Math Problem Statement
Find how much money should be deposited in a bank paying interest at the rate of 6.5%/year compounded quarterly so that at the end of 7 years, the accumulated amount will be $70,000. (Round your answer to the nearest cent.)
Solution
To find the initial deposit amount (present value ) needed to reach $70,000 after 7 years with an interest rate of 6.5% compounded quarterly, we can use the formula for compound interest:
where:
- is the accumulated amount (future value),
- is the initial deposit (present value),
- is the annual interest rate (as a decimal),
- is the number of compounding periods per year,
- is the number of years.
Given values:
- ,
- ,
- (quarterly compounding),
- .
Substitute these values into the formula:
Simplify inside the parentheses:
Calculate the compound factor:
Using the compound factor:
Now, isolate :
Answer:
The initial deposit should be approximately $43,518.24.
Would you like more details, or do you have any questions?
Here are some related questions to expand on this concept:
- How does increasing the interest rate affect the initial deposit required?
- What would happen if the interest was compounded monthly instead of quarterly?
- How does the time duration impact the initial deposit required?
- What is the impact of continuous compounding on the initial deposit?
- How can we calculate the future value if the deposits are made periodically instead of a lump sum?
Tip: For compound interest, the more frequently the interest compounds, the larger the final amount due to the effect of compounding.
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Math Problem Analysis
Mathematical Concepts
Compound Interest
Exponential Growth
Algebra
Formulas
Compound Interest Formula: A = P(1 + r/n)^(nt)
Theorems
Compound Interest Theorem
Suitable Grade Level
Grades 11-12
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