QuestionMay 29, 2025

10. Marai invested 2,000 into a stock that returned 6% annually, compounded quarterly. How many years will it take the stock to triple in value?

10. Marai invested 2,000 into a stock that returned 6% annually, compounded quarterly. How many years will it take the stock to triple in value?
10. Marai invested 2,000 into a stock that returned 6% annually, compounded quarterly. How many years will it take the stock to triple in value?

Solution
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Answer

Approximately 18.85 years Explanation 1. Identify the formula for compound interest Use the formula for compound interest: **A = P \left(1 + \frac{r}{n}\right)^{nt}**, where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. 2. Set up the equation for tripling the investment Marai wants the investment to triple, so A = 3P. Substitute P = 2000, r = 0.06, and n = 4: \[ 3 \times 2000 = 2000 \left(1 + \frac{0.06}{4}\right)^{4t} \] 3. Simplify and solve for t Divide both sides by 2000: \[ 3 = \left(1 + \frac{0.06}{4}\right)^{4t} \] Calculate the base: \[ 1 + \frac{0.06}{4} = 1.015 \] Take the natural logarithm of both sides: \[ \ln(3) = 4t \cdot \ln(1.015) \] Solve for t: \[ t = \frac{\ln(3)}{4 \cdot \ln(1.015)} \] 4. Compute the value of t Calculate using a calculator: \[ t \approx \frac{1.0986}{4 \times 0.0149} \approx 18.85 \]

Explanation

1. Identify the formula for compound interest<br /> Use the formula for compound interest: **$A = P \left(1 + \frac{r}{n}\right)^{nt}$**, where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.<br /><br />2. Set up the equation for tripling the investment<br /> Marai wants the investment to triple, so $A = 3P$. Substitute $P = 2000$, $r = 0.06$, and $n = 4$: <br />\[ 3 \times 2000 = 2000 \left(1 + \frac{0.06}{4}\right)^{4t} \]<br /><br />3. Simplify and solve for $t$<br /> Divide both sides by 2000:<br />\[ 3 = \left(1 + \frac{0.06}{4}\right)^{4t} \]<br /> Calculate the base:<br />\[ 1 + \frac{0.06}{4} = 1.015 \]<br /> Take the natural logarithm of both sides:<br />\[ \ln(3) = 4t \cdot \ln(1.015) \]<br /> Solve for $t$:<br />\[ t = \frac{\ln(3)}{4 \cdot \ln(1.015)} \]<br /><br />4. Compute the value of $t$<br /> Calculate using a calculator:<br />\[ t \approx \frac{1.0986}{4 \times 0.0149} \approx 18.85 \]
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