Question
Shakira has an investment account that she deposits $200.00 into each year. The table shows the amount of interest she earns in dollars and the ending balance each year.
Shakira's Investment Account
Year Deposit (dollars) Interest Earned (2%) (dollars) Ending Balance (dollars)
1
2
200.00 200.00
4.00 8.08
204.00 412.08
Shakira continues to deposit $200.00 each year and earns the same interest rate for several years. Which statement about the ending balance each year is true?
The balance increases by
The balance increases by
2% of the sum of $200.00
2% of $200.00 every year.
and $204.00 every year.
The balance increases by 2% of the prior year's ending balance.
The balance increases by 2% of the sum of the prior year's ending balance and $200.00.
Answers
Answered by
GPT-4o mini
To determine which statement about the ending balance each year is true, let's analyze the pattern of how the balance grows.
1. **Understanding the Account Growth:**
Shakira deposits $200.00 each year, and the interest is compounded at a rate of 2%. The interest earned in the first year is calculated on the initial deposit of $200.00, which is 2% of $200.00 = $4.00.
2. **Second Year Calculation:**
- At the end of Year 1, the ending balance is:
\[
\text{Ending Balance}_1 = \text{Deposit}_1 + \text{Interest}_1 = 200.00 + 4.00 = 204.00
\]
- For Year 2, Shakira deposits another $200.00. Thus, the principal amount at the start of Year 2 is $204.00 (previous balance) + $200.00 (new deposit) = $404.00.
- The interest earned in Year 2 is 2% of $404.00:
\[
\text{Interest}_2 = 0.02 \times 404.00 = 8.08
\]
- Thus, the ending balance at the end of Year 2 becomes:
\[
\text{Ending Balance}_2 = 404.00 + 8.08 = 412.08
\]
3. **General Rule for Each Year:**
For any given year, the balance increases based on the total amount of the prior year's ending balance plus the new deposit ($200.00) and then some interest is added to that total.
**Conclusion:**
From the pattern established, we can deduce:
- The ending balance rises by the interest earned, which is calculated on the sum of the prior year's ending balance and the current year's deposit.
Therefore, the correct statement is:
**The balance increases by 2% of the sum of the prior year's ending balance and $200.00.**
1. **Understanding the Account Growth:**
Shakira deposits $200.00 each year, and the interest is compounded at a rate of 2%. The interest earned in the first year is calculated on the initial deposit of $200.00, which is 2% of $200.00 = $4.00.
2. **Second Year Calculation:**
- At the end of Year 1, the ending balance is:
\[
\text{Ending Balance}_1 = \text{Deposit}_1 + \text{Interest}_1 = 200.00 + 4.00 = 204.00
\]
- For Year 2, Shakira deposits another $200.00. Thus, the principal amount at the start of Year 2 is $204.00 (previous balance) + $200.00 (new deposit) = $404.00.
- The interest earned in Year 2 is 2% of $404.00:
\[
\text{Interest}_2 = 0.02 \times 404.00 = 8.08
\]
- Thus, the ending balance at the end of Year 2 becomes:
\[
\text{Ending Balance}_2 = 404.00 + 8.08 = 412.08
\]
3. **General Rule for Each Year:**
For any given year, the balance increases based on the total amount of the prior year's ending balance plus the new deposit ($200.00) and then some interest is added to that total.
**Conclusion:**
From the pattern established, we can deduce:
- The ending balance rises by the interest earned, which is calculated on the sum of the prior year's ending balance and the current year's deposit.
Therefore, the correct statement is:
**The balance increases by 2% of the sum of the prior year's ending balance and $200.00.**