Understanding the Bluetooth Speaker Financing

This question asks us to calculate the amount of each equal annual installment for a Bluetooth speaker purchased on a hire purchase basis. We are given the cash price, the down payment amount, the number of installments, and the annual interest rate.

Calculating the Balance Amount

First, let's find out how much money needs to be financed after making the initial down payment.

• Cash Price of the speaker = ₹3,500
• Cash Down Payment = ₹1,000
• Balance Amount = Cash Price - Down Payment

Calculation:

Balance Amount = $₹3,500 - ₹1,000 = ₹2,500$

This balance amount of ₹2,500 is the principal amount that will be paid back through installments, with interest.

Method for Installment Calculation

Since the balance amount is to be paid in equal annual installments with compound interest, this is a problem involving the Present Value of an Ordinary Annuity. The balance amount (₹2,500) is equivalent to the present value of the future stream of equal installments.

Applying the Present Value of Annuity Formula

The formula for the Present Value (PV) of an ordinary annuity is:

Where:

• PV = Present Value (the balance amount to be financed) = ₹2,500
• I = Amount of each installment (what we need to find)
• r = Annual interest rate = 12.5% or $0.125$
• n = Number of installments = 3

Let's substitute the known values into the formula:

$₹2,500 = I \times \left[ \frac{1 - (1 + 0.125)^{-3}}{0.125} \right]$

Step-by-Step Calculation:

1. Calculate the value of $(1 + r)$: $1 + 0.125 = 1.125$
2. Calculate $(1 + r)^{-n}$: $(1.125)^{-3} = \frac{1}{(1.125)^3}$ $(1.125)^3 = 1.125 \times 1.125 \times 1.125 = 1.423828125$ So, $(1.125)^{-3} = \frac{1}{1.423828125}$
3. Calculate the numerator of the fraction: $1 - (1.125)^{-3}$ $1 - \frac{1}{1.423828125} = 1 - 0.7023689539... \approx 0.2976310461$
4. Calculate the value inside the brackets (the Present Value Annuity Factor): $\frac{0.2976310461}{0.125} \approx 2.3810483688$
5. Now, substitute this back into the equation: $₹2,500 = I \times 2.3810483688$
6. Solve for I: $I = \frac{₹2,500}{2.3810483688}$

Using fractions for more precision:

Let $r = \frac{12.5}{100} = \frac{1}{8}$. Then $1+r = \frac{9}{8}$.

$PV = I \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right]$

$₹2,500 = I \times \left[ \frac{1 - (\frac{9}{8})^{-3}}{\frac{1}{8}} \right]$

$₹2,500 = I \times 8 \times \left[ 1 - (\frac{8}{9})^3 \right]$

$₹2,500 = I \times 8 \times \left[ 1 - \frac{512}{729} \right]$

$₹2,500 = I \times 8 \times \left[ \frac{729 - 512}{729} \right]$

$₹2,500 = I \times 8 \times \left[ \frac{217}{729} \right]$

$₹2,500 = I \times \frac{1736}{729}$

$I = ₹2,500 \times \frac{729}{1736}$

$I = \frac{1,822,500}{1736}$

$I \approx 1049.827189...$

Final Installment Amount

Rounding the calculated installment amount to two decimal places:

Installment Amount (I) = ₹1,049.83