Question

When purchasing a car and 3 documents worth $15,000 will be paid each to be paid in months 2, 4 and 6. If you wish to pay in 2 installments equal, in months 4 and 8. What should be the amount of these payments? Consider that money changes at a rate of 35% compounded semiannually monthly.

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Answer to a math question When purchasing a car and 3 documents worth $15,000 will be paid each to be paid in months 2, 4 and 6. If you wish to pay in 2 installments equal, in months 4 and 8. What should be the amount of these payments? Consider that money changes at a rate of 35% compounded semiannually monthly.

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Brice
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110 Answers
To solve this problem, we need to find the equivalent value of three payments of $15,000 each, due in months 2, 4, and 6, and then determine the size of two equal payments in months 4 and 8.

### Step 1: Calculate the monthly interest rate
Given a 35% annual interest rate compounded semiannually:
- Semiannual rate = 35% / 2 = 17.5% per half year.

To convert a semiannual rate to a monthly rate when compounded semiannually:
\left(1 + \text{Semiannual rate}\right)^{\left(\frac{1}{6}\right)} - 1

Calculating it:
\left(1 + 0.175\right)^{\left(\frac{1}{6}\right)} - 1 = \left(1.175\right)^{\left(\frac{1}{6}\right)} - 1 \approx 0.0273 \text{ or } 2.73\% \text{ per month}

### Step 2: Calculate the present value of payments at month 4
- **Month 2**: $15,000 due in 2 months discounted to month 4:
PV_{\text{month 2 to 4}} = \frac{15000}{(1.0273^2)} = \frac{15000}{1.0559} \approx 14212.48
- **Month 4**: $15,000 due in 0 months
PV_{\text{month 4}} = 15000
- **Month 6**: $15,000 due in 2 months forward from month 4:
PV_{\text{month 6 to 4}} = \frac{15000}{(1.0273^2)} = \frac{15000}{1.0559} \approx 14212.48

Total present value at month 4:
PV_{\text{total}} = 14212.48 + 15000 + 14212.48 \approx 43424.96

### Step 3: Calculate the amount of each of the two equal payments in months 4 and 8
Let x be the payment amount in months 4 and 8.

x + \frac{x}{(1.0273^4)} = 43424.96
x + \frac{x}{1.1145} = 43424.96
x (1 + \frac{1}{1.1145}) = 43424.96
x (1.7933) = 43424.96
x = \frac{43424.96}{1.7933} \approx 24223.72

### Answer:
Each of the two equal payments to be made in months 4 and 8 should be approximately $24,223.72 to cover the equivalent value of the three payments worth $15,000 each due in months 2, 4, and 6, considering the given interest rate.

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