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## Comparing Fractions

Often on the GRE, we need a quick way to compare the size of two fractions. Sometimes, a quantitive comparative comparison will ask us to compare them directly, and other times, comparing them will be part of a larger problem. So there are a few important rules to note here. First of all, very basic rule, changing numerator, same denominator.

if we keep the same denominator making the numerator bigger makes the fraction bigger. Making the numerator smaller makes the fraction smaller. This is probably pretty obvious, and these are some examples these are probably things most people find pretty easy to see which one is going to be bigger.

Now let’s talk about the case of changing denominators same numerator. This is the one that can confuse people sometimes it’s a little bit anti intuitive. If the numerators are the same then making the denominator. Bigger makes the fraction smaller and making the denominator smaller makes the fraction bigger.

We think about fractions as division,iIf we have a certain number of cookies, well if we divide it by more children then each child is going to get fewer cookies. But if we divide it by fewer children each child is going to get more cookies and that’s basically what we’re doing here. That’s why making the denominator smaller makes the quotient, makes the fraction bigger.

So bigger denominators make for smaller fractions. So for example, we know that 2 5ths has to be bigger than 2 7ths. We know that 3/11ths has to be smaller than 3/10ths, and we know that 7/24ths has to be bigger than seven over 36. These are examples of things you need to be able to see very quickly and be able to compare these fractions very efficiently.

What if we change both the numerator and the denominator? Well, if both the numerator gets bigger and the denominator gets smaller, then the fraction definitely gets bigger. For example, we change from 3/8th to 4/7th, well we’ve made the denominator bigger and we’ve made the numerator bigger and we’ve made the denominator smaller, and so those two rules combine, so definitely 4/7th has to be bigger than 3/8th.

That’s very clear. What if we increase both the numerator and the denominator? Does that make a fraction bigger or smaller? Well, it depends. First of all, it depends on how we increase. First of all, remember that if we multiply both numerator and denominator by the same number, we get an equivalent fraction.

So for example, we start with 3 7ths, multiply both numerator and denominator by 12, we get 36 over 84. Those are two completely identical fractions, the same mathematical value, we can put an equal sign between them. We do not change the value at all in that case. What if we add the same number to both the numerator and the denominator?

So this is an interesting little trick and this might save you time. If you can understand how to apply this. If we add the same number to both th numerator and the denominator. Then the result in fraction is closer to one then was the original fraction so for example. Suppose we have a number line here.

Well if the original fraction is somewhere down here, adding the same thing to the numerator denominator moves it closer to one, so that moves it up, but if the original fraction is somewhere here, adding the same thing to numerator denominator moves the fraction down, and in both cases you could say one is a kind of central attractor pulling the fractions closer.

So let’s talk about this with numbers. Suppose we start with fraction two-fifths, of course two fifths is less than one. Now add 6 to both the numerator and the denominator, we get eight over eleven. Well, have we made things bigger or smaller? Well think about where we are on the number line. Two fifths is less than one, adding the same number to the numerator denominator moves the fraction close to one, which in this case moves it up and so we see that eight elevenths has to be bigger than two fifths.

Again, start with seven over four, seven over four is larger than one of course. It’s an improper fraction. Add one to both the numerator and the denominator, we get eight over five, and we might ask have we made things bigger or smaller? Well again, think about where we are on the number line. On the number line 7/4ths is bigger than one.

Adding one to the numerator denominator moves it closer to one, which means it moves it to a number that is smaller, and so we see that 8/5ths has to be less than seen over four. What if we add something different to the numerator and the denominator? How does the value of the fraction change? Well, suppose for instance we add two to the numerator and five to the denominator.

The resultant fraction will be closer to two-fifths on the number line than was the original fraction. So that means that if we’re starting at here, less than two fifths, adding two to the numerator and five to the denominator will move closer to two fifths, moves it up, makes it bigger.

But if we’re starting over here on the number line, adding two to the numerator and five to the denominator moves it closer to two fifths. Moves it down, so now the central attractor is not one central attractor is two over five. So for example suppose we start with one over eight, one eighth, clearly this is smaller than two fifth.

It has a smaller numerator and a bigger denominator, so this is smaller than two fifths, we’re gonna add two to the numerator and five to the denominator get three over 13 and the question is have we made things bigger or smaller, well think about where we are on the number line, here’s one eighth, less than two fifths. Adding two to the numerator, five to the denominator moves it closer to two fifths which means it moves it up and therefore we see three over 13 has to be bigger than one over eight.

