Comparing more than two fractions using cross multipication

[chetan.kinger45]

Hi,

I know for a fact that more than two fractions can be easily compared by finding a common denominator. (LCM)

I am referring to page 34 of the 5th edition of the MGMAT Fractions, Decimals and Percents strategy guide. Directly quoting from this page, "cross multiplication can save a lot of time when comparing fractions (usually more than two)". Due to lack of an example, I am not sure how to do this.

Let's say that I want to compare 7/9, 4/5 and 8/13, how can I do this using cross multiplication? Is the following approach fool proof?

Step 1 : cross multiply 7/9 and 4/5 to get 35/36 and 36/35
Step 2 : cross multiply 4/5 and 8/13 to get 52/40 and 40/52
Step 3 : 4/5 > 7/9 since 36 > 35
Step 4 : 4/5 > 8/13 since 52>40

Therefore 4/5 is the greatest fraction. This approach is still alright when all we need to find is the greatest or smallest fraction. How can I use this approach to compare fractions in the real sense and arrange them in an ascending or descending order? Also, this example was pretty straight forward since 4/5 was greater than the other fractions. What happens when 4/5 is greater than one fraction and less than another fraction? I have to compare the other two fractions again by cross multiplying? Also, what happens when I have to compare 5 fractions. My permutation combinations will increase with the number of fractions to compare right?

[RonPurewal]

Hi,

I know for a fact that more than two fractions can be easily compared by finding a common denominator. (LCM)

I am referring to page 34 of the 5th edition of the MGMAT Fractions, Decimals and Percents strategy guide. Directly quoting from this page, "cross multiplication can save a lot of time when comparing fractions (usually more than two)". Due to lack of an example, I am not sure how to do this.

This whole method is really the same thing as making a common denominator.

Let's say the fractions are a/b and m/n.
Then, if you make a common denominator of bn, the fractions become an/bn and bm/bn.
Since the bn's are the same, you only need to compare the top parts of the fractions, which are an (the "cross product" from top left to bottom right) and bm (the "cross product" from top right to bottom left).

Note that each cross product corresponds to the fraction whose TOP number it contains.
This is the price you pay for the speed of the method: you have to remember which cross product goes with which of the original fractions. (That's not something I could personally remember, so I just make the common denominators.)

E.g., if you want to compare 4/5 and 7/9, the cross products are 4 * 9 = 36, which corresponds to 4/5, and 7 * 5 = 35, which corresponds to 7/9. (These just reflect the actual fractions with a common denominator, which would be, respectively, 36/45 and 35/45.) So, 4/5 is bigger.

[RonPurewal]

Let's say that I want to compare 7/9, 4/5 and 8/13, how can I do this using cross multiplication? Is the following approach fool proof?

Step 1 : cross multiply 7/9 and 4/5 to get 35/36 and 36/35
Step 2 : cross multiply 4/5 and 8/13 to get 52/40 and 40/52
Step 3 : 4/5 > 7/9 since 36 > 35
Step 4 : 4/5 > 8/13 since 52>40

Therefore 4/5 is the greatest fraction. This approach is still alright when all we need to find is the greatest or smallest fraction. How can I use this approach to compare fractions in the real sense and arrange them in an ascending or descending order?

This is a lot like weighing things on one of those balance scales. You know, those ones that work like see-saws, where you put one object in each of 2 pans. You're just "weighing" fractions instead of physical objects.

[RonPurewal]

Also, this example was pretty straight forward since 4/5 was greater than the other fractions. What happens when 4/5 is greater than one fraction and less than another fraction? I have to compare the other two fractions again by cross multiplying?

It doesn't seem that you're really thinking this through.

Remember, this is just putting things in order -- like putting anything else in order.
(If I'm taller than my cousin, but shorter than my brother, do you need to make my cousin and my brother stand together to tell me which of them is taller? Or do you already know.)

[RonPurewal]

Also, what happens when I have to compare 5 fractions. My permutation combinations will increase with the number of fractions to compare right?

If you want the biggest or smallest one, you only need to make 4 comparisons. Just pick two of them, and take the "winner" against another one, then the "winner" of that one against another one, then the "winner" of that one against the last one.

If you want the second-, third-, or fourth-biggest one, you should just scrap the whole pairwise-comparison thing, and give everybody a common denominator.

[chetan.kinger45]

Thanks for the detailed explanation Ron. To summarize what you said, use cross multiplication when we want the greatest or the smallest fraction. Use LCM when we want to order fractions. Have I got this right?

One thing that is bothering me is that the FDP guide does not mention the LCM approach. Is this covered in some other MGMAT book since it does seem to be a concept that cannot be neglected

[RonPurewal]

Thanks for the detailed explanation Ron. To summarize what you said, use cross multiplication when we want the greatest or the smallest fraction. Use LCM when we want to order fractions. Have I got this right?

