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An investor invested two equal sums of money for year. The first sum was invested at the rate of annual rate of interest and the interest compounding was half-yearly. The second sum was invested at the rate of annual rate of interest and the interest compounding was quarterly. If at the end of the year, the amounts he received are equal, what is the value of in terms of ?
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Q. An investor invested two equal sums of money for year. The first sum was invested at the rate of annual rate of interest and the interest compounding was half-yearly. The second sum was invested at the rate of annual rate of interest and the interest compounding was quarterly. If at the end of the year, the amounts he received are equal, what is the value of in terms of ?
- Denote Principal Amount: Let's denote the principal amount for each investment as . For the first sum with half-yearly compounding, the amount after year at interest rate is calculated using the formula for compound interest: .
- Calculate Amount : For the second sum with quarterly compounding, the amount after year at interest rate is calculated using the formula for compound interest: .
- Calculate Amount A: Since the amounts received from both investments are equal at the end of the year, we can set equal to : .
- Set Amounts Equal: We can cancel out the principal from both sides since it's the same for both investments: .
- Take Square Root: Now, let's take the square root of both sides to simplify the equation: .
- Expand Right Side: Expanding the right side using the binomial theorem gives us: .
- Isolate Terms: Subtract from both sides to isolate the terms with and : $\frac{R}{\(200\)} = \(2\)\left(\frac{r}{\(400\)}\right) + \left(\frac{r}{\(400\)}\right)^\(2\).
- Simplify Right Side: Simplify the right side by combining like terms: \(\frac{R}{200} = \frac{r}{200} + \left(\frac{r}{400}\right)^2\).
- Multiply by \(200\): Now, let's multiply through by \(200\) to solve for \(R\): \(R = r + 200\left(\frac{r}{400}\right)^2\).
- Simplify Second Term: Simplify the second term on the right side: \(R = r + \frac{r^2}{8}\).
- Find Value of R: So, the value of R in terms of r is: \(R = r + \frac{r^2}{8}\).
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