Find two numbers whose sum is 23 and whose product is as large as possible.

Question

Find two numbers whose sum is 23 and whose product is as large as possible.

Answer 1

Since we want the large product as possible, the two numbers must be positive.  This is because if we multiply a negative by a positive, you get a negative product. This will not give us the largest product, but a minimum product.  The sum of two negatives is a negative sum.  We need the sum to be positive.

 

Let x = the first number

Let y = the second number

 

Using the concept and variables, we set up equation based on the description.

 

x + y = 23                    eq1

xy = Product                 eq2

 

 

Substitute eq1 into eq2.  Lets get eq2 in terms of x.  This will allow us to get a function.

 

Product = x(23 - x)

Product = -x² + 23x

 

 

We have a quadratic function.  When the coefficient of a quadratic function's leading term is negative, it indicates that we will have a maximum.  To find the largest product possible, we need to find the maximum value of this function.  We do this by finding the vertex. 

 

Vertex has coordinate (h, k).

 

h = -b / 2a

 

 

where:

a = -1

b = 23

 

Plug in these values to find h.  This will be x coordinate of the maximum value.

 

h = -23 / (2*-1)

h = -23 / (-2)

h = 11.5

 

x = 11.5

 

 

Substitute this value of x into eq1 to find y.

 

y = 23 - x

y = 23 - 11.5

y = 11.5

 

The two numbers that will give the largest product possible are 11.5 and 11.5.

 

 

 

Answer 2

Assuming that you are restricted to integers, the two numbers shall be either the median of the range or the integers to either side of the median. Here the median is 11.5, so the two numbers are 11 and 12. The product is 122.