Answer :
To solve the problem of determining how much you should invest today to achieve a goal of [tex]$20,000 after 12 years with an average yearly growth factor of 1.08, we can use the concept of compound interest. Here's a step-by-step explanation:
1. Understand the Formula:
The future value of an investment can be calculated using the compound interest formula:
\[
\text{Future Value} = \text{Present Value} \times (1 + \text{growth rate})^{\text{number of years}}
\]
In this case, the growth factor is given as 1.08, which means the growth rate is 0.08 (or 8%).
2. Rearrange the Formula:
To find the present value (the amount you need to invest today), rearrange the formula to:
\[
\text{Present Value} = \frac{\text{Future Value}}{(1 + \text{growth rate})^{\text{number of years}}}
\]
3. Substitute the Known Values:
You want a future value of $[/tex]20,000, the growth factor (1 + growth rate) is 1.08, and the number of years is 12. Substitute these values into the formula:
[tex]\[
\text{Present Value} = \frac{20000}{1.08^{12}}
\][/tex]
4. Calculate the Present Value:
By dividing [tex]$20,000 by the value of \(1.08\) raised to the power of 12, you find that the present value, or the amount you need to invest today, is approximately $[/tex]7,942.28.
Thus, the correct equation to use from the options given would be:
[tex]\[
20000 = x(1.08)^{12}
\][/tex]
This calculation tells you that to achieve your goal of [tex]$20,000 in 12 years with the given growth factor, you should invest approximately $[/tex]7,942.28 today.
1. Understand the Formula:
The future value of an investment can be calculated using the compound interest formula:
\[
\text{Future Value} = \text{Present Value} \times (1 + \text{growth rate})^{\text{number of years}}
\]
In this case, the growth factor is given as 1.08, which means the growth rate is 0.08 (or 8%).
2. Rearrange the Formula:
To find the present value (the amount you need to invest today), rearrange the formula to:
\[
\text{Present Value} = \frac{\text{Future Value}}{(1 + \text{growth rate})^{\text{number of years}}}
\]
3. Substitute the Known Values:
You want a future value of $[/tex]20,000, the growth factor (1 + growth rate) is 1.08, and the number of years is 12. Substitute these values into the formula:
[tex]\[
\text{Present Value} = \frac{20000}{1.08^{12}}
\][/tex]
4. Calculate the Present Value:
By dividing [tex]$20,000 by the value of \(1.08\) raised to the power of 12, you find that the present value, or the amount you need to invest today, is approximately $[/tex]7,942.28.
Thus, the correct equation to use from the options given would be:
[tex]\[
20000 = x(1.08)^{12}
\][/tex]
This calculation tells you that to achieve your goal of [tex]$20,000 in 12 years with the given growth factor, you should invest approximately $[/tex]7,942.28 today.
