Edward invested five-ninths of his money at an annual rate of \(2r\%\) compounded semi-annually, and the remaining money

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Thu Aug 20, 2020 9:32 am

Edward invested five-ninths of his money at an annual rate of \(2r\%\) compounded semi-annually, and the remaining money at an annual rate of \(r\%\) compounded annually. If after one year, Edward’s money had grown by one-third, the value of \(r\) is equal to which of the following?

A. \(10\%\)
B. \(15\%\)
C. \(20\%\)
D. \(25\%\)
E. \(33\%\)

Answer: C

Solution:

We can let the total money invested at the beginning be $900. Therefore, 5/9 x 900 = $500 was invested at 2r% annual interest rate, compounded semiannually, and 4/9 x 900 = $400 was invested at r% annual interest rate, compounded annually. Including interest, after one year, the initial $500 becomes 500(1 + r/100)(1 + r/100) dollars. Similarly, including interest, the $400 grows to 400(1 + r/100) dollars after one year. Since the investment had grown by one-third, the total amount of money in the account after one year is 900 x (1 + ⅓) = 900 x 4/3 = $1200. We can create the equation:

500(1 + r/100)(1 + r/100) + 400(1 + r/100) = 1200

Letting y = 1 + r/100 and dividing the equation by 100, we have:

5y^2 + 4y = 12

5y^2 + 4y - 12 = 0

(5y - 6)(y + 2) = 0

y = 6/5 or y = -2

Since r is not negative, y can’t be negative, either. So y can’t be -2. Therefore, y must be 6/5, i.e.,

1 + r/100 = 6/5

r/100 = 1/5

r = 20

Answer: C