Calculate the time (t) for php 2,500 to become php 4,750 at a rate of 15%

To calculate the time \( t \) it takes for an investment of PHP 2,500 to grow to PHP 4,750 at an interest rate of 15%, we'll assume it's compounded annually unless stated otherwise.

We'll use the compound interest formula:

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

Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (PHP 2,500).
- \( r \) is the annual interest rate (15% or 0.15).
- \( n \) is the number of times that interest is compounded per year (assuming 1 for annual compounding).
- \( t \) is the time in years.

Given values:
- \( A = 4,750 \)
- \( P = 2,500 \)
- \( r = 0.15 \)
- \( n = 1 \)

Now, solve for \( t \):

\[
4,750 = 2,500 \left(1 + \frac{0.15}{1}\right)^{1 \cdot t}
\]

\[
4,750 = 2,500 \left(1.15\right)^{t}
\]

Now, divide both sides by 2,500:

\[
\frac{4,750}{2,500} = 1.15^{t}
\]

\[
1.9 = 1.15^{t}
\]

Next, take the natural logarithm (ln) of both sides:

\[
\ln(1.9) = \ln(1.15^{t})
\]

Using the logarithm property \( \ln(a^b) = b\ln(a) \), we get:

\[
\ln(1.9) = t \cdot \ln(1.15)
\]

Solve for \( t \):

\[
t = \frac{\ln(1.9)}{\ln(1.15)}
\]

Let's calculate this:

\[
t = \frac{\ln(1.9)}{\ln(1.15)} \approx \frac{0.64185}{0.13976} \approx 4.59 \text{ years}
\]

So, it will take approximately **4.59 years** for PHP 2,500 to grow to PHP 4,750 at an annual interest rate of 15%.

Would you like more details or have any questions? Here are some related questions you might find useful:

1. How would the time change if the interest rate were compounded quarterly?
2. What would be the future value if the time were 6 years instead?
3. How much interest would you earn after 4.59 years?
4. What is the difference between simple and compound interest for the same scenario?
5. How does the time change if the interest rate increases to 20%?

**Tip:** Remember that the frequency of compounding can significantly affect the growth of an investment.

Learn how to calculate the time it takes for an investment of PHP 2,500 to grow to PHP 4,750 at an annual interest rate of 15%. This problem involves compound interest calculations using specific values for principal, rate, and time.

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