Official Solution: The amount of interest, in dollars, that Samantha will receive in one year is equal to the interest rate multiplied by the principal. For bond \(X\), this product is equal to \(\frac{r_1}{100} \times i_1\). Likewise, for bond \(Y\), this product is equal to \(\frac{r_2}{100} \times i_2\).
The question can be rephrased thus: "Is \(\frac{r_1}{100} \times i_1 \gt \frac{r_2}{100} \times i_2\)?" or, after multiplying through by 100, "Is \(r_1i_1 \gt r_2i_2\)?"
Statement 1: INSUFFICIENT. There is no information about \(i_1\) or \(i_2\).
Statement 2: INSUFFICIENT. We can translate this statement to an inequality:
\(\frac{i_1}{i_2} \gt \frac{r_1}{r_2}\)
Since all of the quantities are positive, we can multiply through without worrying about flipping the inequality symbol, and we get the following:
\(i_1r_2 \gt i_2r_1\)
However, we cannot conclude that \(r_1i_1\) is always larger (or always smaller) than \(r_2i_2\). You can choose numbers to see why this is so.
Statements 1 & 2 together: SUFFICIENT. We want to combine the inequalities in such a way as to get \(r_1i_1\) on one side of the inequality symbol and \(r_2i_2\) on the other side -- if possible. In fact, this combination is possible, and the right way to execute it is first to rearrange the second statement in order to put all the same subscripts on one side. We can start from the product we obtained by cross-multiplying:
\(i_1r_2 \gt i_2r_1\)
Now divide each side by both \(r\)'s. Again, since the interest rates are necessarily positive in this scenario (you cannot be paid "negative interest"), we do not have to worry about flipping the sign. We get the following inequality:
\(\frac{i_1}{r_1} \gt \frac{i_2}{r_2}\)
Finally, we multiply this inequality by the inequality from statement 1. (Normally, this is a dangerous move, but once again, since all the quantities are positive, we are allowed to multiply.) Just make sure that the inequality symbols are in the same direction.
\(\frac{i_1}{r_1} \gt \frac{i_2}{r_2}\)
\(r_1^2 \gt r_2^2\)
We wind up with the inequality we're looking for:
\(r_1i_1 \gt r_2i_2\)
Answer: C
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