Hey, I have been using a preparatory manual by a guy named Jeff Sackmann called Total GMAT Math. I am having a little trouble solving a data sufficiency problem. Perhaps someone can help me out. Here is the problem:

As Z increases from 98 to 99, which of the following must increase?

I. 4 - 3z
II. 4 - 3/z
III. 4 / (3- z^2)

If someone can help me out, it will be greatly appreciated.

As Z increases from 98 to 99, which of the following must increase?

I. 4 - 3z
II. 4 - 3/z
III. 4 / (3- z^2)

Hello Tony,

Thanks for posting your question. I will do my best to answer it. There are essentially, two rules that are at play here.

The first rule is:
for a fixed positive numerator of a ratio, as the denominator increases in size, the ratio decreases in size. For example, lets fix the numerator, say 3. Then, 3/5 < 3/4 < 3/2< 3/1. You see, as the denominator increases the value of the fraction decreases.

The second rule is:
when the ratio is negative, the first rule goes in the opposite direction. In other words -3/5 > - 3/4 > -3/2 > -3/1.

Now, lets try and use these two rules to solve this problem. (I) 4- 3z : This is certainly not increasing since 3z is getting larger as z increases. Therefore, we are subtracting a larger quantity from 4 as z increases. The value of 4 - 3z is getting smaller. For example, 4 - 3(98.5) = -291.5, and 4 - 3(98.8) = -292.4. Of course there is no need to do any time consuming numerical values for this. Just know that as z gets bigger 4 - 3z gets smaller. Its simple enough.

Lets look at (II). 4 - 3/z. Clearly, according to our rules, as z gets bigger 3/z will get smaller. So as we go up the number line we are subtracting increasingly smaller numbers from 4. so 4 - 3/z is increasing.

Finally, lets look at (III). Again, we will use our above mentioned rules 4 / 3- z^2. Lets focus on the denominator since our numerator is positive and fixed. Our denominator is clearly negative since any value of z between 98 and 99 will be much larger than 3 when squared. So we have the ratio of 4 to an increasingly more negative denominator. If the denominator is becoming a larger negative number as we move up the number line. Normally, a larger positive denominator would make the ratio smaller, however, because its negative, the second rule tells us that the fraction is actually increasing. Lets look at a small numerical illustration 4/ 3 - (98.5^2) = -0.000412 where as 4 / 3 -(98.8^2) = -0.00410 . Clearly, the numbers are increasing.

So the answer is (II) and (III) both work.