Finding Two Numbers with Mean and Third Proportional

To solve this problem, we need to use the concept of mean and third proportional. Let's first define these terms:

Mean proportional: It is a relationship between two numbers in which the ratio of the first number to the mean is equal to the ratio of the mean to the second number. Mathematically, if a and b are two numbers, then their mean proportional is x, such that:

a : x = x : b

Third proportional: It is a relationship between three numbers in which the ratio of the first number to the second is equal to the ratio of the second number to the third. Mathematically, if a, b, and c are three numbers, then their third proportional is x, such that:

a : b = b : x

Solution:

Let's assume the two numbers we are looking for are a and b. We are given that:

- Mean proportional between a and b is 18, which means:

a : 18 = 18 : b

- Third proportional between a and b is 144, which means:

a : b = b : 144

We can use these two equations to find the values of a and b. Here's how:

From the first equation, we can simplify it to get:

a x b = 18 x 18

a x b = 324

From the second equation, we can simplify it to get:

a x 144 = b x b

a x 144 = b^2

Now we can substitute a x b from the first equation into the second equation:

a x 144 = (a x b) x b

a x 144 = 324 x b

a = (324 x b) / 144

a = 2.25 x b

Substituting this value of a into the second equation, we get:

(2.25 x b) x 144 = b^2

b^2 = 324 x b

b = 324 / 18

b = 18

Now we can find the value of a using the first equation:

a x 18 = 18 x 18

a = 18

Therefore, the two numbers we are looking for are 18 and 324/18 = 18.

Conclusion:

In conclusion, we have found two numbers such that mean proportional between them is 18 and third proportional between them is 144. The two numbers are 18 and 324/18 = 18. We used the concept of mean and third proportional to solve the problem.