True Discount in General Aptitude

For a better understanding of the true discount concept, consider the situation where if person P goes to purchase a mobile which costs an amount A. The shopkeeper says that if he pays the amount now he can take it at the same cost, else if he opts to buy on credit he needs to pay some i% interest.  
If he pays the present worth on that instant by not opting at the buy on credit. The amount of interest saved by not opting for the 2nd method is the true discount.

Definition – The amount of interest saved by paying the present worth is called the true discount, i.e the difference between the amount and present worth is called the true discount.
So the important formulas that can be derived are :

TRUE DISCOUNT (T.D) = AMOUNT(A) – PRESENT WORTH(P.W)

Where the amount is the value after the addition of interest to the present value, so the true discount is the same as simple interest.

TRUE DISCOUNT (T.D) = (P.W * Duration(N)* Rate(R)/100

Where P.W is the present worth and duration is the period and rate is the % rate of interest.

PRINCIPAL WORTH (P.W) = (100 * T.D)/(Duration(N)* Rate(R))

Now T.D in terms of amount can also be derived by substituting the 3rd formula in 1st formula:

TRUE DISCOUNT (T.D) = (Amount(A) * Duration(N) * Rate(R))/(100+ (Duration(N)* Rate(R)))

Example 1 – Calculate the present worth of R.s 20,000 due 2 years and at the rate of interest of 10% per annum.
Solution – In the question, the given data is:
Amount (A)= 20,000
Duration (N) = 2 years
Rate of Interest (R) = 10%
Present Worth (P.W) = ?
We know that –  
T.D = A – P.W
A = P.W + T.D
So, for calculating True Discount (T.D)
T.D = (P.W * N *  R)/100
T.D = (P.W * 2* 10)/100 = P.W/5
so, A = P.W + P.W /5 = 6 * P.W/5 
P.W = 5 * A/6 = (5 * 20,000)/6 = Rs 16,666.66
Principal Worth (P.W) is 16,666.66

Example 2 – Calculate the % rate of interest for an amount of Rs10,000 due 6 months and the true discount hence is  Rs1000.
Solution – In the question, the given data is:
Amount (A)= 10,000
Duration (N) = 6 months = 6/12 year = 1/2 year
Rate of Interest (R) = ?
Now we know that
T.D = A – P.W
P.W = A – T.D
P.W = 10,000 – 1000 = 9000
P.W = (100 * T.D)/(N * R)
so,9000 = (100 * 1000)/(1/2*R)
R = (200 * 1000)/9000
Rate of interest (R) = 22.22%