The two procedures are basically variants of the same thing, so you needn't draw such strict boundaries.

Basically, the "cross multiplication" method has one primary advantage: for most people, it's faster.
It has two primary disadvantages: (a) it can't be used to compare more than two fractions at a time, and (b) it's not self-explanatory -- i.e., you have to memorize which product goes with which of the two original fractions.
There's no way I would remember that reliably enough to trust on a high-stakes exam, so I would personally just make the common denominator every time.

[RonPurewal]

One thing that is bothering me is that the FDP guide does not mention the LCM approach. Is this covered in some other MGMAT book since it does seem to be a concept that cannot be neglected

Make sure you're not expecting the strategy guides to provide you with a complete "road map" to every possible problem-solving situation.
(If that were possible, then there would be a lot of perfect scores on this exam.)

The concept of a common denominator is definitely in the guides. It's not possible, though, to describe every single thing you would ever want to do with a common denominator; there are potentially thousands of such applications. More generally, while the set of mathematical concepts tested on the gmat is limited -- and certainly discussed exhaustively in the guides -- there are basically infinitely many different ways to combine those concepts to make problems.

As an analogy, a beginner's guitar course can teach all of the major chords that one might play, but it certainly can't teach all of the different ways those chords can be strung together into a song.

Still, I'm surprised that the concept of comparing fractions isn't in there somewhere. Are you sure? (I have no access to the guides right now.) If it's not in the fdp book, maybe it's in the foundations of math book instead?

[chetan.kinger45]

Ron, don't get me wrong. I love the MGMAT guides because they are comprehensive. You said that LCM is a better approach for ordering fractions. The statement that I quoted says that cross multiplication can be used to compare more than two fractions. All I am saying is that in this specific case, it would have really made sense if the book at least gave an example on when to compare more than two (maybe 5) fractions using cross multiplication because there is a dedicated section for this topic in the book. The MGMAT books have a reputation for being comprehensive so IMO, an example supporting a claim is a must.

I also understand that LCM has thousands of applications. But it wouldn't harm to mention this in a section that explicitly talks about comparing fractions since LCM has been an age old way of doing this efficiently.

[RonPurewal]

Ron, don't get me wrong. I love the MGMAT guides because they are comprehensive. You said that LCM is a better approach for ordering fractions.

Careful"”no, I didn't say that. I said that I, personally, would probably go with the LCM, but not because it's "better". I just don't trust that I would remember which cross-product corresponds to which fraction.

It's very possible to have a situation with multiple (more than two) fractions in which cross-multiplication is still faster.
E.g., if you have to decide which is the largest fraction among, say, 3/8, 4/11, 5/14, 7/20, and 9/26, then you are definitely not going to want to make an LCM of those (!!). You'd be much better off comparing two of them at a time, and keeping the "winner" of each "fight".

[RonPurewal]

The statement that I quoted says that cross multiplication can be used to compare more than two fractions.

Yep. See the previous post"”it may even be the most efficient way.

Just remember: Your goal is not to find the "best" or "fastest" way. Nope. That kind of thinking isn't productive; in fact, it's anti-productive"”it will make you throw methods away (if you label them as "worse" or "slower"), rather than expand your skill set.

Your goal is simple: Find as many ways as possible to do things.
If you can do the cross-products, great. If you can do the LCM, great.
If you can do both, equally comfortably, then that's twice as great as either one individually.
Diversification is the name of the game.

[RonPurewal]

All I am saying is that in this specific case, it would have really made sense if the book at least gave an example on when to compare more than two (maybe 5) fractions using cross multiplication because there is a dedicated section for this topic in the book. The MGMAT books have a reputation for being comprehensive so IMO, an example supporting a claim is a must.

I see your point, but this is a test with essentially endless potential combinations of concepts. Even with a fairly limited palette of concepts, you can create an infinite variety of problems.

Though I didn't mention it, there's also an empirical aspect: The guides are based on what we've actually seen in real problems from GMAC.
If this isn't mentioned, the omission may be on purpose. If GMAC rarely, or never, does something in its official problems, then there would be no reason to expound on that topic, especially at length, in the guides.

In the case of comparing fractions, it's extremely unlikely that this would be the essence of a GMAC problem, because it's too "textbook". If there's one thing common to GMAC's problems (except at the lowest levels), that one thing is that they don't resemble "homework problems" from textbooks.
A straight comparison of fractions is something that could have been taken directly from an 8th grade homework assignment. As such, it's less likely to appear wholesale on the exam. If it does appear, it will probably be integrated into a problem with other stuff.