Question

Among all pairs of numbers whose sum is $ 100 $ , how do you find a pair whose product is as large as possible. (Hint: express the product as a function of $ x $ ) ?

Answer 1

Hint: For solving this particular question we have to use the concept of parabola that is the standard parabolic equation is always represented as $ y = a{x^2} + bx + c $ and the expression $ \dfrac{{ - b}}{{2a}} $ gives the x-coordinate and this vertex always gives the maximum value of the parabolic equation.

Complete step by step solution:
It is given that pairs of numbers having sum $ 100 $ ,
Now, we have to let certain quantities such as ,
Let be a variable $ x $ which will represent the first number.
Since pairs of numbers having sum $ 100 $ ,
We can say that ,
The second number is represented by $ 100 - x $ ,
Now , we have
First number as $ x $ and
Second number as $ 100 - x $ .
Now, multiplication of the two number is given as
 $
   = x(100 - x) \\
   = 100x - {x^2} \\
  $
Or we can write it as
 $ = - {x^2} + 100x $
The above equation represents the equation of parabola ,
And for finding the maximum value , we have to recall the property of parabola which says parabola has a maximum value at its vertex.
Therefore, we have to find the vertex of the parabola, we know that the standard parabolic equation is always represented as $ y = a{x^2} + bx + c $ and the expression $ \dfrac{{ - b}}{{2a}} $ gives the x-coordinate , and when we substitute x-coordinate in the original equation we get the y-coordinate.
Therefore, x-coordinate of the parabola is $ \dfrac{{ - b}}{{2a}} = - \dfrac{{100}}{{ - 2}} = 50 $
Therefore, the first number is $ 50 $ ,
And the second number is $ 100 - x = 100 - 50 = 50 $
Hence, we get the required result.
So, the correct answer is “50”.

Note: The equation for a parabola can also be written in vertex form as $ y = a{(x - h)^2} + k $ where $ (h,k) $ is the vertex of a parabola. The point where a parabola has zero gradient is known as the vertex of the parabola.