​(Related **to Checkpoint**​ 5.5)**** ​ (Solving **for**

***n***​)

How many years will it take for****

​$480480

to grow to

​$1 comma 059.711,059.71

if​ it's invested at

88

percent compounded​ annually?

Question content area bottom

Part 1

The number of years it**** will take**** for****

​$480480

to grow to

​$1 comma 059.711,059.71

at

88

percent compounded annually is

enter your response here

years. ​ (Round to one decimal​ place.)

To solve this problem, we will use the compound interest formula:

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

Where:
- $ A $ is the future value of the investment/loan, which is \$1,059.71,
- $ P $ is the initial principal balance, which is \$480.48,
- $ r $ is the annual interest rate (as a decimal), which is 88% or 0.88,
- $ n $ is the number of times the interest is compounded per year (since it's compounded annually, $ n = 1 $),
- $ t $ is the number of years, which is what we are solving for.

The equation simplifies to:

\[
1059.71 = 480.48(1 + 0.88)^t
\]

### Step 1: Isolate the exponential term
Divide both sides of the equation by 480.48:

\[
\frac{1059.71}{480.48} = (1.88)^t
\]

\[
2.2057 \approx (1.88)^t
\]

### Step 2: Solve for $ t $ using logarithms
Take the natural logarithm of both sides:

\[
\ln(2.2057) = t \cdot \ln(1.88)
\]

Now, calculate the values of the logarithms:

\[
\ln(2.2057) \approx 0.789
\]
\[
\ln(1.88) \approx 0.631
\]

### Step 3: Solve for $ t $
Now solve for $ t $:

\[
t = \frac{0.789}{0.631} \approx 1.25
\]

So, it will take approximately **1.3 years** for \$480.48 to grow to \$1,059.71 at 88% interest compounded annually.

---

Would you like more details on how logarithms were used? Here are some related questions to explore:

1. What if the interest rate was compounded monthly instead of annually?
2. How does the formula change for different compounding frequencies?
3. What would the investment amount be after 5 years at 88% interest?
4. How does the growth change if the interest rate is reduced to 50%?
5. What happens if the initial investment was increased to \$1,000?

**Tip:** When solving compound interest problems, logarithms are essential for isolating the time variable.

How many years will it take for $480.48 to grow to $1,059.71 if it's invested at 88% compounded annually? Round to one decimal place.

Find out how long it will take for $480.48 to grow to $1,059.71 with an 88% annual interest rate compounded annually. This problem uses key algebraic concepts such as compound interest formulas and logarithms. Suitable for high school students in Grades 10-12.

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