Aptitude : True Discounts

(1) A woman purchased a horse for Rs. 4000 and sold it the same day for Rs. 4800, allowing the buyer a credit of 3 years. If the rate of interest be 12% per annum, then the woman has a gain of:

• (a)500 Rs
• (b)525.5 Rs
• (c)555.5 Rs
• (d)583.7 Rs

Answer

Correct Answer : (d) 583.7 Rs
The woman sold the horse for Rs. 4800 but the money she gets isn’t immediate; it will be paid after 3 years. This is essentially like lending Rs. 4800 to the buyer at an interest rate of 12% per annum for 3 years.
To find out how much she is actually getting immediately (in present terms), we need to calculate the present value of Rs. 4800 received after 3 years.
The formula for the present value (PV) is:
PV = FV / (1 + r/n)^(nt)
Where:
FV = Future Value (4800)
r = interest rate per period (12% or 0.12 as a decimal)
n = number of compounding periods per year (since it’s not specified, we assume it’s compounded annually so n=1)
t = time the money is invested for in years (3)
So, PV = 4800 / (1 + 0.12/1)^(1*3)
After solving this, we get:
PV = 4800 / (1 + 0.12)^3
PV = 4800 / 1.404928
PV ? Rs. 3416.30
So, in present terms, the woman is getting Rs. 3416.30 for the horse. Since she purchased the horse for Rs. 4000, she actually has a loss, not a gain.
Loss = Cost Price – Present Value of Selling Price
Loss = 4000 – 3416.30 = Rs. 583.70

(2) A retailer owes a wholesaler Rs. 15,035 due 2 years hence. The retailer wants to settle the account after 4 months. If the rate of interest 15% per annum, how much cash should he pay?

• (a)13017 Rs
• (b) Rs
• (c) Rs
• (d) Rs

Answer

Correct Answer : (a) 13017 Rs
The retailer owes Rs. 15,035 to the wholesaler, which is due 2 years later. However, he wants to settle the account after 4 months (which is 1/3 of a year).
Given the time value of money (money available now is worth more than the same amount in the future due to its potential earning capacity), the amount that he should pay now would be less than Rs. 15,035. This is because the wholesaler would have earned interest on the owed amount over the remaining period (1 2/3 years).
This is a Present Value (PV) problem where we need to find out how much the future amount (Rs. 15,035) is worth today. We use the formula for the present value:
PV = FV / (1 + r/n)^(nt)
Here:
FV = Future Value = Rs. 15,035
r = interest rate per period = 15% or 0.15 in decimal form
n = number of compounding periods per year (since it’s not specified, we assume it’s compounded annually so n = 1)
t = time the money is invested or borrowed for in years = 1 2/3 years = 1.67 years
Substituting these values in the formula, we get:
PV = 15035 / (1 + 0.15)^(1*1.67)
After calculating, we get:
PV ? Rs. 13017.17
So, the retailer should pay approximately Rs. 13017.17 now to settle the account that is due 2 years hence.

(3) A man wants to sell his car. There are two offers, one at Rs. 22,000 cash and the other a credit of Rs. 24,880 to be paid after 10 months, money being at 20% per annum. Which is the better offer?

• (a)30000 Rs
• (b)22,000 Rs
• (c)24880 Rs
• (d)0 Rs

Answer

Correct Answer : (b) 22,000 Rs
The first offer is for Rs. 22,000 cash, so its present value is Rs. 22,000.
The second offer is for Rs. 24,880, but the money is to be received after 10 months, not immediately. Thus, we need to calculate the present value of this offer to see how much it is worth today.
To find the present value (PV), we use the formula:
PV = FV / (1 + r/n)^(nt)
where:
FV = Future Value = Rs. 24,880
r = interest rate per period = 20% or 0.20 in decimal form
n = number of compounding periods per year (since it’s not specified, we assume it’s compounded annually so n = 1)
t = time the money is invested or borrowed for in years = 10/12 years (since 10 months is equivalent to 10/12 years)
Substituting these values in the formula, we get:
PV = 24,880 / (1 + 0.20)^(1*(10/12))
After calculating, we get:
PV ? Rs. 21,461.37
Comparing the two present values:
Present value of the first offer: Rs. 22,000
Present value of the second offer: Rs. 21,461.37
The first offer of Rs. 22,000 cash is the better offer as its present value is higher.

(4) If Rs. 15 be allowed as true discount on a bill of Rs. 165 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:

• (a)25 Rs
• (b)40 Rs
• (c)30 Rs
• (d)55 Rs

Answer

Correct Answer : (c) 30 Rs
Given that Rs. 15 is allowed as a true discount on a bill of Rs. 165 for a certain time, we can write this as follows:
Simple Interest = Principal * Rate * Time
Where:
Simple Interest is the discount given (Rs. 15)
Principal is the sum due (Rs. 165)
Rate is the annual interest rate
Time is the period for which the sum is due.
We are trying to find the discount allowed on the same sum (Rs. 165) due at the end of double the time. Since the rate is the same, we only need to multiply the original discount (Rs. 15) by 2. This is because the discount (which is effectively simple interest) increases linearly with time. If the time doubles, the discount also doubles.
Therefore, the discount allowed on the same sum due at the end of double the time is: Rs. 15 * 2 = Rs. 30.

(5) A woman buys a piece of jewelry for Rs. 2950 in cash and sells it for Rs. 3200 at a credit of 2 years. If the rate of interest is 12% per annum, the woman incurs a loss or profit , if so how much ?

• (a)-398 Rs
• (b)352 Rs
• (c)398 Rs
• (d)425 Rs

Answer

Correct Answer : (a) -398 Rs
The woman sells the jewelry for Rs. 3200, but she allows the buyer a credit of 2 years. This is essentially like her lending Rs. 3200 to the buyer at an interest rate of 12% per annum for 2 years. So we need to calculate the present value of Rs. 3200 that she’ll receive after 2 years.
The formula for the present value (PV) is:
PV = FV / (1 + r/n)^(nt)
Here:
FV = Future Value (Rs. 3200)
r = interest rate per period (12% or 0.12 as a decimal)
n = number of compounding periods per year (since it’s not specified, we assume it’s compounded annually so n=1)
t = time the money is invested for in years (2)
So, PV = 3200 / (1 + 0.12)^(1*2)
After solving this, we get:
PV = 3200 / (1.12)^2
PV = 3200 / 1.2544
PV ? Rs. 2551.97
The present value of Rs. 3200 received after 2 years is approximately Rs. 2551.97. This is the effective amount the woman gets for the jewelry in today’s terms.
Since she purchased the jewelry for Rs. 2950, we can now determine whether she made a gain or a loss.
Gain (or loss) = Present Value of Selling Price – Cost Price = Rs. 2551.97 – Rs. 2950 = – Rs. 398.03.
The negative value indicates that the woman incurs a loss. So, the woman has a loss of approximately Rs. 398.03 due to allowing the buyer a credit of 2 years.