Another example, start with three over four, three over four is definitely than 2/5ths. Because we’ve made the denominator, the numerator bigger and the denominator smaller, so this is larger than two fifths, we’re gonna add two to the numerator five to the denominator, get five ninths, have we gotten bigger or smaller, well think about where we are on the number line, three fourths is up here, larger than two fifths.

Adding two to the numerator and five to the denominator moves us down. Moves us to five ninths so we see that five ninths has to be less than three fourths. So it allows us very efficiently to compare these two fractions. Now I’ll state the general rule. This general rule involves a lot of algebra.

So this might be a bit confusing, but I’ll go though this for the people who like the algebra. If we start with a fraction, a over b and then add p to the numerator and q to the denominator, the resulting fraction which will be a plus p over b plus q, will be closer. To p over q, then was the original fraction.

So if the original fraction is smaller than p over q then doing this makes it bigger. If the original fraction is larger than p over q then doing this makes it smaller and so again whatever we’re adding, we’re adding p to the numerator and q to the denominator. That fraction p over q becomes a central attractor, fractions here move closer to it, fractions here move closer to it, so the ones here get larger, and the ones here get smaller.

If you see to apply this rule, it can be a very powerful shortcut. If we subtract numbers from both the numerator and the denominator, the effect is the opposite, so I would recommend think in terms of adding, think in the direction that you will be adding, and that will be, make it easy as to apply the rule, if we add to the numerator and subtract from the denominator or vice versa, then whichever fraction has a larger numerator and a smaller denominator is bigger.

Now, here’s another, completely different trick for comparing two fractions. This is also a really time-saving shortcut. Suppose in a problem, we have to compare quickly the size of 7/11 and 5/8. Two relatively close fractions and decide which one’s bigger. Using the calculator then writing the decimals down to compare them would be time consuming.

Instead, write the fractions with a double question mark sign between them. This double question mark could represent greater than or less than, so we don’t know. Seven over 11, is it gonna be larger than or smaller than 5/8th? We don’t know, so we’re just writing double question marks. Now simply cross multiply.

So we get seven over eight on one side, five, five times 11 on the other side, seven times eight is 56, five times 11 is 55. Of course 56 is larger so of course. The inequality has to point in the same direction before we cross multiply. Cross multiplying for positive numbers cross multiplying is not gonna change the numbers of the inequality, and therefore this means that 7 over 11 has to be bigger then 5 over 8.

So again this can be an enormously simplifying shortcut to compare two fractions. Here’s a practice question. Pause the video here and then we’ll talk about this. Okay, a few things to notice.

We noticed that essentially what we’ve done, is we’ve added 100 to the numerator and 200 to the denominator, and of course, 100 over 200 equals one half. Well, fi, 4/5ths is larger than one half. So when we start with 4/5ths and add one in the numerator and one in the denominator, or anything else in that ratio, the fraction gets smaller, and so we can see, just by applying this trick, that column A, must be bigger.

And again, since four fifths is up here, adding. One to the numerator and two to the denominator, or anything in that ratio, is gonna move it closer to 1/2, which makes it smaller. And so the, the fraction in B is smaller, and the fraction in A is larger. Here’s another practice question, pause the video, and then we’ll talk about this.

At a certain school in December 2013, there were six teachers and 200 students. On January 2nd, the day after new year, one teacher and 35 students joined the school, and no one left the school, and so we want to compare the teacher-student ratio in December, and the teacher-student ratio in late January. Well this is tricky.

The teacher student ratio in December is 6/200. And I’m just gonna simplify this right away to 3/100. So that was the old ratio, and then we’re gonna add one and 35. Well, here’s the thing. We know that, suppose we make a number line. Here’s zero, and here’s one over 35.

Well, if one over thirty-five is bigger than this fraction, then. Adding one to the numerator and thirty five to the denominator will make the fraction bigger. But if one over thirty five is smaller than three over a hundred, then adding one to the numerator and thirty five to the denominator will make it smaller.

So really it comes down to the question of which one is bigger? Is it three over a hundred or one over thirty five? I’m gonna cross multiply, three times 35 is 105, and, of course, one times 100 is 100. That means this is bigger. That means three over 100 is bigger than one over 35.

Adding one to the numerator and 35 to the denominator makes the fraction smaller. So this new ratio is going to be smaller, and so that means that column A, is bigger. In summary, with the same denominator. Bigger numerator means the bigger fraction a bigger fraction.

That’s a really basic rule, that you have to have that at your finger tips. With the same numerator, bigger denominator means a smaller fraction. This is a little bit trickier to see but you have to have this one absolutely at your fingertips, these first two you have to know inside-out. We talked about the rules for adding numbers to the numerator and the denominator.

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