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.

Algebra ->  Expressions -> 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.      Log On


   

Question 77956: Which of the fractions 4567890123/780123456 and 4567890124/780123458 is greater. Give a algebric proof i.e. which avoids multiplications of the numbers involved.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If we let a=4567890123 and b=780123456 we have a first fraction of:

a%2Fb

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:

%28a%2B1%29%2F%28b%2B2%29

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

a%2Fb=%28a%2B1%29%2F%28b%2B2%29

Now cross multiply

a%28b%2B2%29=b%28a%2B1%29

Now distribute

ab%2B2a=ab%2Bb

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:

a%2Fb%3E%28a%2B1%29%2F%28b%2B2%29

Which means the first fraction is larger, ie

4567890123%2F780123456%3E4567890124%2F780123458