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Word Problems/Comparing Fractions

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Question
Hello:

Can you verify whether or not the following method that I used to compare fractions is a reliable method?

Which fraction is larger  5/9 or 3/14?

I determined which fraction is closer to the whole number 1.

5/9 is closer to 1 by adding only 4, 1/9.  With 3/14, 11, 1/14 are needed to equal 1.  So I reasoned that the fewer same denominator terms needed to get to 1 would indicate that that fraction is larger.  The more same denominator terms are needed would indicate that that fraction is smaller. 5/9 needs only 4, 1/9 and 3/14 needs 11, 1/4. So 5/9 is larger than 3/14.

Is this method always reliable in determining which fraction is larger or smaller?   It is much easier to compare fractions using this method than to convert to decimals or to change both fractions to the same denominator. Nevertheless, I have never seen this method explained in any textbooks.

I thank you for your reply.

Answer
A way that works every time is to multiply the top and bottom of each fraction by the denominator of the other fraction.  This gives (14/14)(5/9) and (3/14)(9/9).
Multiplying these out gives 60/126 and 27/126.

Since they both have the same denominator, it can be seen the 60 is greater than 70,
so its the first fraction that is bigger.

Another way to do it is to note that 5/9 = 0.5555... and 3/14 0.285714285714...

Doing this shows that the first fraction is greater than the second fraction.

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