Calculating the Cost of a Television Set with Monthly Payments and Compound Interest

Mrs. Remoto wishes to purchase a television set, paying it off through 6 monthly installments starting at the end of the month. The monthly payment is Php 3000, and the interest is 9%, compounded semi-annually. How can we determine the cost of the television set?

Understanding the Problem

When Mrs. Remoto makes monthly payments on a television set, the payment terms, including an interest rate and the compounding period, affect the total cost of the television. We need to compute the present value of the annuity to find the original cost of the set.

Step-by-Step Calculation

Step 1: Convert the Annual Interest Rate to a Monthly Interest Rate

Since the interest is compounded semi-annually, we first need to find the effective monthly interest rate:

Semi-annual interest rate:

[ i frac{9}{2} 4.5% ]

Effective monthly interest rate:

[ r_{text{monthly}} left(1 0.045right)^{frac{1}{6}} - 1 ]

[ r_{text{monthly}} approx 0.007389 ]

Step 2: Calculate the Present Value of the Annuity

The present value (PV) of an annuity can be calculated using the following formula:

[ PV PMT times left( frac{1 - left(1 r_{text{monthly}}right)^{-n}}{r_{text{monthly}}} right) ]

Where:

[ PMT ] 3000 (monthly payment) [ r_{text{monthly}} ] 0.007389 (effective monthly interest rate) [ n ] 6 (total number of payments)

Note that since we are using the effective monthly interest rate, the compounding occurs monthly, and the total number of payments is 6.

Step 3: Calculate the Present Value

Substituting the values into the formula:

[ PV 3000 times left( frac{1 - left(1 0.007389right)^{-6}}{0.007389} right) ]

Calculate the term inside the parentheses:

[ left(1 0.007389right)^{-6} approx 0.956659 ]

[ 1 - 0.956659 0.043341 ]

[ frac{0.043341}{0.007389} approx 5.865 ]

Thus, the present value is:

[ PV 3000 times 5.865 17595 ]

Conclusion: The cost of the television set is approximately Php 17595.

Important Considerations

It's important to note that if the interest were not compounded semi-annually, a simpler approach could be used simply by multiplying the monthly payment by the total number of installments. However, in this case, we must account for the semi-annual compounding, which adds a level of complexity to the calculation.

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Additional Tips

For similar problems, always ensure you convert interest rates and payment terms to the appropriate units before making calculations. Additionally, remember to use the present value of annuity formula when dealing with monthly installments and compounding interest.