Comparing Fractions

[KWF]

Is it possible to determine which fraction is larger than another by using its numerator?

Example: 1/7 is larger than 1/9, but when the numerators are larger than 1, as in 3/7 and 4/9, can the larger fraction be determined by just using the numerators 3 and 4 without changing 3/7 and 4/9 to fractions
having the same denominator?

 

[pka]

Is it possible to determine which fraction is larger than another by using its numerator?

Example: 1/7 is larger than 1/9, but when the numerators are larger than 1, as in 3/7 and 4/9, can the larger fraction be determined by just using the numerators 3 and 4 without changing 3/7 and 4/9 to fractions
having the same denominator?

\(\displaystyle 3\cdot 9 <4\cdot 7\\\text{ so }\\\dfrac{3}{7}<\dfrac{4}{9}\)

 

[Ishuda]

The short answer to your question is no.

The longer answer can be proved but requires more than just Arithmetic so I'll just just give a couple of examples and 'wave my hands' for the proof. Take the example you gave; comparing 3/7 and 4/9, we see that 3/7 is the smaller so the 'rule' (if there were one) would have to include something like if numerator 1 is larger than numerator 2 then the fraction with numerator 1 is largest. That is, 4 is larger than 3, so 4/9 is larger than 3/7.

However consider the two fractions 3/7 and 4/77. If our rule above were true, then, since 4 is larger than 3, 4/77 would be larger than 3/7. However, obviously it is not.

These two examples show that you can change the size of a fraction sufficiently just by changing the denominator. Thus the denominator must be taken into account when determining which fraction is larger.

btw: I did provide a proof in a different post but was told it was inappropriate for the Arithmetic sub forum and it was deleted.

 

[mmm4444bot]

btw: I did provide a proof in a different post but was told it was inappropriate for the Arithmetic sub forum and it was deleted.

The abstract proof was bounced back by private message. Please feel free to post it on either the Intermediate Algebra or Math Odds & Ends board. :cool: