At what nominal rate of interest, compounded yearly, will money double in eight years?
Understand the Problem
The problem asks us to find the nominal interest rate at which money will double in eight years when compounded yearly. The solution utilizes the compound interest formula to solve for the interest rate, r.
Answer
The nominal interest rate is approximately $9.05\%$. The exact answer is $r = \sqrt[8]{2} - 1 \approx 0.0905$.
Answer for screen readers
The nominal rate of interest is approximately $9.05%$.
Steps to Solve
- Write down the compound interest formula
The future value $A$ of an investment $P$ compounded yearly at an interest rate $r$ for $t$ years is given by:
$$A = P(1 + r)^t$$
- Set up the equation for doubling the money
We want to find the interest rate $r$ such that the money doubles in 8 years. This means $A = 2P$ and $t = 8$. Substituting into the formula:
$$2P = P(1 + r)^8$$
- Simplify the equation
Divide both sides by $P$:
$$2 = (1 + r)^8$$
- Isolate $1+r$
Take the 8th root of both sides:
$$\sqrt[8]{2} = 1 + r$$
- Solve for $r$
Subtract 1 from both sides:
$$r = \sqrt[8]{2} - 1$$
- Calculate the approximate value of $r$
$$r \approx 1.0905 - 1$$ $$r \approx 0.0905$$
- Convert to percentage
Multiply by 100 to express as a percentage:
$$r \approx 0.0905 \times 100 = 9.05%$$
The nominal rate of interest is approximately $9.05%$.
More Information
The doubling time is independent of the principal amount $P$.
Tips
A common mistake is forgetting to subtract 1 after taking the 8th root, resulting in $1+r$ being mistaken for $r$. Another mistake is not converting the decimal to a percentage.
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