To find the difference in the balance between simple interest and compound interest, we need to calculate both types of interest for the investment amount of $900 over a period of 5 years.
Step 1: Calculate Simple Interest (SI)
The formula for simple interest is:
\[ SI = P \times r \times t \]
where:
- \( P = 900 \) (principal)
- \( r = 0.05 \) (interest rate)
- \( t = 5 \) (time in years)
Now, substituting the values:
\[ SI = 900 \times 0.05 \times 5 = 900 \times 0.25 = 225 \]
The total amount with simple interest is:
\[ A_{SI} = P + SI = 900 + 225 = 1125 \]
Step 2: Calculate Compound Interest (CI)
The formula for compound interest is:
\[ A = P (1 + r)^t \]
where:
- \( P = 900 \) (principal)
- \( r = 0.06 \) (interest rate)
- \( t = 5 \) (time in years)
Now, substituting the values:
\[ A = 900 \times (1 + 0.06)^5 = 900 \times (1.06)^5 \]
Calculating \( (1.06)^5 \):
\[ (1.06)^5 \approx 1.338225 \]
Now, substituting this into the formula:
\[ A \approx 900 \times 1.338225 \approx 1204.4025 \]
Rounding to two decimal places, we get:
\[ A_{CI} \approx 1204.40 \]
Step 3: Find the Difference
Now, we find the difference between the compound interest amount and the simple interest amount:
\[ \text{Difference} = A_{CI} - A_{SI} = 1204.40 - 1125 = 79.40 \]
Conclusion
The difference in the balance between simple interest at 5% and compound interest at 6% when $900 is invested for a period of 5 years is:
\[ \boxed{79.40} \]