What is the maximum product of two positive numbers whose sum is 96?

Question

I really dont get this 

Answer 1

Problem statement:  "What is the maximum product of two positive numbers whose sum is 96?"

 

Keyword:  maximum, product, positive numbers, sum

  sum:  addition

  product:  multiplication

  positive numbers:  numbers greater than zero (which, by the way, is neither positive nor negative)

  maximum:  the greatest value; a function has, at most, one value of the dependent variable for each allowable value of the input variables.

 

If "I really don't get this" is related to the definitions, then review them."  If it is a matter of finding a process to determine the answer, then read on.

 

The allowable input values (that is, the range) of this function is all positive numbers whose sum is 96.  Right away, that limits the values. Since they must be "positive," that means each number must be greater than 0;  since the sum must be 96 that means each value must be individually less than 96.  This concept will be used often in later math, so understanding it now is vey important.

 

Now, of the various numbers that are greater than 0 and less than 96, the products (that is, the output or domain of this function) goes from just greater than 0 to some unknown value (that the problem wants us to find) and then back down to just greater than 0.  The fact that the products do this is also an important concept, so, if you need to calculate them, find:

  1 * 95 = ??

  2 * 94 = ??

  3 * 93 = ??

 . . .

  93 * 3 = ??

  94 * 2 = ??

  95 * 1 = ??

 

[Note: the products of the above expressions are duplicates, increasing and then decreasing.  Since the problem stated "positive numbers," we must allow fractions even though I did not illustrate that.]

 

So, do you understand the problem, now?  [note:  re-read the above explanation if you need to because now I'm going to proceed to finding the answer.]

 

How to find the answer [algebra method]:

Look again at the list of products above (with answers increasing, then decreasing].  A important omitted answers are:

...

46         50        2300
47         49        2303
48         48        2304
49         47        2303
50         46        2300

...

Yep, the product increases to a maximum, then it decreases.  Now, you must decide whether only integers are allowed and whether there is a maximum between two of the values listed.  If the problem assumes that "numbers" means integers, then 48 and 48 are them.  Now, we would certainly look between two pairs of values if their products were the same, but because this is only algebra, not calculus, we can "keep it simple."

 

So, put a note in your math diary/journal (you will need this for later math):  "The maximum product of two positive numbers occurs at their mid-point."

 

Example of how to apply this rule:  "If the sides of a square add up to x, what is the length of a side that gives the square the maximum area possible?"

 

[Note:  if you take calculus, you will learn how to find formulas for the maximum and the minimum of complicated functions -- this is very handy, for example, when you must minimize the time a project takes, minimize the cost you must pay, maximize the profit you are going to earn, maximize the grade you are going to earn, ...  A story:  one day my high school son was standing in the kitchen talking about his summer job as a sales clerk.  He said, "Dad, I know the prices just as well as any of the full-time people and I do well with customers, but full-time sales people earn twice as much as I do.  If I earned twice as much per hour, then I'd only have to work half as many hours."]

 

 

 

Answer 2

Hi Elen!

 

If one number is x, the other is (96 - x)...since if you add those two together, you will get 96..and the product is x(96 - x) = 96x - x^2

Since this is a quadratic function (x to the power of 2), it is a parabola. More specifically, it is a parabola opening downwards, because there is a negative in front of the x squared. Because it downward facing the maximum value of x is going to be where the vertex is. The x value of the vertex can be found by taking the negative of b (which is the number in front of teh x) over a squared (which is the number in front of the x squared which in this case is 1)

 

So x = -96 / 2(-1) = -96/-2 = 48

 

Since as we first stated one number is x, the first number is 48

The second number will be 48, because we stated the second number to be 96-x

Therefore 48 times 48 will give you the maximum product 2304.

 

Hope this helps. :)

I am located a distance from you, but I also do online tutoring with a tablet and stylus if you need help on anything else! Take care.