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At what interest rate will P2,050 amount P3,875 in 4 years and 6 montha, if interest is compounded semi-annually? Round off to the nearest hundredths.​

Sagot :

Answer:

To determine the interest rate, we can use the compound interest formula:

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

Where:

- \( A \) is the future value of the investment (3,875)

- \( P \) is the principal amount (2,050)

- \( r \) is the annual interest rate (to be found)

- \( n \) is the number of times interest is compounded per year (semi-annually, so \( n = 2 \))

- \( t \) is the time the money is invested for in years (4.5 years)

Plugging in the values:

\[ 3,875 = 2,050 \left(1 + \frac{r}{2}\right)^{2 \times 4.5} \]

This simplifies to:

\[ 3,875 = 2,050 \left(1 + \frac{r}{2}\right)^9 \]

Dividing both sides by 2,050:

\[ \left(1 + \frac{r}{2}\right)^9 = \frac{3,875}{2,050} \]

\[ \left(1 + \frac{r}{2}\right)^9 = 1.8902 \]

To solve for \( r \), take the 9th root of both sides:

\[ 1 + \frac{r}{2} = \left(1.8902\right)^{\frac{1}{9}} \]

Calculating the 9th root of 1.8902:

\[ 1 + \frac{r}{2} \approx 1.0747 \]

Subtracting 1 from both sides:

\[ \frac{r}{2} \approx 0.0747 \]

Multiplying both sides by 2:

\[ r \approx 0.1494 \]

Converting to a percentage and rounding to the nearest hundredth:

\[ r \approx 14.94\% \]

So, the annual interest rate is approximately 14.94%.