Multiplying 2 inequalities?

[Whit Garner]

Hi!

I've been reviewing the "Challenge" questions online and have hit a few where the solution involved multiplying one inequality times another inequality (ex. Samantha's Bonds, 3/23/09) I've put the question (and the part I'm concerned about) below and wondered if anyone could point me in the direction of some extra guidance for when (and why) we are allowed to do this.


Samantha invests i1 dollars in bond X, which pays r1 percent simple interest annually, and she invests i2 dollars in bond Y, which pays r2 percent simple interest annually. After one year, will she have earned more interest, in dollars, from bond X than from bond Y?

(1) r12 > r22

(2) The ratio of i1 to i2 is larger than the ratio of r1 to r2.

**Solution Given:
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.

Thanks for any advice!!

[RonPurewal]

hmm.

i actually don't really know of any good sources off the top of my head. if i were to try to find them, i would essentially be doing exactly what you would be doing in the same situation -- i would just be using google to try to find sources.
so, if you want to try to dig up formal sources on this, your best bet is probably to just plug away to a search engine.

with the positive numbers, however, it should be somewhat straightforward to see why it's okay to multiply the inequalities when the "alligators" face the same way.
if you don't see why that's the case, think about the situation with concrete numbers and/or a concrete situation:
imagine that person A works fewer hours, at a lower wage per hour, then person B. in this case, it should be quite obvious that person A will make less money than person B.
...but that's all you're actually doing with the inequality:
you have w1 < w2 (the wages)
and h1 < h2 (the hours)
and you're concluding that w1 times h1 < w2 times h2.

i don't really see the gmat doing this with numbers of mixed signs, so you're probably off the hook on that one. if it does happen, in a worst case scenario, you can always investigate the situation by plugging in different combinations of numbers and watching what happens.

[sharok50]

I think simple logic works better. as all the values are positive,

Larger amount X Larger rate > Smaller amount X smaller rate

So i1 X r1 > i2 X r2