If P500,000 is invested at 8% interest compounded quarterly, how many years will it take for this amount to accumulate to P900,000?

Steps to Solve

1. Identify the variables in the compound interest formula

The compound interest formula is given by: $$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$ where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial investment).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time in years.

In this problem:

• $A = 900000$
• $P = 500000$
• $r = 0.08$
• $n = 4$ (since the interest is compounded quarterly)

2. Rearrange the formula to solve for time (t)

We need to isolate $t$ in the compound interest formula. First, divide both sides by $P$: $$ \frac{A}{P} = \left(1 + \frac{r}{n}\right)^{nt} $$

Next, take the natural logarithm of both sides: $$ \ln\left(\frac{A}{P}\right) = nt \cdot \ln\left(1 + \frac{r}{n}\right) $$

Now, isolate $t$: $$ t = \frac{\ln\left(\frac{A}{P}\right)}{n \cdot \ln\left(1 + \frac{r}{n}\right)} $$

3. Substitute the values into the formula

Now we will substitute the known values into the equation: $$ t = \frac{\ln\left(\frac{900000}{500000}\right)}{4 \cdot \ln\left(1 + \frac{0.08}{4}\right)} $$

4. Calculate the values

First, simplify $\frac{A}{P}$: $$ \frac{900000}{500000} = 1.8 $$

Next, calculate $1 + \frac{r}{n}$: $$ 1 + \frac{0.08}{4} = 1 + 0.02 = 1.02 $$

Now plug these values into the equation: $$ t = \frac{\ln(1.8)}{4 \cdot \ln(1.02)} $$

5. Compute the logarithms

Now compute the logarithms:

• $\ln(1.8) \approx 0.5878$
• $\ln(1.02) \approx 0.0198$

Substituting these values in: $$ t = \frac{0.5878}{4 \cdot 0.0198} $$

6. Final Calculation for t

Calculate: $$ t \approx \frac{0.5878}{0.0792} \approx 7.41 $$