Question
What equal payments in 3 years and 5 years would replace payments of $42,500 and $60,000 in 7 years and 9 years, respectively? Assume money can earn 3.36% compounded quarterly. Round to the nearest cent
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Answer to a math question What equal payments in 3 years and 5 years would replace payments of $42,500 and $60,000 in 7 years and 9 years, respectively? Assume money can earn 3.36% compounded quarterly. Round to the nearest cent
Hermann
4.6128 Answers
To solve this problem, we first need to calculate the present value of the future payments and then determine the equal payments at the 3-year and 5-year marks that have the same total present value when discounted back to the present at the given interest rate of 3.36% compounded quarterly.
### Step 1: Calculate the quarterly interest rate
Given that the annual interest rate is 3.36%, the quarterly interest rate is:
r = \frac{3.36\%}{4} = 0.84\%
Converting to decimal form, we get:
r = 0.0084
### Step 2: Calculate the number of quarters for each payment
- 7 years = 28 quarters
- 9 years = 36 quarters
- 3 years = 12 quarters
- 5 years = 20 quarters
### Step 3: Calculate the present value (PV) of each future payment
The present value formula is given by:
PV = \frac{P}{(1 + r)^n}
whereP r n
Calculating the present values of the future payments:
- Present value of42,500 due in 7 years is
- Present value of60,000 due in 9 years is
### Step 4: Calculate the total present value
Adding the present values of the two future payments gives a total present value of $78,024.14.
### Step 5: Calculate the present value of the payments in 3 years and 5 years
We need to find equal payments that, when discounted back to the present at the interest rate, will match the total present value from step 4.
### Step 6: Solve for the equal payments (P
Setting up the equation:
P \times \left( \frac{1}{(1 + 0.0084)^{12}} + \frac{1}{(1 + 0.0084)^{20}} \right) = 78024.14
By solving the equation, we find that each equal payment should be approximately $44,573.97.
Therefore, the equal payments required at the 3-year and 5-year marks to match the total present value of the original future payments are approximately\
### Step 1: Calculate the quarterly interest rate
Given that the annual interest rate is 3.36%, the quarterly interest rate is:
Converting to decimal form, we get:
### Step 2: Calculate the number of quarters for each payment
- 7 years = 28 quarters
- 9 years = 36 quarters
- 3 years = 12 quarters
- 5 years = 20 quarters
### Step 3: Calculate the present value (PV) of each future payment
The present value formula is given by:
where
Calculating the present values of the future payments:
- Present value of
- Present value of
### Step 4: Calculate the total present value
Adding the present values of the two future payments gives a total present value of $78,024.14.
### Step 5: Calculate the present value of the payments in 3 years and 5 years
We need to find equal payments that, when discounted back to the present at the interest rate, will match the total present value from step 4.
### Step 6: Solve for the equal payments (
Setting up the equation:
By solving the equation, we find that each equal payment should be approximately $44,573.97.
Therefore, the equal payments required at the 3-year and 5-year marks to match the total present value of the original future payments are approximately
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