a and b are two numbers such that their mean proportion is 6 and third proportion is 48. What is the value of a and b?

3, 6

Mean proportional = 6 

√ab = 6,

⇒ ab = 36  ..(1.)

Third proportional = 48

⇒ 48a = \( {b }^{2 } \)   ..(2.)

⇒ From eq. (1.) a = \(\frac{36}{b}\) and put into (2.)

⇒ \( {b }^{3 } \) = 48 x 36 

⇒ \( {b }^{3 } \) = 1728 

⇒ b = 12.

⇒ Putting in (1.)

⇒ a = \(\frac{36}{12}\)

⇒ a = 3.

Therefore, a and b are 3 and 12 respectively.