Question

question 3
becky tells her friend about the rule of 72 and her friend immediately tries it out with her own example. becky doesnt see all of her friends math but does see $f(4) = 18$ written on her friends paper.
a. what do you know about the friends investment based on this?
b. is the friends math correct based on the rule of 72?

Explanation:

Step1: Recall Rule of 72 formula

The Rule of 72 formula is $\text{Time to double} = \frac{72}{\text{Annual interest rate (in \%)}}$

Step2: Interpret f(4)=18 for part a

Here, $f(4)=18$ means if the time (in years) is 4, the interest rate is 18%, or vice versa. Using the Rule of 72:
If time = 4 years: $\text{Interest rate} = \frac{72}{4} = 18\%$. This means the investment with an 18% annual interest rate will double in 4 years, or an investment with a 4% rate will double in 18 years.

Step3: Verify the math for part b

Check the two cases:
Case 1: Rate = 18%, Time to double should be $\frac{72}{18} = 4$ years.
Case 2: Rate = 4%, Time to double should be $\frac{72}{4} = 18$ years.
Both match $f(4)=18$.

Answer:

a. We know either:

• The investment has an 18% annual interest rate, and it will take 4 years for the investment to double in value.
• OR the investment has a 4% annual interest rate, and it will take 18 years for the investment to double in value.

b. Yes, the friend's math is correct based on the Rule of 72.