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A person bought a television set paying Rs. $20,000$ in cash and promised to pay Rs. $1,000$ at the end of every month for the next $2$ years. If the money is worth $12\% $ p.a. converted monthly, what is the cash price of the television set? [${(1.01)^{ - 24}} = 0.7884$ ].

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Last updated date: 16th Jun 2024
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Answer
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Hint: Present value is the current value of the future sum of the money given in the specified rate of return. It also states that an amount of money today is worth more than the same amount of money in the future. Use the general formula for present value $ = \dfrac{c}{i}[1 - {(1 + i)^{ - n}}]$.

Complete step by step solution: The initial down payment $ = Rs.{\text{ 20,000}}$
The monthly payment, c$ = Rs.1000$
Duration of the instalments, n $ = {\text{24 months}}$
Rate of the Interest $ = 12\% $ per annum
Rate of the Interest, i $ = 1\% $ per month
Convert percentage in the form of fraction
$
  \therefore 1\% = \dfrac{1}{{100}} \\
  \therefore 1\% = 0.01 \\
 $
Apply the formula –
Present value $ = \dfrac{c}{i}[1 - {(1 + i)^{ - n}}]$
Place all the known values-
\[
  Present{\text{ }}value = \dfrac{{1000}}{{0.01}}[1 - {(1 + 0.01)^{ - 24}}] \\
  Present{\text{ }}value = \dfrac{{1000}}{{0.01}}[1 - {(1.01)^{ - 24}}] \\
 \]
[Place - ${(1.01)^{ - 24}} = 0.7884$(given)]
Present Value $
   = \dfrac{{1000}}{{0.01}}[1 - 0.7884] \\
    \\
 $
Simplify the Left hand side of the equation -
Present Value
                      $
   = \dfrac{{1000}}{{0.01}} \times 0.2116 \\
   = 21160 \\
 $
Present Value $ = Rs.{\text{ 21160}}$
Total Price is equal to the sum of the initial down payment and the present value.
Total Price $ = Initial{\text{ down payment + present value}}$
$
  Total\Pr ice = {\text{20000 + 21160}} \\
  Total\Pr ice = \;{\text{41160 }} \\
 $
The required solution is - The cash price of the television set is $41,160$ Rupees.

Note: In other words present value shows that the amount received in the future is not as worth as an equal amount received today. Always remember the relation among the present value and the principal amount. Always convert the percentage rate of interest in the form of fraction or the decimals and then substitute further for the required solutions.