Question

Let a and b be natural numbers such that a>b. Which of the following rational expressions are greater?

(a²-b²)/(a-b) or (a²+b²)/(a+b)

Answer 1

Given that,


a > b

Now,

(a² - b²)/(a - b)

We can factorise a² - b²

=> (a² - b²)/(a - b)

= (a + b)(a - b)/(a - b)

Cancelling (a - b)

= (a + b)

=> (a² - b²)/(a - b) = (a + b)

Now, (a² + b²) = (a + b)² - 2ab

=> (a² + b²)/(a + b)

= {(a + b)² - 2ab}/(a + b)

=> (a + b)²/(a + b) - 2ab/(a + b)

=> (a + b) - 2ab/(a + b)

Now the least natural number is 1
So Let us put b as 1 and a as 2

we get,

2ab/(a + b)

= 2(2)(1)/(2 + 1)

= 4/3

=> a + b - 4/3

Now since,

a + b > a + b - 4/3 (since the sum is greater than the difference for natural numbers)

=> (a² - b²)/(a - b) > (a² + b²)/(a + b)


NOTE :- We have taken the value just to know whether the outcome is greater or less.

Hope you understand