When discussing fractions, I often find a very interesting area of confusion among students. I’ll ask them to compare, for example, 1/3 and 1/4. Almost everyone correctly answers that 1/4 is smaller. “How do you know?” I’ll ask. And this is where the confusion starts. More often then not, most of the class will say something like, “When the denominator is larger, the fraction is smaller.”
But is that always true?
Well, in this case, sure, it’s true. But what if I asked to compare, for example, -1/3 and -1/4? Now, it would be incorrect to say the number with the larger denominator is smaller. Now that we’re on the negative side of the number line, everything is reversed. -1/4 is actually larger than -1/3.
The confusion is only compounded when we get to fractions involving exponents. Let’s say b is a proper fraction (i.e. a number between 0 and 1). It’s easy enough to see that, for example, b^5 will be less than b^3. For example, (1/2)^5 = 1/32 is less than (1/2)^3 = 1/8.
But students often extend this to a general rule and say something like, “The higher the exponent, the smaller the fraction.” Of course, this isn’t always true. If b were a negative proper fraction (i.e. between -1 and 0), then b^5 would actually be larger than b^3 (e.g. -1/32 > -1/8).
So how to work around this confusion? Simply put: Get rid of the terms “larger” and “smaller”. They’ll only cause trouble, because the rules switch once you go from positive to negative or vice versa. Instead, think about the fractions in terms of their relationship to zero.
Let’s look at our first example. We found that 1/4 is smaller than 1/3, because the fractions are both positive. Then we found that -1/4 is larger than -1/3, because the fractions are both negative. What’s the common thread? In each case, the number with the larger denominator is closer to zero! You can see this on a number line:
Only after you’ve figured out which fraction is closer to zero should you deal with “bigger” or “smaller”. Once you’ve determined that -1/4 is closer to zero than -1/3, you can easily figure out that -1/4 is larger, because being closer to zero on the negative side indicates a larger value. You could also view this as the absolute value getting smaller and smaller.
This proves especially helpful when dealing with exponents of negative fractions. Let’s use -1/2 as our example. As we increase the integer powers, weird stuff starts to happen:
(-1/2)^1 = -1/2
(-1/2)^2 = +1/4
(-1/2)^3 = -1/8
(-1/2)^4 = +1/16
Notice that the terms are now alternating between positive and negative values. But the common thread is that the terms always get closer to zero. In other words, the absolute value gets smaller and smaller as the exponents get larger. You can cut out a lot of confusion by sticking to the principle of distance from zero.
Let’s say you’re told that b is a non-zero value between -1 and 1, meaning that b is either a negative or positive proper fraction. If you’re asked to compare b^5 and b^3, you can’t say that either one is always smaller or larger, since they could be either positive or negative. But you can say that the fraction with the higher exponent (b^5) will always be closer to zero. So if you are then given information saying that b is negative, you will know that b^5 is larger (i.e. closer to zero on the negative side).
They probably won’t be so generous as to ask you to compare b^5 and b^4. In that case, b^5 is negative, since we’re taking an odd power of a negative number, and b^4 is positive, since we’re taking an even power of a negative number. But if they brought in absolute values and asked you to compare |b^5| and |b^4|, you could easily apply the principle we just discussed! Whether b is positive or negative, it must be true that |b^5| < |b^4|, because a higher exponent on a fraction will always move the fraction closer to zero, meaning its absolute value will decrease.
Hopefully this will alleviate some of the confusion about fractions as you go forward with your GMAT studies!