6. Prove the following

For the real numbers \(a\) \(=\) 20 and \(b\) \(=\) 55, if \(a\) is increased by \(1\) and \(b\) is decreased by \(1\), then show that the  increment in product is always equal to \(b\) \(-\) \(a\) \(-\) \(1\).

 

 

Given, \(a\) \(=\) 20 and \(b\) \(=\) 55.

 

Their product \(ab\) \(=\) .

 

By the given condition, the new product is given by \((a-1)(b+1)\)\((a+1)(b-1)\)\(a(b-1)\).

 

The new product \(=\)

 

The difference in products \(=\)

 

By the distributive property, the increment in product is given by a - 1b - 1b - a - 1.

 

The increment in product using the obtained property \(=\) .

 

Difference in product is not equal tois equal to Increment in product

 

Hence, proved.