Math Problem Statement
How much would you need to deposit in an account now in order to have $3000 in the account in 15 years? Assume the account earns 5% interest compounded daily.
$
Solution
To determine how much you need to deposit now in order to have $3000 in an account in 15 years, with an interest rate of 5% compounded daily, we can use the formula for compound interest:
where:
- is the amount of money accumulated after years, including interest.
- is the principal amount (the initial amount of money).
- is the annual interest rate (decimal).
- is the number of times that interest is compounded per year.
- is the time the money is invested for in years.
Given:
- (since interest is compounded daily)
We need to find . Rearranging the formula to solve for :
Substituting the given values into the formula:
Now, let's calculate this step-by-step.
- Calculate :
- Add 1 to this value:
- Raise this to the power of :
- Calculate the power:
- Finally, divide 3000 by this result:
So, you would need to deposit approximately $1418.53 now in order to have $3000 in the account in 15 years, assuming the account earns 5% interest compounded daily.
Would you like any further details or have any questions?
- What is compound interest, and how does it differ from simple interest?
- Why is it important to know the frequency of compounding in interest calculations?
- How does the number of compounding periods per year affect the final amount?
- Can you explain how the compound interest formula is derived?
- How would the required deposit change if the interest rate were different?
- What if the interest were compounded monthly instead of daily?
- What are some real-world applications of compound interest calculations?
- How can understanding compound interest help in personal finance planning?
Tip: When dealing with compound interest problems, carefully consider the compounding frequency, as it significantly impacts the amount of interest earned over time.
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Math Problem Analysis
Mathematical Concepts
Compound Interest
Interest Rates
Exponential Growth
Formulas
Compound Interest Formula
Theorems
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Suitable Grade Level
Advanced High School
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