Question
You expect to receive $4,300 upon your graduation and will invest your windfall at an interest rate of .67 percent per quarter until the account is worth $5,650. How many years do you have to wait until you reach your target account value?
Multiple Choice
11.01 years
8.94 years
9.54 years
10.32 years
10.22 years
Multiple Choice
11.01 years
8.94 years
9.54 years
10.32 years
10.22 years
Answers
Answered by
GPT-4o mini
To determine how many years you need to wait until your investment grows from $4,300 to $5,650 at an interest rate of 0.67% per quarter, we can use the formula for compound interest:
\[
A = P(1 + r)^n
\]
Where:
- \(A\) is the amount of money accumulated after n quarters, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the interest rate per period (quarter).
- \(n\) is the number of periods (quarters).
In this case:
- \(P = 4300\)
- \(A = 5650\)
- \(r = 0.0067\) (since 0.67% as a decimal is 0.0067)
- We need to find \(n\).
Rearranging the formula to solve for \(n\):
\[
n = \frac{\log(A/P)}{\log(1 + r)}
\]
Substituting in the values we have:
\[
n = \frac{\log(5650 / 4300)}{\log(1 + 0.0067)}
\]
Calculating each part:
1. \( A / P = 5650 / 4300 \approx 1.31395 \)
2. \( \log(1.31395) \approx 0.11900 \) (Note: This value may slightly differ depending on the calculator used)
3. \( 1 + r = 1 + 0.0067 = 1.0067 \)
4. \( \log(1.0067) \approx 0.002898 \)
Now plug these values in:
\[
n \approx \frac{0.11900}{0.002898} \approx 41.0 \text{ quarters}
\]
Now, converting quarters to years:
\[
\text{Years} = \frac{41.0}{4} \approx 10.25 \text{ years}
\]
Now, rounding to two decimal places, we find that you would need to wait approximately **10.25 years** to reach your target account value.
Among the provided choices, the closest is **10.22 years**.
\[
A = P(1 + r)^n
\]
Where:
- \(A\) is the amount of money accumulated after n quarters, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the interest rate per period (quarter).
- \(n\) is the number of periods (quarters).
In this case:
- \(P = 4300\)
- \(A = 5650\)
- \(r = 0.0067\) (since 0.67% as a decimal is 0.0067)
- We need to find \(n\).
Rearranging the formula to solve for \(n\):
\[
n = \frac{\log(A/P)}{\log(1 + r)}
\]
Substituting in the values we have:
\[
n = \frac{\log(5650 / 4300)}{\log(1 + 0.0067)}
\]
Calculating each part:
1. \( A / P = 5650 / 4300 \approx 1.31395 \)
2. \( \log(1.31395) \approx 0.11900 \) (Note: This value may slightly differ depending on the calculator used)
3. \( 1 + r = 1 + 0.0067 = 1.0067 \)
4. \( \log(1.0067) \approx 0.002898 \)
Now plug these values in:
\[
n \approx \frac{0.11900}{0.002898} \approx 41.0 \text{ quarters}
\]
Now, converting quarters to years:
\[
\text{Years} = \frac{41.0}{4} \approx 10.25 \text{ years}
\]
Now, rounding to two decimal places, we find that you would need to wait approximately **10.25 years** to reach your target account value.
Among the provided choices, the closest is **10.22 years**.
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