Question

question 2
becky wants her investment to double in 8 years. what yearly return rate would she need to achieve this?

Explanation:

Step1: Use Rule of 72 approximation

The Rule of 72 states that the approximate annual return rate $r$ to double an investment is given by:
$$r \approx \frac{72}{t}$$
where $t$ is the time in years.

Step2: Substitute $t=8$ years

$$r \approx \frac{72}{8}$$

Step3: Calculate exact rate (compound interest)

For exact compound interest, use the doubling formula $A = P(1+r)^t$, where $A=2P$.
$$2P = P(1+r)^8$$
Divide both sides by $P$:
$$2 = (1+r)^8$$
Take the 8th root of both sides:
$$1+r = 2^{1/8}$$
Calculate $2^{1/8}$:
$$1+r \approx 1.0905$$
Subtract 1 to find $r$:
$$r \approx 1.0905 - 1 = 0.0905$$

Answer:

Approximately 9.05% (or ~9% using the Rule of 72 approximation)