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To solve the problem of how long it will take money to quadruple at the given interest rates, compounded quarterly, we use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( A \) is the final amount (quadruple means \( A = 4P \))
- \( P \) is the initial principal
- \( r \) is the annual interest rate (as a decimal)
- \( n \) is the number of times the interest is compounded per year (quarterly means \( n = 4 \))
- \( t \) is the time in years

We want to solve for \( t \) when the money quadruples, i.e., \( A = 4P \).

### Step-by-Step Solution

We divide both sides by \( P \):

\[
4 = \left(1 + \frac{r}{n}\right)^{nt}
\]

Take the natural logarithm of both sides:

\[
\ln(4) = nt \ln\left(1 + \frac{r}{n}\right)
\]

Solve for \( t \):

\[
t = \frac{\ln(4)}{n \ln\left(1 + \frac{r}{n}\right)}
\]

Now, let's solve for \( t \) for both (A) and (B).

### For (A) 4.1% compounded quarterly:
- \( r = 0.041 \)
- \( n = 4 \)

### For (B) 9.1% compounded quarterly:
- \( r = 0.091 \)
- \( n = 4 \)

I'll calculate both now.The time it will take for the money to quadruple at the given rates, compounded quarterly, is:

- (A) 4.1%: **34.00 years** (rounded to two decimal places)
- (B) 9.1%: **15.41 years** (rounded to two decimal places)

Would you like further details or have any questions?

Here are 5 related questions you might consider:
1. How would the time change if interest was compounded monthly instead of quarterly?
2. What would the formula look like for continuous compounding?
3. How long would it take to triple the money under the same conditions?
4. How would doubling the interest rate affect the time to quadruple the money?
5. What happens if the interest rate is compounded annually instead?

**Tip:** The logarithm function in finance is useful for solving exponential growth problems, such as compound interest.

How long will it take money to quadruple if it is invested at the following rates?
(A) 4.1% compounded quarterly
(B) 9.1% compounded quarterly

Find out how long it takes for an investment to quadruple when compounded quarterly at rates of 4.1% and 9.1%. Learn the mathematical concepts of exponential growth and logarithms using the compound interest formula.

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