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Math  /  Algebra

QuestionQuestion 2 An industriat oven can be bought directly from the factory at a cost of 100,000 pesos if paid in cash. On the installment basis, a purchaser should pay 20,000 down payment and 12 quarterly installments, the first due at the end of the first year after purchase. if money is worth 15%15 \% compounded monthly, determine the quarterly installment.

Studdy Solution

STEP 1

1. The cost of the industrial oven is 100,000 pesos if paid in cash.
2. On installment, a down payment of 20,000 pesos is made, and 12 quarterly installments are required.
3. The first installment is due at the end of the first year.
4. The interest rate is 15% compounded monthly.
5. We need to determine the amount of each quarterly installment.

STEP 2

1. Determine the present value of the remaining amount after the down payment.
2. Convert the monthly interest rate to an effective quarterly interest rate.
3. Use the present value of an annuity formula to find the quarterly installment amount.

STEP 3

Calculate the remaining amount to be financed after the down payment:
Remaining amount=100,000−20,000=80,000 pesos \text{Remaining amount} = 100,000 - 20,000 = 80,000 \, \text{pesos}

STEP 4

Convert the annual interest rate to a monthly interest rate:
imonthly=15%12=0.0125 i_{\text{monthly}} = \frac{15\%}{12} = 0.0125
Convert the monthly interest rate to an effective quarterly interest rate:
iquarterly=(1+0.0125)3−1 i_{\text{quarterly}} = (1 + 0.0125)^{3} - 1
Calculate the effective quarterly interest rate:
iquarterly=(1.0125)3−1≈0.037854 i_{\text{quarterly}} = (1.0125)^{3} - 1 \approx 0.037854

STEP 5

Use the present value of an annuity formula to find the quarterly installment amount. The present value of the annuity is 80,000 pesos, and there are 12 installments:
P=PV×iquarterly1−(1+iquarterly)−n P = \frac{PV \times i_{\text{quarterly}}}{1 - (1 + i_{\text{quarterly}})^{-n}}
Where: - P P is the quarterly installment. - PV=80,000 PV = 80,000 is the present value. - iquarterly≈0.037854 i_{\text{quarterly}} \approx 0.037854 is the quarterly interest rate. - n=12 n = 12 is the number of installments.
Substitute the values into the formula:
P=80,000×0.0378541−(1+0.037854)−12 P = \frac{80,000 \times 0.037854}{1 - (1 + 0.037854)^{-12}}
Calculate the quarterly installment:
P≈80,000×0.0378541−(1.037854)−12 P \approx \frac{80,000 \times 0.037854}{1 - (1.037854)^{-12}}
P≈3,028.321−0.641 P \approx \frac{3,028.32}{1 - 0.641}
P≈3,028.320.359 P \approx \frac{3,028.32}{0.359}
P≈8,436.53 P \approx 8,436.53
The quarterly installment is approximately:
8,436.53 pesos \boxed{8,436.53 \, \text{pesos}}

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