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Question 1179081: Nick has been depositing $ 190 in a savings account every three months for the past three years. This account paid 3 % convertible quarterly. Nick has just made the last deposit. Nick is buying a car for $ 14000. He is taking out a car loan. He will use the accumulated value of his savings account as the downpayment on the car. The loan is at 3 % convertible semiannually and has a term of 4 years. Find the size of Nick's monthly car loan payment.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break this problem into two parts:
**Part 1: Calculate the Accumulated Value of Nick's Savings Account**
1. **Deposits:** $190 every three months (quarterly)
2. **Time Period:** 3 years
3. **Interest Rate:** 3% convertible quarterly (0.03/4 = 0.0075 per quarter)
4. **Number of Deposits:** 3 years * 4 quarters/year = 12 deposits
We can use the future value of an ordinary annuity formula:
FV = P * [((1 + r)^n - 1) / r]
Where:
* FV = Future Value
* P = Periodic Payment ($190)
* r = Interest Rate per Period (0.0075)
* n = Number of Periods (12)
FV = 190 * [((1 + 0.0075)^12 - 1) / 0.0075]
FV = 190 * [(1.0075^12 - 1) / 0.0075]
FV = 190 * [(1.093806897 - 1) / 0.0075]
FV = 190 * [0.093806897 / 0.0075]
FV = 190 * 12.50758627
FV ≈ $2376.44
**Part 2: Calculate Nick's Monthly Car Loan Payment**
1. **Car Price:** $14,000
2. **Down Payment:** $2376.44 (from the savings account)
3. **Loan Amount:** $14,000 - $2376.44 = $11,623.56
4. **Loan Term:** 4 years
5. **Interest Rate:** 3% convertible semiannually (0.03/2 = 0.015 per 6 months)
6. **Number of Semiannual Periods:** 4 years * 2 = 8
7. **Number of Monthly Payments:** 4 years * 12 months/year = 48
First, we need to find the equivalent monthly interest rate.
* (1 + 0.015)^2 = 1.030225. This is the effective annual rate.
* (1.030225)^(1/12) = 1.002470126. This is the monthly multiplier.
* Monthly Interest Rate = 1.002470126 - 1 = 0.002470126
Now, we can use the loan payment formula:
M = P * [r(1 + r)^n] / [(1 + r)^n - 1]
Where:
* M = Monthly Payment
* P = Loan Amount ($11,623.56)
* r = Monthly Interest Rate (0.002470126)
* n = Number of Monthly Payments (48)
M = 11623.56 * [0.002470126 * (1.002470126)^48] / [(1.002470126)^48 - 1]
M = 11623.56 * [0.002470126 * 1.127493457] / [1.127493457 - 1]
M = 11623.56 * [0.002785721] / [0.127493457]
M = 11623.56 * 0.021850327
M ≈ $253.97
**Answer:**
Nick's monthly car loan payment will be approximately $253.97.
Answer by ikleyn(52787)  (Show Source):
You can put this solution on YOUR website! .
Nick has been depositing $ 190 in a savings account every three months for the past three years.
This account paid 3 % convertible quarterly. Nick has just made the last deposit.
Nick is buying a car for $ 14000. He is taking out a car loan. He will use the accumulated value
of his savings account as the downpayment on the car. The loan is at 3 % convertible semiannually
and has a term of 4 years. Find the size of Nick's monthly car loan payment.
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As the problem is worded, printed and presented, it seems illogical to me.
Indeed, it says that the car loan is compounded semiannually.
The problem says nothing about the periodicity of the Nick payments.
By default, it means that Nick pays semiannually for the car loan.
But then the problem suddenly asks about the size of Nick's monthly car loan payments.
It is out of the human logic to present the problem in this form.
Then @CPhill in his solution finds the size of the Nick's monthly payments,
assuming that the monthly loan rate is equivalent to the semiannual rate.
But in reality, the bank does not perform monthly compounding for this loan -
-it performs semiannual payments for accumulated 6-month deposits.
I understand that the problem's creator wants very much to create a new problem about a loan.
I saw many attempts at this forum to create such problems.
But very often, such attempts do not go along the human logic; instead, they go precisely against it.
Exactly as this one does not go along the human logic - it goes against it.
My opinion is that for this problem its formulation and the @CPhill' solution
both are out of the human logic.
They both are in area of absurdism, and the right place for both is a garbage bin.
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