Math Problem Statement
Solution
To solve this problem, we need to use the formula for the present value of a compound interest investment:
Where:
- = Present Value (the amount you should deposit now),
- = Future Value (the amount you want to have in the future, which is $12,000),
- = Annual interest rate (7%, or 0.07),
- = Number of times the interest is compounded per year (semiannually means ),
- = Number of years (5 years in this case).
Step-by-step solution:
-
Given values:
- ,
- ,
- ,
- .
-
Substitute these into the formula:
-
Simplify the equation:
-
Calculate :
-
Calculate the present value (PV):
So, you should deposit approximately $8,505.39 now to have $12,000 in 5 years.
Would you like more details on any step?
Here are some related questions:
- How does the frequency of compounding affect the amount you need to deposit?
- What happens if the interest rate changes to 6% instead of 7%?
- How would the formula change if the interest is compounded quarterly?
- How does the present value formula differ from the future value formula?
- What is the effect of increasing the investment period from 5 years to 10 years?
Tip: Always double-check whether interest is compounded annually, semiannually, or at another interval—it greatly affects the result!
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Math Problem Analysis
Mathematical Concepts
Compound Interest
Present Value
Exponential Growth
Formulas
PV = FV / (1 + r/n)^(nt)
Theorems
Compound Interest Formula
Suitable Grade Level
Grades 10-12
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