SOLUTION: A second-hand car was bought for ₱100,000 downpayment and monthly installments of ₱10,000 to be paid for 4 years. What is the cash price of the car if money is worth 12% co

Question 1193609: A second-hand car was bought for ₱100,000 downpayment and
monthly installments of ₱10,000 to be paid for 4 years. What is the
cash price of the car if money is worth 12% compounded monthly?
Found 2 solutions by yurtman, ikleyn:
Answer by yurtman(42)   (Show Source): You can put this solution on YOUR website!
To calculate the cash price of the car, we need to find the present value of the monthly installment payments and add the downpayment.
---
### Step 1: Formula for Present Value of an Annuity
The formula for the present value of an annuity is:
\[
PV = M \cdot \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- $ PV $ = present value of the monthly installments (the amount equivalent to these payments today)
- $ M $ = monthly payment ($ 10,000 $)
- $ r $ = monthly interest rate ($ \frac{12\%}{12} = 0.01 $)
- $ n $ = total number of payments ($ 4 \, \text{years} \times 12 = 48 $)
---
### Step 2: Calculate the Present Value of Installments
Substitute the known values:
\[
PV = 10,000 \cdot \frac{1 - (1 + 0.01)^{-48}}{0.01}
\]
#### Calculate Step-by-Step:
1. $ (1 + 0.01)^{-48} = 1.01^{-48} \approx 0.608437 $
2. $ 1 - 0.608437 = 0.391563 $
3. $ \frac{0.391563}{0.01} = 39,156.30 $
So:
\[
PV \approx 39,156.30
\]
---
### Step 3: Add the Downpayment
The total cash price of the car is:
\[
\text{Cash Price} = \text{Downpayment} + PV
\]
Substitute the values:
\[
\text{Cash Price} = 100,000 + 39,156.30 = 139,156.30
\]
---
### Final Answer:
The cash price of the car is approximately **₱139,156.30**.
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
A second-hand car was bought for ₱100,000 downpayment and
monthly installments of ₱10,000 to be paid for 4 years. What is the
cash price of the car if money is worth 12% compounded monthly?
~~~~~~~~~~~~~~~~~~~~~~~~~~

        Calculations in the post by @yurtman are incorrect.
        I came to provide correct calculations.

To calculate the cash price of the car, we need to find the present value of the monthly installment payments and add the downpayment. Step 1: Formula for Present Value of an Annuity The formula for the present value of an annuity is: PV = Where: PV = present value of the monthly installments (the amount equivalent to these payments today) M = monthly payment (10,000) r = monthly interest rate (0.12/12} = 0.01) n = total number of payments (48 = 4*12) -- Step 2: Calculate the Present Value of Installments Substitute the known values: PV = = 379739.60 <<<---=== about one order more than in the post by @yurtman I performed this calculation using my Excel spreadsheet So: PV = 379739.60 --- Step 3: Add the Downpayment The total cash price of the car is: Cash Price = Downpayment + PV = 100,000 + 379739.60 = 479739.60 --- Final Answer: The cash price of the car is approximately 479739.60.

Solved correctly.

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