# Quickly Determine Which Fraction Is Greater

Which is greater?

$\frac{3}{8}$ or $\frac{5}{13}$

Most people would approach this question by converting the fractions into decimals or putting both the fractions to a common denominator. While there’s nothing wrong with those methods, there’s a simpler and more elegant way of solving this problem.

## Solution

The secret is cross-multiplication! Just multiply the left numerator with the right denominator and multiply the right numerator with the left denominator. Afterwards, compare the two products.

The side of $\frac{5}{13}$ has a larger product (40) than the side of $\frac{3}{8}$ (39). We can now conclude that $\frac{5}{13}$ is greater than $\frac{3}{8}$. In decimal, $\frac{3}{8}$ = 0.375 and $\frac{5}{13}$ = 0.385.

That’s it! You don’t have to do anything else.

I promise that this method works with any two fractions. However, note that you can only use this method for comparing two fractions. Furthermore, this method also works for negative fractions.

Here are some exercises for you to practice your new-found skill.

Which is greater?

1. $\frac{4}{7}$ or $\frac{7}{13}$

2. $\frac{6}{7}$ or $\frac{9}{11}$

3. $\frac{8}{11}$ or $\frac{5}{7}$

## Proof

For those of you who want to see the proof before trusting me, here it is.

Theorem: Given integers p, q, r and s, where q and s are positive, we have,

If and only if ps < rq

The statement remains true if ‘<‘ is replaced with ‘>’ or ‘=’.

Proof:  First, we write,

where ε < 0, = 0 or > 0 according to whether the inequality is ‘<‘, ‘>’ or ‘=’ respectively.

Multiplying both sides by qs, we get,

ps = qr + ε1            where ε1 = εqs                  Eq. 2

ε1 has the same sign as ε, since qs is positive.

To prove the converse, we start with equation 2. Since q and s are positive, qs is positive so we can divide Eq. 2 through by qs (or equivalently, multiply by the reciprocal of qs). This gives Eq. 1. Again, ε has the same sign as ε1, since qs is positive.

Don’t worry if you don’t understand the proof. You can use this trick with no problem without knowing how it works.

### Posted by Edmark M. Law

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is edmarklaw@learnfunfacts.com

## 52 thoughts on “Quickly Determine Which Fraction Is Greater”

1. That’s smooth. I can do basic operations so fast that tricks sometimes don’t occur to me. But this is definitely one that’s straight forward enough to pass on to some of my students.

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1. This is faster than changing the fractions into common denominators in many cases.

Some people may find this useful.

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2. Here’s my favorite math “trick”:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321

So, your mission, should you decide to accept it:

What is 111111111 x 111111111?

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1. In base 16, this works up to a 15-digit number. So,

111,111,111,111,111 × 111,111,111,111,111 = 123456789ABCDEFEDCBA

Thus, in base 60, this works up to a 59-digit number.

Another cool fact about 11: The number arrays of the Pascal’s triangle are basically the powers of 11, i.e. 1, 11, 121, 1331, etc.

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3. Love it! Now I have to figure out a way for this to come up in normal conversation so I can wow my family and friend. (Yes, one friend — but a good one!) After a while, I’ll give you credit. How about six weeks, does that sound reasonable? Seriously, this was great stuff. Thanks!

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1. I’m glad that you liked it.

I think that conversatiins about fractions often come up but maybe it’s just me.

That sounds reasonable enough :).

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1. Edmark, My money says that despite how smart I think you are, you don’t realize how funny you are. I wish you lived next door to me, because I’d make a bet with you to see how fast you could engage someone in a conversation about fractions. If you win, you get to carve my turkey on Thanksgiving (a high honor in my home) and earn the first slice of pie for dessert! I’m laughing thinking about a conversation about fractions. Then again, I bet the moment you slice that pie, you’d start another one!

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2. I’d definitely take on that challenge.

Then while slicing the pie, I’ll also talk about fractions with you.

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4. Hey,
thanks for taking the time to stop by my site and liking my post!

Charles

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5. Wish I’d known this one when teaching my kids their fractions!

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6. Anonymous says:

This was so helpful. I was just struggling this evening trying to help my daughter with this concept. This is a much easier way to explain this to her.

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7. jveeds says:

Great tip…but it still requires some pesky multiplication!

