Question

The mean proportion between two numbers is 6 and their third proportion is 48. Find the two numbers.

Answer 1

Given: Let the two numbers be x and y. Mean proportion between x and y = 6 and the third proportional to x and y = 48.

Step-wise calculation:

1. Mean proportional 6 between x and y gives x : 6 = 6 : y

⇒ xy = 36   ...(I)

2. Third proportional 48 to x and y gives x : y = y : 48

⇒ y² = 48x   ...(II)

3. From (I), `x = 36/y`. 

Substitute into (II):

`y^2 = 48 xx (36/y)` 

⇒ y³ = 48 × 36 

= 1728

4. y³ = 1728

⇒ `y = root(3)(1728)`

⇒ y = 12

5. `x = 36/y`

= `36/12`

= 3

The two numbers are 3 and 12.