Find two numbers that differ by 10 and have a product that is a minimum

Question

Must be at least two numbers

 

Answer 1

Let the two numbers be X and Y  

 

They differ by 10, so  Y - X = 10  

 

Their product is P = X*Y  

 

If you solve the first equation for Y and substitute that value for Y in the second equation you will get a quadratic.  Then use what you know about quadratics (Where is the vertex?  Is the vertex a maximum or minimum?) to find what value of X will answer the question.  Once you know what X is use the first equation to find Y.  

 

Once you find the 2 numbers see what their product is and check that your answer makes sense by comparing it to some other possible pairs of numbers (like 10 and 20 which have a product of 200, or 0 and 10 which have a product of 0) to make sure your product is lower.