RenB wrote:
Larry invested $20,000 in an investment that earned an annual interest of r percent when compounded quarterly. On the same day, Mark invested $20,000 in an investment that earned an annual interest of q percent compounded semi-annually. Did Mark earn more interest than Larry at the completion of one year of their investments?
(1) q/200 > r/400
(2) Peter, whose investment of $20,000 earned an annual interest of q percent compounded annually, earned more interest than Larry at the end of one year of the investment.
Amount at the end of the first year
Larry = \(20,000 * (1+\frac{r}{400})^4\)
Mark = \(20,000 * (1+\frac{q}{200})^2\)
Question
\(20,000 * (1+\frac{q}{200})^2 > 20,000 * (1+\frac{r}{400})^4 \)
Dividing the 2000 on both sides
\((1+\frac{q}{200})^2 > (1+\frac{r}{400})^4 \)
Taking square root on both sides
\((1+\frac{q}{200})> (1+\frac{r}{400})^2 \)
Statement 1(1) \(\frac{q}{200} > \frac{r}{400}\)Case 1:
- \(\frac{q}{200} = 0.5\)
- \(\frac{r}{400} = 0.1\)
\(1+\frac{q}{200}\) = 1.5
\((1+\frac{r}{400})^2 = (1+0.1)^2 = 1.21\)
\((1+\frac{q}{200})> (1+\frac{r}{400})^2 \) -- Yes !
Case 2:
- \(\frac{q}{200} = 0.5\)
- \(\frac{r}{400} = 0.3\)
\(1+\frac{q}{200}\) = 1.5
\((1+\frac{r}{400})^2 = (1+0.3)^2 = 1.69\)
\((1+\frac{q}{200})> (1+\frac{r}{400})^2 \) -- No !
Statement 1 is not sufficient. Hence, we can eliminate A and D.
Statement 2(2) Peter, whose investment of $20,000 earned an annual interest of q percent compounded annually, earned more interest than Larry at the end of one year of the investment.
The formula for calculating the future value of an investment with compound interest is:
\(A = P(1+\frac{r}{n})^{nt}\)
So, for the same principal value, the more frequently n (compounding periods per year) increases, the more granular the compounding becomes, and the more interest one can potentially earn over time.
Statement 2 tells us that Peter, whose investment of $20,000 earned an annual interest of \(q\) percent compounded annually, earned more interest than Larry at the end of one year of the investment.
Both Peter and Mark invested the same principal value and the investments were compounded using the same rate of interest (i.e. \(q\) percent). As the principal values and the rate of interest of both investments were the same and one investment was compounded more frequently than the other, we can conclude that Mark earned more interest than Peter in one year.
After one year -
- Interest Earned by Mark > Interest Earned by Peter
- Interest Earned by Peter > Interest Earned by Larry
Therefore
Interest Earned by Mark > Interest Earned by Larry
Option B