Two numbers have a sum of 10. Find the largest possible value of their product?

Question

How do I set up an equation? I dont want to do trial and error! Please help

Answer 1

5+5=10

5x5=25 which is the biggest number

Answer 2

Here's how to do it with an equation, and not trial-and-error:

You have two numbers, A and B. A+B=10, and you want the max of A*B. Notice that you can write this as A(10-A).

If you were to graph values of A on the x-axis against values of A(10-A), the peak of this graph would tell you the value of A such that A(10-A) was at a max. Peaks happen when the slope of the tangent line is zero. Therefore, take the derivative of the function and set it equal to zero, then solve for A:

d/dA ( A(10-A) ) = 0
d/dA ( 10A - A² ) = 0
10 - 2A = 0
10 = 2A
A = 5

So the two numbers are 5 and 5.

Answer 3

The trial n error method is very easy,,
1 2 3 4 5 6 7 8 9
1+9 = 10, product = 9
2+8 = 10, product = 16
3+7 = 10, product = 21
4+6 = 10. product = 24...

Therefore the ans. is 24...
Simple as dat !

Answer 4

You can use the AM-GM inequality. Let's call the numbers x an y, then the AM-GM inequality says

(x+y)/2 >= sqrt(xy)

We know that the sum x+y = 10. So

sqrt(xy) <= 5
xy <= 25

Hence, the product must be less than 25. The maximum value 25 is reached if x = y. Thus, if x = y, then x =5 and so is y. The answer is that both values must be 5.

Answer 5

Write the numbers as 5+x and 5-x.
We're not restricting the numbers to be integers. x can be any real number.
The two numbers add up to 5+x + 5 - x = 10

Now multiply them:
(5+x)(5-x) = 25 - x^2

The product is maximum when - x^2 is as small as possible. Since the square cannot be negative, the smallest value is 0 when x = 0

Therefore the two numbers are 5+0 = 5 and 5-0 = 5

Answer 6

24 (6, 4) if the nos are distinct and if u can use the same no it is 25 (5, 5)

i.e if x & y, are two nos then x + y = 10 and if x = y then x^2 = 10

in first case divide the no. by 2 and x = 10 + 1and y = 10 -1
in the second case , divide the no. by 2

It is applicable for even no (10)
if it is odd sum (11) then divide it by two and the product of the adjacent whole nos gives u the largest that is 5 and 6 are adjacent whole nos for 5.5

Answer 7

Hay friend its very easy.

Whenever the sum of twoo number is constant the product will be maxuimum when both are equal.


So obviously answe will be 5 and 5.
So number are 5 and 5 and the maximum product is 25.

have a nice time
bikash Jain

Answer 8

25=5*5

Answer 9

5 and 5 are the numbers with the biggest product that add up to ten. The equation is really simple so trial and error doesn't take long so it is ok to try.

Answer 10

Lets try and make this as complicated as possible..


first lets create the quadratic x^2 since we are trying to find two numbers when multipled gives us the maximum product.

Now lets take the derivative of this function to find the tangent near the peak....

x^2dx=2x.

now we need to have this equal to 10..so

2x=10...dividing by 2. x=5.

plugging this into the equation 5^2=25.


Jon