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1. For what it’s worth, it’s easier than converting the fractions to decimals.

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8. Had a terrible teacher in 4th grade who never really taught so I missed learning about fractions. Wish I’d read this a thousand years ago when I was young. Thanks! 😀

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9. My daughter and I thank you so much. Standardized tests make most crazy; but the convoluted methods used today drive me nuts. A great support.

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10. Fascinating! I’m going to follow your blog and get more useful party tricks like this! 😉

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11. Inese Poga Art plus Life says:

Just amazing! I always loved such mathematic solutions.

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12. Hey thanks for your objectivity. I didn’t want to make it out like I was being critical – I’ve been thoroughly enjoying your posts. What I’m told by people perhaps more savvy than to write me off as an idiot, is that I complicate things. I don’t score well on tests, for example, because I’m always looking for a hidden motive or trick question.

I’m in a better community environment now than I was for a number of years, where people are distinctly more respectful of each other – a University community but in a small town in a largely rural area. I think the density with which people are packed together in the urban areas, at least in some parts of the United States, has the effect of putting people more on the defensive, and they tend to lash out at others more frequently. You’re absolutely right – one should consider the source of name-calling, and not internalize that kind of character assassination.

Thank you for posting these “fun facts.” 🙂

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1. No problem. I can accept criticisms and learn from them :).

I always say that standardized tests are overrated. While they can be effective for testing comprehension of the subjects being tested, they can’t test a person’s character, outside the box creativity, common sense and life skills that well. And those are the real skills anyone need to succeed and be happy in life. I personally know a lot of those who got straight A’s during their studies but are practically useless at comimg up with their own ideas…

That’s good for you!

Thanks for reading and you’re welcome.

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13. OK I think I get it. The missing link was – I wasn’t making the connection that you multiply in an upward direction – (as the illustration shows) not downward. I mean, your arrows point both ways, but the actual figures, the products, are on the up-side, not the down-side. That’s what, for me, shows that it couldn’t be “either way” (which would be illogical, and leads to unknowing.)

All that said (albeit perhaps unintelligbly) I do think I’m dyslexic. I mix up things of “twos” all the time – like which letters come before or after each other in a word like “weird” – is it “wierd?” etc. People call me an idiot savant – or sometimes, just an idiot. 😦

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1. Thank you for pointing it out. I agree that having the arrows point both ways can cause confusion. That’s a mistake on my part and I’ll have to correct it.

Having dyslexia doesn’t make one an idiot. I have a dyslexic friend who is an excellent mathematician.

Who are those people anyway to decide for you that you’re an idiot. Those people should just be ignored.

For you to find a flaw in this post just proved that you’re not an idiot.

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14. All right – say I’ve known about cross-multiplication, but I never can remember which side means what. Before I even turned the page, I knew I was supposed to multiply 3 x 13 and 5 x 8, and I knew the two products (39 and 40) and that 40 was greater than 39. But how do I get from knowing that 40 is greater than 39, to knowing that 5/13 is greater than 3/8? How do I remember which side is which? Am I just dumb, or dyslexic?

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1. You don’t have to remember “which side is which”. The corss multiplication is straightforward. Whichever side has the larger product has the larger fraction. I have marked the process with colors red and green. This tells you which side you have to put the products.

Other examples

1/2 or 2/3?

1 x 3 = 3 (this goes to the side of 1/2)
2 x 2 = 4 (this goes to the side of 2/3)

So, 2/3 is larger.

2/5 or 3/4?

2 x 4 = 8 (this goes to the side of 2/5)
3 x 5= 15 (this goes to the side of 3/4)

So, 3/4 is larger.

Do you see the pattern now?

If you have any more questions, don’t hesitate to ask.

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15. Good one, Edmark. My 10-year old daughter said that she has been wasting her time by converting the fractions…..ﾍ(=￣∇￣)ﾉ

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1. I’ve just written a similar piece on a favourite maths trick of mine, would really appreciate it if you could give it a read!

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16. I thought so hard about it, made me want cake 🙂 Seriously, thanks for getting my brain going this morning!

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17. That’s brilliant! Thanks for sharing it. I’ll try to impress my friends before letting on where I got it from… 😉

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1. No. I’m more inteterested in the research side of things, and I’m not that good at teaching.

I have been invited a few times to do lectures at math conventions, though I seldom do it anymore…

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