High School

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A buys a shop for a ₹48750 down payment and ₹60000 after one year. If money is worth 10% per year compounded half-yearly, find the purchase price of the shop.

(a) ₹94280
(b) ₹103170
(c) ₹45530
(d) ₹54420

Answer :

To find the purchase price of the shop, we need to calculate the present value of all future payments, considering the compounded interest.

Here's how you can do it step-by-step:

  1. Given Information:

    • Down Payment = ₹48750
    • Future Payment = ₹60000 due after 1 year
    • Interest Rate = 10% per annum compounded half-yearly.

  2. Convert Annual Interest Rate to Half-Yearly:

    • The half-yearly rate [tex]r[/tex] is [tex]\frac{10\%}{2} = 5\%[/tex].
    • As a decimal, this is 0.05.

  3. Number of Compounding Periods:

    • Since the interest is compounded half-yearly, there will be 2 compounding periods in a year.

  4. Calculate Present Value of the Future Payment:

    • Present Value formula for compound interest:
      [tex]PV = \frac{FV}{(1 + r)^n}[/tex]
    • For our problem:
      [tex]PV = \frac{60000}{(1 + 0.05)^2}[/tex]
    • [tex]PV = \frac{60000}{(1.05)^2}[/tex]
    • [tex]PV = \frac{60000}{1.1025}[/tex]
    • [tex]PV \approx 54420 \, \text{(rounded to the nearest integer)}[/tex]

  5. Calculate Total Purchase Price:

    • Add the down payment to the present value of the future payment:
      [tex]\text{Total Purchase Price} = 48750 + 54420 = 103170[/tex]

Therefore, the purchase price of the shop is ₹103170.

The correct option is (b) ₹103170.

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