Comparing more than two fractions using cross multiplication

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sgnikc45: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Fri Jan 10, 2014 8:37 pm

by sgnikc45 » Sat Jan 25, 2014 7:30 am

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?

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Source: Beat The GMAT — Problem Solving | Sat Jan 25, 2014 7:30 am

bubbliiiiiiii: Legendary Member; Posts: 979; Joined: Tue Apr 14, 2009 1:38 amFollowed by: 12 members; Location: Hyderabad, India; Thanked: 49 times; GMAT Score:700

by bubbliiiiiiii » Sat Jan 25, 2014 8:17 am

sgnikc45 wrote: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?

Method 1: The LCM method (for illustrative purpose)

compare 7/9, 4/5 and 8/13

LCM is 9 * 5 * 13

Multiplying each by LCM => 7*5*13, 4*9*13, 8*5*9
=> 35*13, 36*13, 40*9
=> 4/5 is greatest.

Method 2: Cross multiplication.

compare 7/9, 4/5 and 8/13

First we compare fraction 1 and 2 i.3., 7/9 and 4/5

by cross multiplication method we get: 35 and 36 => 4/5 is greatest

Now, we do cross multiplication between the greatest of first two and third i.e., 4/5 and 8/13

=> 64 and 40 => 4/5 is greater.

Coming to how to arrange the three in ascending or descending order:

Lets do the cross multiplication between the left out two i.e., 7/9 and 8/13 => 91 and 72

So the ascending order is 8/13, 7/9 and 4/5

Regards,

Pranay

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sgnikc45: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Fri Jan 10, 2014 8:37 pm

by sgnikc45 » Sat Jan 25, 2014 8:46 am

Thanks a tonne bubbliiiii. An important point that I missed and you pointed out correctly is that we have to compare the greatest fraction with the next fraction.

The key to using the cross multiplication technique (which indeed seems to be faster) is to always compare the greatest fraction till now with the next fraction in the list. Once we find the greatest fraction, we try finding the next greatest fraction and compare it with the remaining fractions and so on.. We can easily reverse the order and find the least fraction first if we want to.. Have I understood this correctly?

Also, I think you made a mistake in cross multiplying 4/5 with 8/13. We should compare 52 and 40 and not 64 and 40 right?

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bubbliiiiiiii: Legendary Member; Posts: 979; Joined: Tue Apr 14, 2009 1:38 amFollowed by: 12 members; Location: Hyderabad, India; Thanked: 49 times; GMAT Score:700

by bubbliiiiiiii » Sun Jan 26, 2014 4:49 am

sgnikc45 wrote:Thanks a tonne bubbliiiii. An important point that I missed and you pointed out correctly is that we have to compare the greatest fraction with the next fraction.

The key to using the cross multiplication technique (which indeed seems to be faster) is to always compare the greatest fraction till now with the next fraction in the list. Once we find the greatest fraction, we try finding the next greatest fraction and compare it with the remaining fractions and so on.. We can easily reverse the order and find the least fraction first if we want to.. Have I understood this correctly?

Also, I think you made a mistake in cross multiplying 4/5 with 8/13. We should compare 52 and 40 and not 64 and 40 right ?

Glad it helped you.

Yes .. you got it correct.

yes .. 52 and 40 should be compared. Thanks for pointing it out.

Regards,

Pranay

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Jan 26, 2014 9:04 am

sgnikc45 wrote:
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?

Using the cross-multiplication technique can be quite time consuming if you have 3 or more fractions to compare.
In most cases, it's much faster to convert the fractions to decimals.

To help with this, all students should commit the following conversions to memory:
1/2 = 0.5
1/3 â‰ˆ 0.33
1/4 = 0.25
1/5 = 0.2
1/6 â‰ˆ 0.17
1/7 â‰ˆ 0.14
1/8 = 0.125
1/9 â‰ˆ 0.11

So, if 1/9 â‰ˆ 0.11 then 7/9 â‰ˆ 0.77
Likewise, if 1/5 = 0.2 then 4/5 = 0.8 (aka 0.80)
This leaves us with 8/13. We can use long division to convert this to a decimal, or we can use a technique I discuss in the following BTG article: https://www.beatthegmat.com/mba/2013/01/ ... o-percents

Let's take 8/13 and find a decimal that's APPROXIMATELY equivalent. In particular, let's find an equivalent fraction with 100 in the denominator. That is, 8/13 = ?/100

Aside: I'm doing this because it's very easy to take a fraction with 100 in the denominator and covert it to a decimal. For example, 19/100 = 0.19

Let's focus on the denominators here: 8/13 = ?/100
What must we multiply 13 by to get 100?
Well, 13 x 8 = 104
So, to get a product of 100, we must multiply 13 by a number a LITTLE BIT LESS THAN 8
Since we're creating an equivalent fraction, we must also multiply the numerator (8) by a number a LITTLE BIT LESS THAN 8
When we do this, we get a numerator that's a little bit less than 64.
In other words, 8/13 â‰ˆ (a little bit less than 64)/100 â‰ˆ a little bit less than 0.64

Now, that we've converted 7/9, 4/5 and 8/13 to 0.77, 0.80 and â‰ˆ0.64, it's easy to compare them.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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