Carcass wrote:
If P dollars borrowed or invested for \(t\) years at the interest rate \(r\) compounded \(n\) times per year, then the future value (a) will be \(A=P(1+\frac{r}{n})^{nt}\)
Two individuals, M and N invested P dollars with the following conditions and both gained the same future value at the end of the investment period:
M invested $P at the interest rate \(\frac{r-5}{100}\) for 6 years compounded 4 times per year
N invested $P at the interest rate \(\frac{r+5}{100}\) for 4 years compounded 6 times per year
Quantity A |
Quantity B |
\(r\) |
\(25\) |
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
\(A_M = P [1 + \frac{(r-5)}{(100)(4)}]^{(4)(6)}\)
\(A_N = P [1 + \frac{(r+5)}{(100)(6)}]^{(6)(4)}\)
As per the question, \(A_M = A_N\)
i.e. \(P [1 + \frac{(r-5)}{(100)(4)}]^{(4)(6)} = P [1 + \frac{(r+5)}{(100)(6)}]^{(6)(4)}\)
\([1 + \frac{(r-5)}{(100)(4)}]^{(4)(6)} = [1 + \frac{(r+5)}{(100)(6)}]^{(6)(4)}\)
Taking \(24^{th}\) root both sides;
\([1 + \frac{(r-5)}{(100)(4)}] = [1 + \frac{(r+5)}{(100)(6)}]\)
\(\frac{(r-5)}{(100)(4)} = \frac{(r+5)}{(100)(6)}\)
\(\frac{(r-5)}{4} = \frac{(r+5)}{6}\)
\(6r - 30 = 4r + 20\)
\(2r = 50\)
\(r = 25\)
Col. A: \(25\)
Col. B: \(25\)
Hence, option C