SOLUTION: Which of the fractions 4567890123/780123456 and 4567890124/780123458 is greater. Give a algebric proof i.e. which avoids multiplications of the numbers involved.

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If we let a=4567890123 and b=780123456 we have a first fraction of:



Since the 2nd fraction has a numerator of 1 added to 4567890123 and a denominator of 2 added to 780123456 we have a 2nd fraction:



So lets assume these two fractions are equal (they're not, but lets place a relation between them)



Now cross multiply



Now distribute



Since the left side has 2 "a" terms, the left side is larger (there are more larger terms on the left side). In other words, this is true:



Which means the first fraction is larger, ie