Math Problem Statement
A savings account balance 4 months ago was $549.23. No deposits or withdrawals were made. Money earns 2.49% compounded monthly. a) What was the balance 2 years ago? b) How much interest was earned between 2 years ago and 4 months ago?
Input the balance in the Answer box; the interest in the second box. Show 2 decimals. Example: Answer: 23456.78; units: 1234.56
Solution
To solve this problem, we’ll use the formula for compound interest:
where:
- is the amount of money in the account after years,
- is the initial principal balance (the amount 2 years ago, which we are solving for),
- is the annual interest rate (2.49% or 0.0249),
- is the number of times interest is compounded per year (monthly, so ),
- is the time in years (in this case, 1 year and 8 months from two years ago to four months ago).
Step-by-Step Solution
a) Finding the Balance 2 Years Ago
- Let’s assume the amount two years ago was .
- We know that four months ago, the balance was $549.23.
- The time between two years ago and four months ago is 20 months (1 year and 8 months), which is years.
Setting up the equation for compound interest with :
Now, let’s calculate .
b) Finding the Interest Earned
After finding , we can calculate the interest earned between two years ago and four months ago using the formula:
Calculating these values now:Answer: 526.93; units: 22.30
The balance 2 years ago was $526.93, and the interest earned between 2 years ago and 4 months ago was $22.30.
Would you like further details on any part of the solution?
Here are some related questions for further exploration:
- What would the balance be in the account 3 years from now at the same interest rate?
- How would a different compounding frequency, such as quarterly, affect the interest earned?
- If a withdrawal had been made, how would the calculations adjust?
- What if the interest rate had changed after one year—how would we calculate the new balance?
- How does compounding monthly compare to compounding annually for the same interest rate?
Tip: Compounding more frequently generally leads to slightly higher interest due to the effect of earning interest on interest more often.
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Math Problem Analysis
Mathematical Concepts
Compound Interest
Exponential Growth
Financial Mathematics
Formulas
Compound Interest Formula: A = P(1 + r/n)^(nt)
Theorems
Exponential Growth Formula
Suitable Grade Level
Grades 10-12
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