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.