Input the sum of two numbers to instantly calculate their maximum product.
Let the two numbers be \(x\) and \(y\). If their sum is \(S\), the goal is to find the maximum product \(P = x \cdot y\).
For a fixed sum \(S\), the product \(P = x \cdot y\) is maximized when the two numbers are as close as possible. Algebraically, the maximum product occurs when the two numbers are equal.
If \(x + y = S\), then setting \(x = y = \frac{S}{2}\) yields the maximum product: \( P_{\text{max}} = x \cdot y = \left( \frac{S}{2} \right) \cdot \left( \frac{S}{2} \right) = \frac{S^2}{4} \)
Given the sum \(S\), the maximum product \(P_{\text{max}}\) is calculated as: \( P_{\text{max}} = \frac{S^2}{4} \)
Solution:
Substitute \(S = 124\) into the formula:
\( P_{\text{max}} = \frac{124^2}{4} = \frac{15376}{4} = 3844 \)
Result: The maximum product is 3844.
Solution:
Substitute \(S = 426\) into the formula:
\( P_{\text{max}} = \frac{426^2}{4} = \frac{181476}{4} = 45369 \)
Result: The maximum product is 45369.