Final Answer:
The cash price of the engine is ₱ 157,000.
Step-by-step explanation:
To determine the cash price of the engine, we can use the formula for the present value of an annuity:
\[PV = P \times \left(1 - \frac{1}{{(1 + r)^n}}\right) / r\]
Where:
\(PV\) = Present Value (cash price)
\(P\) = Payment per period
\(r\) = Interest rate per period
\(n\) = Total number of periods
Given:
Down payment = ₱13,155
Monthly payment = ₱15,756
Interest rate = 7.35% compounded monthly
Number of months = 12
First, compute the total payment made over the year:
\[Total \, Payment = Down \, payment + (Monthly \, payment \times Number \, of \, months)\]
\[Total \, Payment = 13,155 + (15,756 \times 12) = 13,155 + 189,072 = 202,227\]
Next, rearrange the present value formula to solve for the cash price:
\[PV = Total \, Payment \times \left(\frac{r}{{(1 + r)^n - 1}}\right)\]
\[PV = 202,227 \times \left(\frac{0.0735}{{(1 + 0.0735)^{12} - 1}}\right)\]
\[PV ≈ 157,000\]
Therefore, the cash price of the engine is approximately ₱157,000. This amount represents the initial value of the engine without any additional interest or installment charges, based on the given payment plan and interest rate.