CrackVerbalGMAT wrote:
One way to solve this question is to know that if you add the same constant to the numerator and denominator of a positive fraction, then the value of the fraction increases and will always tend to 1.
These comments are not correct, so I'll clarify.
First, a "fraction" is just a way to write a number, by using a numerator and a denominator (as opposed to, say, using decimal notation). When we call a number a "fraction", we're not saying anything about how big the number is -- "18/5" is just as much a fraction as "1/2" is. You both mean to describe a fraction with an overall value specifically between 0 and 1. Then if the numerator and denominator are both positive, when we add a positive number k to both the numerator and denominator, the overall value of the fraction will increase. So if we start with, say, the fraction 2/3, then if k > 0, it is always true that
2/3 < (2 + k)/(3 + k) < 1
But if you start with a fraction bigger than 1, with a positive numerator and denominator, the opposite of the above is true -- adding a positive number to the numerator and denominator will decrease the fraction's value. 3/2 is larger than 6/5, for example.
Second, it's not technically correct to say that this principle applies to "positive fractions" (even with the restriction that the "fraction" has a value between 0 and 1). What matters is not whether the fraction itself is positive. What matters is that the numerator and denominator are both individually positive. The fraction (-1)/(-5) is a "positive fraction", since it is equal to 1/5, but if you add 2 to the numerator and denominator of (-1)/(-5), you get 1/(-3), which is now smaller than what we started with: its value does not increase.
The general rule is this: if x, y, a and b are all positive, then one of the following three things is always true:
x/y < (x + a)/(y + b) < a/b
or
x/y > (x + a)/(y + b) > a/b
or
x/y = (x + a)/(y + b) = a/b
so in other words, when everything is positive, if we add a to the numerator and b to the denominator of a fraction, the fraction's overall value moves towards a/b. That's a principle everyone learns, usually in a very different way, when studying weighted averages (at least from a source that explains weighted averages well). But if there might be negative values anywhere, none of the above applies.
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