- The general compound interest formula is \(A = P(1 + \frac{r}{n})^{nt}\), where \(n\) represents the number of compounding periods per year.
- By matching this formula with the given expression \(5,000(1 + \frac{0.04}{12})^{12t}\), we identify \(P = 5,000\), \(r = 0.04\), and \(n = 12\).
- The value \(12\) appears both as the denominator in the base and as the coefficient of \(t\) in the exponent, confirming it is the compounding frequency.
- The final answer is \(\boxed{12}\).
Explanation
- Problem Analysis and Formula Comparison
The problem asks to identify the number of times per year interest is compounded, represented by the variable \(n\), in the given compound interest expression: \[5,000\left(1+\frac{0.04}{12}\right)^{12t}.\] To solve this, we compare the given expression to the general compound interest formula: \[A = P\left(1+\frac{r}{n}\right)^{nt},\] where:
- \(A\) is the final amount in the account.
- \(P\) is the principal amount (the initial deposit), which is \(5,000\).
- \(r\) is the annual interest rate (as a decimal), which is \(0.04\).
- \(n\) is the number of times the interest is compounded per year.
- \(t\) is the time the money is invested, in years.
- \(nt\) is the total number of compounding periods over the duration \(t\).
- Identifying the Compounding Frequency
By examining the provided expression \(5,000\left(1+\frac{0.04}{12}\right)^{12t}\) and comparing it to the general formula \(P\left(1+\frac{r}{n}\right)^{nt}\), we can align each component:
- The principal \(P\) corresponds to \(5,000\).
- The rate \(r\) corresponds to \(0.04\).
- The compounding frequency \(n\) appears in two places: as the denominator of the fraction \(\frac{r}{n}\) inside the parentheses, and as the coefficient of the time variable \(t\) in the exponent \(nt\).
In our specific expression, \(12\) occupies both of these positions: \(\frac{0.04}{12}\) and \(12t\). Therefore, the value representing the number of times per year the interest is compounded is \(12\).
- Final Conclusion
Based on the analysis, we have mapped the components of the given expression to the standard compound interest formula. The compounding frequency \(n\) is clearly indicated as \(12\). This means the interest is compounded \(12\) times per year (which corresponds to monthly compounding). Thus, the required value is \(\boxed{12}\).
Examples
Understanding how compound interest is calculated is essential for managing personal finances. For example, if you open a savings account with a 3% annual interest rate compounded quarterly, your money grows slightly faster than if it were compounded annually. By identifying the compounding frequency in an expression like the one we analyzed, you can compare different bank offers to ensure you are choosing the account that maximizes the return on your savings over time.