Question
Suppose $1000 deposited for six years in an account paying 5% interest compounded annually.
a) Find the final amount in the account
b) Find the actual amount of interest earned
a) Find the final amount in the account
b) Find the actual amount of interest earned
Answers
GPT-4o mini
To calculate the final amount in the account and the actual amount of interest earned on a $1000 deposit at 5% interest compounded annually for six years, we can use the compound interest formula:
\[
A = P(1 + r)^n
\]
where:
- \( A \) is the final amount in the account,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (in decimal form),
- \( n \) is the number of years the money is invested or borrowed.
Given:
- \( P = 1000 \),
- \( r = 0.05 \) (which is 5% expressed as a decimal),
- \( n = 6 \).
**a) Finding the final amount in the account:**
\[
A = 1000(1 + 0.05)^6
\]
Calculating \( (1 + 0.05)^6 \):
\[
(1.05)^6 \approx 1.340095
\]
Now substituting this value back into our formula:
\[
A = 1000 \times 1.340095 \approx 1340.10
\]
So, the final amount in the account is approximately \(\$1340.10\).
**b) Finding the actual amount of interest earned:**
The interest earned can be found by subtracting the principal from the final amount:
\[
\text{Interest} = A - P = 1340.10 - 1000 = 340.10
\]
Thus, the actual amount of interest earned over six years is approximately \(\$340.10\).
### Summary
- Final amount in the account: \(\$1340.10\)
- Actual amount of interest earned: \(\$340.10\)
\[
A = P(1 + r)^n
\]
where:
- \( A \) is the final amount in the account,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (in decimal form),
- \( n \) is the number of years the money is invested or borrowed.
Given:
- \( P = 1000 \),
- \( r = 0.05 \) (which is 5% expressed as a decimal),
- \( n = 6 \).
**a) Finding the final amount in the account:**
\[
A = 1000(1 + 0.05)^6
\]
Calculating \( (1 + 0.05)^6 \):
\[
(1.05)^6 \approx 1.340095
\]
Now substituting this value back into our formula:
\[
A = 1000 \times 1.340095 \approx 1340.10
\]
So, the final amount in the account is approximately \(\$1340.10\).
**b) Finding the actual amount of interest earned:**
The interest earned can be found by subtracting the principal from the final amount:
\[
\text{Interest} = A - P = 1340.10 - 1000 = 340.10
\]
Thus, the actual amount of interest earned over six years is approximately \(\$340.10\).
### Summary
- Final amount in the account: \(\$1340.10\)
- Actual amount of interest earned: \(\$340.10\)