \times (300)}{(400) - (300)}$
⇒ Sum due = Rs. 1200
Now,
Rate = $\dfrac{(100) \times (S.I.)}{(Sum due ) \times (T)}$
⇒ Rate = $\dfrac{(100) \times (400)}{(1200 ) \times (3)}$
⇒ Rate = $\dfrac{40000}{3600}$
⇒ Rate = 11.11 %
Therefore, Sum due = Rs. 1200 and Rate = 11.11 %

Exercise :

Type 2:

The difference between the simple interest and true discount on a certain sum of money for 6 months at 12$\dfrac{1}{2}$% per annum is Rs. 30. Find the sum?

Solution:
Given that,
Time = 6 months =$\dfrac{1}{2}$ of year
Rate =12 $\dfrac{1}{2}$% = $\dfrac{25}{2}$%
Consider,
True discount = $\dfrac{A∗R∗T}{100+(R∗T)}$
=>True discount = $\dfrac{x∗\dfrac{25}{2}∗\dfrac{1}{2}}{100+(\dfrac{25}{2}∗\dfrac{1}{2})}$
=>True discount =x*$(\dfrac{25}{4})*(\dfrac{4}{425})$
=>True discount =$\dfrac{25x}{425}$
=>True discount =x/17
Now,
Simple interest =(x*$\dfrac{25}{2} \times \dfrac{1}{2} \times \dfrac{1}{100}$)
=> Simple interest=$\dfrac{x}{16}$
Therefore,
$\dfrac{x}{16} –\dfrac{ x}{17}$ = 30
⇒ 17x – 16x = 30 * 16 * 17
⇒ x = 8160
Therefore, sum due = Rs. 8160.

Exercise :

Type 3 :

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 gains Rs. 50. What is the amount of the bill, if the money be worth 12(1/2)% ?

Solution:
Let the sum be x. Then,
⇒ $ \left[\dfrac{x}{2} + \dfrac{\dfrac{x}{2} * 100}{100 + (\dfrac{25}{2} * 2)}\right] –\dfrac{x∗100}{100+(\dfrac{25}{2}∗1)} = 50$
⇒ $\dfrac{x}{2}+\dfrac{2x}{5}–\dfrac{8x}{9} = 50$
⇒ $\dfrac{45x+36x–80x}{90} = 50$
⇒ $\dfrac{x}{90} = 50$
⇒ x = 4500
Therefore, amount of the bill = Rs. 4500.

Exercise :

Type 4:

The true discount on a bill due 9 months hence at 16 % per annum is Rs. 600. Find the amount of the bill and its present worth?

Solution:
Given,
Rate = 16 %
Time = 9 months = $\dfrac{3}{4}$ years
Let amount be Rs.x. Then,
Consider,
True discount =$\dfrac{x \times R \times T}{100 + (R \times T)}$
⇒ 600 = $\dfrac{x∗16∗\dfrac{3}{4}}{100+(16∗\dfrac{3}{4})}$
⇒ 600 = $\dfrac{12x}{100+12}$
⇒ 600 = $\dfrac{3x}{28}$
⇒ 3x = 600 x 28
⇒ x = $\dfrac{600 \times 28}{3}$
⇒ x = 5600
Therefore, Amount = Rs. 5600
Present Worth = Rs (5600 – 600) = Rs. 5000

Exercise :

Type 5 :

Find the present worth of Rs. 950 due 2 years hence at 6% per annum. Also find the discount?

Solution:
Given that,
Amount = RS. 950
Time = 2 years
Rate = 6%
Now,
Present worth = Rs. $\dfrac{100∗A}{100+(R∗T)}$
⇒ Present worth = Rs. $\dfrac{100∗950}{100+(6∗2)}$
⇒ Present worth = Rs. $\dfrac{100∗950}{112}$
⇒ Present worth = Rs. 848.21
Therefore,
True discount = amount – present worth = Rs. (950 – 848.21)
= Rs. 101.78
≅ Rs. 102

Exercise :