Question 1172965: A certain household would like to buy a washing machine set payable for 6 months starting at the end of the month. How much is the cost of the washing machine if the monthly payment amounts to $3000 and interest is 9% compounded semi-annually?
Thank you so much in advance.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the cost of the washing machine:
**Understanding the Problem**
* **Monthly Payments:** The household will pay $3000 per month for 6 months.
* **Interest:** The interest rate is 9% per year, compounded semi-annually.
* **Goal:** We need to find the present value of these payments, which represents the initial cost of the washing machine.
**Calculations**
Here's how we'll break down the calculation:
1. **Semi-annual Interest Rate:**
* The annual interest rate is 9% (0.09).
* Since it's compounded semi-annually, the interest rate per period is 0.09 / 2 = 0.045.
2. **Adjusting for Monthly Payments:**
* Since the payments are monthly and the interest is compounded semi annually, this calculation becomes complex. The provided code assumes that the semi annual interest rate can be devided by the number of compounding periods, however this is incorrect.
* To get a more accurate result, it is better to calculate the effective monthly rate.
* The effective annual rate is (1 + 0.045)^2 = 1.092025. Meaning the effective annual rate is 9.2025%.
* To get the monthly rate we calculate (1.092025)^(1/12) - 1 = 0.00739268. Meaning the monthly rate is 0.739268%.
3. **Present Value of an Ordinary Annuity:**
* We'll use the present value of an ordinary annuity formula:
* PV = PMT * \[1 - (1 + r)^-n] / r
* Where:
* PV = Present Value (the cost of the washing machine)
* PMT = Monthly Payment ($3000)
* r = monthly interest rate (0.00739268)
* n = Number of Payments (6)
4. **Applying the Formula:**
* PV = 3000 * \[1 - (1 + 0.00739268)^-6] / 0.00739268
* PV = 3000 * \[1 - (1.00739268)^-6] / 0.00739268
* PV = 3000 * \[1 - 0.956795] / 0.00739268
* PV = 3000 * \[0.043205] / 0.00739268
* PV = 3000 * 5.84422
* PV = 17532.66
**Answer**
The cost of the washing machine is approximately $17532.66.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
A certain household would like to buy a washing machine set payable for 6 months starting at the end of the month.
How much is the cost of the washing machine if the monthly payment amounts to $3000 and interest is 9% compounded semi-annually?
Thank you so much in advance.
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In this problem, the payments are monthly, but the compounding is semi-annually.
In his solution, @CPhill tries to accommodate the solution, using/introducing/assuming
that the compounding is monthly, too.
But in reality, it is not so - - - therefore, this scheme DOES NOT WORK
I saw many problems of this kind, where the math composers construct such non-traditional
schemes, trying to create new problems.
But the answer is one and is UNIQUE - you can't go against nature.
So, all such attempts are doomed to failure - - - and this one too.
The solution by @CPhill is no an exception.
It is incorrect and represents an unsuccessful attempt.
It is much better to understand the whole truth as it is and to say the truth as it is,
than to deceive yourself and all other around.
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