Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
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 (r1/100) × i1. Likewise, for bond Y, this product is equal to (r2/100) × i2.
The question can be rephrased thus: “Is (r1/100) × i1 > (r2/100) × i2?” or, after multiplying through by 100, “Is r1i1 > r2i2?”
Statement 1: INSUFFICIENT. There is no information about i1 or i2.
Statement 2: INSUFFICIENT. We can translate this statement to an inequality:
i1 / i2 > r1 / r2
Since all of the quantities are positive, we can multiply through without worrying about flipping the inequality symbol, and we get the following:
i1r2 > i2r1
However, we cannot conclude that r1i1 is always larger (or always smaller) than r2i2. 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 r1i1 on one side of the inequality symbol and r2i2 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:
i1r2 > i2r1
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:
i1 / r1 > i2 / r2
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.
i1 / r1 > i2 / r2
r12 > r22
We wind up with the inequality we’re looking for:
i1r1 > i2r2
The correct answer is C.