Example 1: Find the prsent wroth of Rs. 930 due 3 years hance at 8% per annum. Also find the discount.
Solution:

P.W. = [latex]\frac{100 \times Amount}{100 + (R \times T)}[/latex] = Rs[latex][\frac{100 \times 930}{100 + (8 \times 3)}][/latex] = Rs. [latex]\frac{100 \times 930}{124}[/latex] = Rs. 750.
T.D. = (Amount) - (P.W.) = Rs. (930 -750) = Rs. 180.

Example 2: If the true discount on a sum due 2 years hence atr 14% per anuum be Rs. 168, the sum due is:
Solution:

P.W. = [latex]\frac{100 \times T.D.}{R \times T}[/latex] = [latex]\frac{100 \times 168}{14 \times 2}[/latex] = 600
Sum =(P.W. +T.D.) = Rs. (600 +168) = Rs. 768

Example 1: If Rs. 10 be allowed as true discount on a bill of Rs. 110 due at the end of a certain time, then the discount allowed on the same sum due at the end of double the time is:
Solution:

S.I on Rs. (110 -10) for a certain 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. ([latex]\frac{20}{120}[/latex] x 110) = Rs. 18.33.

Example 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 intrest being the same?
Solution:

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. ([latex]\frac{10}{250}[/latex] x 260) = Rs. 10.40.

Example 1: The true discount on a certain sum of money due 3 years hence is Rs. 250 and the simple intrest on the same sum for the same time and at the same rate is Rs. 375. Find the sum and the rate percent
Solution:

T.D. = Rs. 250 and S.I. = Rs. 375.
Sum due = [latex]\frac{S.I. \times T.D}{(S.I.) - (T.D.)}[/latex]
= Rs. [[latex]\frac{375 \times 250}{375 - 250}[/latex]] = Rs. 750.
Rate = ([latex]\frac{100 \times 375}{750 \times 3}[/latex])% = 16 [latex]\frac{2}{3}[/latex]%

Example 2: The simple intrest 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:
Solution:

Sum = [latex]\frac{S.I. \times T.D.}{S.I. - T.D.}[/latex] = [latex]\frac{85 \times 80}{(85 - 80)}[/latex] = Rs. 1360.

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 = [latex]\frac{100 * A}{100 + (R * T)}[/latex] = [latex]\frac{100 * T.D}{100 + (R * T)}[/latex]
2. True discount = [latex]\frac{P.W * R * T}{100}[/latex] = [latex]\frac{A * R * T}{100 + (R * T)}[/latex]
3. Sum = [latex]\frac{S.I. * T.D.}{S.I. - T.D.}[/latex]
4. Simple interest - true discount = Simple interest on true discount.
5. When the sum is put at compound interest, then Present worth = [latex]\frac{A}{(1 + \frac{R}{100})^T}[/latex]

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. [latex]\frac{S.I. * T.D.}{S.I. - T.D.}[/latex]
⇒ Sum due = Rs. [latex]\frac{400 * 300}{400 - 300}[/latex]
⇒ Sum due = Rs. 1200
Now,
Rate = [latex]\frac{100 * S.I.}{Sum due * T}[/latex]
⇒ Rate = [latex]\frac{100 * 400}{1200 * 3}[/latex]
⇒ Rate = [latex]\frac{40000}{3600}[/latex]
⇒ Rate = 11.11 %
Therefore, Sum due = Rs. 1200 and Rate = 11.11 %

2. the difference between the simple interest and true discount on a certain sum of money for 6 months at 12[latex]\frac{1}{2}[/latex]% per annum is Rs. 30. Find the sum?
Solution:

Given that,
Time = 6 months =[latex]\frac{1}{2}[/latex] of year
Rate = 12[latex]\frac{1}{2}[/latex]% = [latex]\frac{25}{2}[/latex]%
Consider,
True discount = [latex]\frac{A * R * T}{100 + (R * T)}[/latex]
⇒ True discount = [latex]\frac{x * \frac{25}{2} * \frac{1}{2}}{100 + (\frac{25}{2} * \frac{1}{2})}[/latex]
⇒ True discount = [latex]x * \frac{25}{4} * \frac{4}{425}[/latex]
⇒ True discount = [latex]\frac{25x}{425}[/latex]
⇒ True discount = [latex]\frac{x}{17}[/latex]
Now,
Simple interest = ([latex]x * \frac{25}{2} * \frac{1}{2} * \frac{1}{100})[/latex]
⇒ Simple interest = [latex]\frac{x}{16}[/latex]
Therefore,
[latex]\frac{x}{16}[/latex] - [latex]\frac{x}{17}[/latex] = 30
⇒ 17[latex]x[/latex] - 16[latex]x[/latex] = 30 * 16 * 17
⇒ [latex]x[/latex] = 8160
Therefore, sum due = Rs. 8160.

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[latex]\frac{1}{2}[/latex]%?
Solution:

Let the sum be [latex]x[/latex]. Then,
⇒ [[latex]\frac{x}{2}[/latex] + \frac{\frac{x}{2} * 100}{100 + (\frac{25}{2} * 2)}] - [latex]\frac{x * 100}{100 + (\frac{25}{2} * 1)}[/latex] = 50
⇒ [latex]\frac{x}{2} + \frac{2x}{5} - \frac{8x}{9}[/latex] = 50
⇒ [latex]\frac{45x + 36x - 80x}{90}[/latex] = 50
⇒ [latex]\frac{x}{90}[/latex] = 50
⇒ [latex]x[/latex] = 4500
Therefore, amount of the bill = Rs. 4500.

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 = [latex]\frac{3}{4}[/latex] years
Let amount be Rs.[latex]x[/latex]. Then,
Consider,
True discount = [latex]\frac{x * R * T}{100 + (R * T)}[/latex]
⇒ 600 = [latex]\frac{x * 16 * \frac{3}{4}}{100 + (16 * \frac{3}{4})}[/latex]
⇒ 600 = [latex]\frac{12x}{100 + 12}[/latex]
⇒ 600 = [latex]\frac{3x}{28}[/latex]
⇒ [latex]3x[/latex] = 600 x 28
⇒ [latex]x[/latex] = [latex]\frac{600 x 28}{3}[/latex]
⇒ [latex]x[/latex] = 5600
Therefore, Amount = Rs. 5600
Present Worth = Rs (5600 - 600) = Rs. 5000

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. [latex]\frac{100 * A}{100 + (R * T)}[/latex]
⇒ Present worth = Rs. [latex]\frac{100 * 950}{100 + (6 * 2)}[/latex]
⇒ Present worth = Rs. [latex]\frac{100 * 950}{112}[/latex]
⇒ Present worth = Rs. 848.21
Therefore,
True discount = amount - present worth = Rs. (950 - 848.21) = Rs. 101.78 ≅ Rs. 102