- Calculate the number of years by dividing the total interest \(6.42 \times 10^4\) by the annual interest \(802.50\).
- Find that the division results in \(80\).
- Convert \(80\) into scientific notation as \(8 \times 10^1\).
- Confirm the result matches the option \(\boxed{8 \times 10^1}\).
Explanation
- Understanding the Problem and Data Analysis
To determine the number of years required for the bank deposit to yield a total of \(\$6.42 \times 10^4\) in interest, given an annual yield of \(\$802.50\), we first set up the relationship between total interest, annual interest, and the number of years. Let \(I_{total}\) be the total interest, \(I_{annual}\) be the interest earned per year, and \(n\) be the number of years. The relationship is given by:
\[n = \frac{I_{total}}{I_{annual}}.\]
We are given \(I_{total} = 6.42 \times 10^4 = 64,200\) and \(I_{annual} = 802.50\).
- Calculating the Number of Years and Converting to Scientific Notation
We need to calculate the value of \(n\):
\[n = \frac{64,200}{802.50}.\]
Performing the division, we find:
\[n = 80.\]
Now, we must express this result in scientific notation. Scientific notation follows the form \(a \times 10^b\), where \(1 \leq |a| < 10\) and \(b\) is an integer. For the number \(80\), we can write it as:
\[80 = 8 \times 10^1.\]
- Conclusion
Comparing our result, \(8 \times 10^1\), with the provided options:
- \(1.25 \times 10^2\)
- \(8 \times 10^4\)
- \(8 \times 10^2\)
- \(8 \times 10^1\)
The correct option is \(8 \times 10^1\). Thus, it will take \(8 \times 10^1\) years for the deposit to yield the specified total interest. The final answer is \(\boxed{8 \times 10^1}\).
Examples
Understanding scientific notation and division is essential for financial planning and data analysis. For example, if you are calculating how long it will take to reach a savings goal, such as saving for a home down payment with fixed monthly or annual deposits, you apply these same principles to estimate the timeframe. This skill is helpful in everyday tasks like project management, budgeting, and long-term financial forecasting, allowing you to quickly scale and compare large quantities.