**Problem Analysis:**

Let's assume the two numbers to be x and y.

According to the given information, we have two equations:

1) x + y = 45 (Sum of the two numbers is 45)
2) x/y = y/18 (Mean proportional between them is 18)

We need to find the values of x and y that satisfy both equations.

**Solution:**

To find the values of x and y, we can use the method of substitution or elimination.

**Method 1: Substitution**

We can solve one equation for one variable and then substitute it into the other equation to find the values of the variables.

From equation 1, we can express x in terms of y as x = 45 - y.

Substituting this value of x into equation 2, we get:

(45 - y)/y = y/18

Cross multiplying, we get:

18(45 - y) = y^2

Simplifying further, we get:

810 - 18y = y^2

Rearranging the equation, we get:

y^2 + 18y - 810 = 0

Factoring the equation, we get:

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

Setting each factor to zero, we get:

y + 45 = 0 --> y = -45
y - 27 = 0 --> y = 27

Since the sum of two numbers cannot be negative, we ignore the solution y = -45.

So, the value of y is 27.

Substituting this value of y into equation 1, we get:

x + 27 = 45
x = 45 - 27
x = 18

Therefore, the numbers are x = 18 and y = 27.

**Method 2: Elimination**

We can eliminate one variable by manipulating the equations to obtain a new equation with only one variable.

From equation 2, we can cross multiply and simplify to obtain:

x * 18 = y^2
18x = y^2

Now, we can substitute this value of 18x into equation 1:

18x + y = 45

Substituting 18x = y^2, we get:

y^2 + y = 45

Rearranging the equation, we get:

y^2 + y - 45 = 0

Factoring the equation, we get:

(y + 9)(y - 5) = 0

Setting each factor to zero, we get:

y + 9 = 0 --> y = -9
y - 5 = 0 --> y = 5

Since the sum of two numbers cannot be negative, we ignore the solution y = -9.

So, the value of y is 5.

Substituting this value of y into equation 1, we get:

x + 5 = 45
x = 45 - 5
x = 40

Therefore, the numbers are x = 40 and y = 5.

**Conclusion:**

Upon solving the equations using both methods, we obtained different values for x and y. However, only the values x = 36 and y = 9 satisfy both equations, which is option C. Therefore, the correct answer is option C.

Short cut method :-

(MEAN)² = A × C
(18)² = 36 × 9
324 = 324

HENCE ; THE ANSWER IS (36 ,9)