The sum of two numbers is 45 and the mean proportional betweem them is 18.the numbers are

x = first number

y = second number

x + y = 45 Subtract x to both sides

x + y - x = 45 - x

y = 45 - x

square root ( x * y ) = 18 Take square of both sides

x * y = 18 ^ 2

x * y = 324

x * ( 45 - x ) = 324

x * 45 - x * x = 324

45 x - x ^ 2 = 324 Subtract 324 to both sides

45 x - x ^ 2 - 324 = 324 - 324

45 x - x ^ 2 - 324 = 0

- x ^ 2 + 45 x - 324 = 0 Multiply by - 1

x ^ 2 - 45 x + 324 = 0

The solutions are :

x = 9 and x = 36

For x = 9

y = 45 - x = 45 - 9 = 36

For x = 36

y = 45 - x = 45 - 36 = 9

So numbers are 9 and 36

9 + 36 = 45

square root ( 9 * 36 ) = square root ( 324 ) = 18

P.S

If you don't know how to solve equation :

x ^ 2 - 45 x + 324 = 0

In google type :

quadratic equation online

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Free Online Quadratic Equation Solver: Solve by Quadratic...

When page be open in rectangle type :

x ^ 2 - 45 x + 324 = 0

and click option :

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You will see solution step - by - step

Well, it seems we have a mathematical mystery on our hands! Let's see if we can solve it with a little humor.

Let's call one of the numbers "x" and the other number "y".

According to the information given, we know that:

x + y = 45 (the sum of the two numbers is 45)

And we also know that:

x / y = y / 18 (the mean proportional between them is 18)

Hmm, this is getting a bit tricky. It seems like these numbers are playing hard to get!

But fear not, I'll summon my trusty interpreter and see if we can make any sense of this equation nonsense.

Interpreter: "Hey, numbers, can you give us a clue about your values?"

Numbers: *silence*

Interpreter: "Numbers, don't be shy! We just want to get to know you better, especially if you're 18!"

Numbers: *more silence*

Oh my, it seems the numbers aren't in the mood to reveal themselves! Maybe they're off on a secret vacation together, who knows?

But on a serious note, without further information or additional clues, it's impossible to determine the exact values of x and y. Sorry to disappoint, dear questioner. Math can sometimes be as mysterious as a clown pulling an endless stream of handkerchiefs out of his pocket!

Let's call the first number x and the second number y.

According to the given information:

1) The sum of the two numbers is 45:
x + y = 45

2) The mean proportional between them is 18:
x/y = y/18

To solve this system of equations, we can use the method of substitution.

From equation 2), we can rewrite it as x = y^2/18.

Substituting this value of x into equation 1), we have:

(y^2/18) + y = 45

To simplify this equation, we can multiply both sides by 18 to get rid of the denominators:

y^2 + 18y = 810

Rearranging this equation, we get:

y^2 + 18y - 810 = 0

Now we can solve this quadratic equation. We can factorize it or use the quadratic formula:

(y + 45)(y - 27) = 0

This gives us two possible values for y: y = -45 or y = 27.

If y = -45, then substituting this value back into equation 1) gives us x = 90.

If y = 27, then substituting this value back into equation 1) gives us x = 18.

Therefore, the two numbers could be either 90 and -45, or 18 and 27.

To find the two numbers, let's assign variables to them. Let's call the first number 'x' and the second number 'y'.

According to the given information, the sum of the two numbers is 45. This can be expressed as:

x + y = 45

The mean proportional between the two numbers is 18. The mean proportional is calculated by finding the square root of the product of the two numbers. Mathematically, this can be represented as:

sqrt(x * y) = 18

To find the numbers, we can solve these two equations simultaneously.

Let's rearrange the first equation to solve for x:

x = 45 - y

Now substitute this value of x in the second equation:

sqrt((45 - y) * y) = 18

Squaring both sides of the equation to eliminate the square root:

((45 - y) * y) = 18^2
(45y - y^2) = 324
y^2 - 45y + 324 = 0

This equation is a quadratic equation in terms of 'y'. We can solve it by factoring or using the quadratic formula. Factoring this equation, we can find that:

(y - 36) * (y - 9) = 0

So, either y - 36 = 0 or y - 9 = 0. This means that y can be either 36 or 9.

If we substitute y = 36 back into the first equation, we get:

x + 36 = 45
x = 45 - 36
x = 9

If we substitute y = 9 back into the first equation, we get:

x + 9 = 45
x = 45 - 9
x = 36

Therefore, the two numbers can be 9 and 36, or 36 and 9.