Number Theory/Comparing Fractions
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: Hello Kenneth
I'm sorry, but your argument does not work. What you are saying is that because 4/9 has a smaller numerator than 11/14, it follows that 4/9 is smaller than 11/14, and so 5/9 is larger than 3/14. Try 1/2 and 5/9. Your argument compares 1/2 with 4/9, and concludes that 1/2 is greater than 5/9, clearly false.
But there is a good way. It involves what is called a determinant which is a square array of numbers.
c d has the value a*d - b*c. 1/2 - 5/9 = (1*9 - 5*2)/18. The numerator can be written as a determinant.
Determinants stay the same value if one row is subtracted from another, or one column subtracted from another. So subtract row 1 from row 2 twice to give
The value is now 1*-1 - 0*5 = -1 < 0. So 1/2 is less than 5/9.
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I want to thank you for your reply. Is there a simple reason why my idea is not always reliable? I was surprised how often it works.
I thank you for your reply and assistance!
A very important criterion in mathematics is that it takes only one counter-example to disprove a method or theorem. I have demonstrated that.
I remember my dad showing me a method for doing an easier calculation. He said, divide one factor by a number and multiply the other by the same number. For instance. To multiply 25 by 36, divide 36 by 4 and multiply 25 by 4. 900. Easy. But he failed to impress it only worked for multiplication. I used it in school at about the age of seven for a division problem and of course, obtained the wrong answer.