An investor invested two equal sums of money for $1$ year. The first sum was invested at the rate of $R\%$ annual rate of interest and the interest compounding was half-yearly. The second sum was invested at the rate of $r\%$ 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 $R$ in terms of $r$ ?

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Grade 8

Interpreting Linear Expressions

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Q. An investor invested two equal sums of money for

1

year. The first sum was invested at the rate of

R\%

annual rate of interest and the interest compounding was half-yearly. The second sum was invested at the rate of

r\%

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

R

in terms of

r

Denote Principal Amount: Let's denote the principal amount for each investment as $P$ . For the first sum with half-yearly compounding, the amount $A_1$ after $1$ year at $R\%$ interest rate is calculated using the formula for compound interest: $A_1 = P(1 + \frac{R}{200})^2$ .
Calculate Amount $A_1$ : For the second sum with quarterly compounding, the amount $A_2$ after $1$ year at $r\%$ interest rate is calculated using the formula for compound interest: $A_2 = P(1 + \frac{r}{400})^4$ .
Calculate Amount A $2$ : Since the amounts received from both investments are equal at the end of the year, we can set $A_1$ equal to $A_2$ : $P(1 + \frac{R}{200})^2 = P(1 + \frac{r}{400})^4$ .
Set Amounts Equal: We can cancel out the principal $P$ from both sides since it's the same for both investments: $(1 + \frac{R}{200})^2 = (1 + \frac{r}{400})^4$ .
Take Square Root: Now, let's take the square root of both sides to simplify the equation: $1 + \frac{R}{200} = \left(1 + \frac{r}{400}\right)^2$ .
Expand Right Side: Expanding the right side using the binomial theorem gives us: $1 + \frac{R}{200} = 1 + 2\left(\frac{r}{400}\right) + \left(\frac{r}{400}\right)^2$ .
Isolate Terms: Subtract $1$ from both sides to isolate the terms with $R$ and $r$ : $\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: $R = r + \frac{r^2}{8}$.