Question

An article is sold for Rs. 500 cash or for Rs. 150 cash down payments followed by 5 equal monthly instalments. If the rate of interest charged is \(18 \%\) p.a., compute the monthly instalment : (a) \(63.07 \%\) (b) \(37.06 \%\) (c) \(75.0 \%\) (d) \(73.06 \%\)

Short answer

Answer: The monthly instalment amount is Rs. 73.06.

Step-by-step solution

Find the monthly interest rate

Given the annual rate of interest, i = 18%. To find the monthly interest rate, we divide the annual interest rate by 12. So, the monthly interest rate, r = (i/12)/100. r = (18/12)/100 = 0.015

Calculate the present value of the article

The present value, PV, of the article is the amount for which it is sold cash (Rs. 500) minus the down payment (Rs. 150). PV = 500 - 150 = 350

Calculate the monthly instalment using the present value formula

Using the present value formula for future cash flows, we get: PV = P * [(1 - (1+r)^(-n)) / r] Here, n = number of monthly installments, which is 5, and PV = 350. Solving for P yields: 350 = P * [(1 - (1+0.015)^(-5)) / 0.015] P = 350 / [(1 - (1+0.015)^(-5)) / 0.015] P = 73.06 The monthly instalment, P, is Rs. 73.06. So, the correct answer is (d) \(73.06\%\).

Key concepts

Interest Rate

Interest rates play a crucial role in installment calculations. An interest rate is basically the cost of borrowing money or the reward for saving. In our exercise, we have an annual interest rate of 18%. When calculating installments, we often need to convert this annual rate into a rate applicable to a shorter term, such as monthly.

To convert an annual rate to a monthly rate, divide the annual rate by 12. For example, the monthly interest rate here is calculated as follows:

• Annual rate = 18%
• Monthly rate = 18% / 12 = 1.5% per month

The monthly rate is expressed in decimal form for precise calculations, becoming 0.015. This adjusted rate is essential for accurate installment computations.

Present Value

The concept of present value (PV) allows us to determine how much future payments are worth today. It is crucial when dealing with payments over time, such as installments. In the context of our exercise, we start by calculating the present value of the article after making a down payment.

The initial selling price is Rs. 500, but a down payment of Rs. 150 is made immediately. Therefore, the present value of the remaining amount is Rs. 350 (500 - 150). This Rs. 350 is the amount to be covered by the installment plan. Knowing the present value helps us understand what future payments are essentially worth today, which in turn helps us to set fair installment amounts.

Monthly Instalment Formula

The monthly installment formula is a mathematical expression used to determine the fixed payment you need to make each month to repay a debt over a certain period. This involves several components such as the present value, interest rate, and number of periods.

The formula used is derived from the present value of an annuity equation: \[PV = P \cdot \left(\frac{1 - (1+r)^{-n}}{r}\right)\]Where:

• PV is the present value
• P is the monthly installment
• r is the monthly interest rate
• n is the number of installments

In our exercise, substituting the known values into the formula (PV = 350, r = 0.015, n = 5) allows us to solve for P:

• P = 350 / \(\left(\frac{1 - (1+0.015)^{-5}}{0.015}\right)\)

Thus, the monthly installment, P, is Rs. 73.06.

Quantitative Aptitude

Quantitative aptitude refers to the ability to solve numerical problems efficiently and accurately. In the context of installment calculations, it covers skills in understanding and applying mathematical concepts such as interest rate calculations, present value analysis, and using formulas to find solutions.

This exercise involves breaking down a problem into smaller, manageable steps:

• Converting annual interest rates to monthly rates
• Understanding present value and its significance
• Using formulas to compute installment amounts

Mastering quantitative aptitude enables you to handle various real-life financial problems, such as loans or investments, with confidence. Practicing such problems helps in developing strong numerical skills essential for academic and everyday financial decisions.