We appreciate your visit to You ve set a goal of having tex 10 000 tex after 15 years If you estimate that your investment account will have an average. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To determine how much you should invest today in order to have [tex]$10,000 after 15 years, given an average yearly growth factor of 1.09, you can use the concept of compound interest.
Here's a step-by-step explanation:
1. Understand the Compound Interest Formula: The general formula for compound interest, when your goal is to find the principal (\(x\)) you should invest today, is:
\[
\text{{Future Value}} = \text{{Principal}} \times (\text{{Growth Factor}})^{\text{{Number of Years}}}
\]
Rearranging to solve for the principal, we have:
\[
\text{{Principal}} = \frac{\text{{Future Value}}}{(\text{{Growth Factor}})^{\text{{Number of Years}}}}
\]
2. Identify the Given Values:
- Future Value: This is your investment target, which is $[/tex]10,000.
- Growth Factor: This is 1.09, representing a 9% average yearly growth.
- Number of Years: This is 15 years, your investment duration.
3. Set Up the Equation:
Using the values identified, you set up the equation as:
[tex]\[
10000 = x \times (1.09)^{15}
\][/tex]
4. Correct Answer Choice:
Look for the equation that matches the above setup. The correct choice from the options provided is:
[tex]\[
\text{{c) }} 10000 = x \times (1.09)^{15}
\][/tex]
This equation represents the relationship between the amount you invest today (x), the growth factor, and your future investment target after 15 years.
Here's a step-by-step explanation:
1. Understand the Compound Interest Formula: The general formula for compound interest, when your goal is to find the principal (\(x\)) you should invest today, is:
\[
\text{{Future Value}} = \text{{Principal}} \times (\text{{Growth Factor}})^{\text{{Number of Years}}}
\]
Rearranging to solve for the principal, we have:
\[
\text{{Principal}} = \frac{\text{{Future Value}}}{(\text{{Growth Factor}})^{\text{{Number of Years}}}}
\]
2. Identify the Given Values:
- Future Value: This is your investment target, which is $[/tex]10,000.
- Growth Factor: This is 1.09, representing a 9% average yearly growth.
- Number of Years: This is 15 years, your investment duration.
3. Set Up the Equation:
Using the values identified, you set up the equation as:
[tex]\[
10000 = x \times (1.09)^{15}
\][/tex]
4. Correct Answer Choice:
Look for the equation that matches the above setup. The correct choice from the options provided is:
[tex]\[
\text{{c) }} 10000 = x \times (1.09)^{15}
\][/tex]
This equation represents the relationship between the amount you invest today (x), the growth factor, and your future investment target after 15 years.
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