How many years at 4 percent compounded quarterly to grow $1 to $7

To find how many years it takes for $1 to grow to $7 at an interest rate of 4% compounded quarterly, we can use the compound interest formula:

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

Where:
- $ A $ is the final amount (in this case, $7),
- $ P $ is the initial principal (in this case, $1),
- $ r $ is the annual interest rate (4% or 0.04),
- $ n $ is the number of times the interest is compounded per year (quarterly, so $ n = 4 $),
- $ t $ is the number of years.

First, plug in the known values:

\[
7 = 1 \left(1 + \frac{0.04}{4}\right)^{4t}
\]

Simplify:

\[
7 = \left(1 + 0.01\right)^{4t}
\]

\[
7 = \left(1.01\right)^{4t}
\]

To solve for $ t $, take the natural logarithm (ln) of both sides:

\[
\ln(7) = \ln\left((1.01)^{4t}\right)
\]

\[
\ln(7) = 4t \cdot \ln(1.01)
\]

Now solve for $ t $:

\[
t = \frac{\ln(7)}{4 \cdot \ln(1.01)}
\]

I'll calculate this next.It will take approximately **48.89 years** for $1 to grow to $7 at a 4% annual interest rate, compounded quarterly.

Would you like more details, or do you have any questions?

Here are five related questions:
1. What if the interest rate were compounded monthly instead of quarterly?
2. How would the time change if the rate increased to 5%?
3. What would happen if you started with $100 instead of $1?
4. How does the formula change for continuous compounding?
5. How much would $1 grow in 20 years with the same rate and compounding frequency?

**Tip:** The more frequent the compounding, the faster the investment grows, given the same interest rate.

Find out how long it takes for $1 to grow to $7 at a 4% interest rate, compounded quarterly. This detailed solution uses the compound interest formula and logarithms to show that it takes approximately 48.89 years. Suitable for high school students in Grades 10-12.

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