Question

what will be the mean proportional for a and b ? (take mean proportion as x)
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Answer 1

Answer:

We will learn how to find the mean and third proportional of the set of three numbers.

If x, y and z are in continued proportion then y is called the mean proportional (or geometric mean) of x and z.

If y is the mean proportional of x and z, y^2 = xz, i.e., y = +√xz.

For example, the mean proportion of 4 and 16 = +√4×16  = +√64 = 8

If x, y and z are in continued proportion then z is called the third proportional.

For example, the third proportional of 4, 8 is 16.

 

Solved examples on understanding mean and third proportional

1. Find the third proportional to 2.5 g and 3.5 g.

Solution:

Therefore, 2.5, 3.5 and x are in continuous proportion.

 2.53.5 = 3.5x

⟹ 2.5x = 3.5 × 3.5

⟹ x = 3.5×3.52.5

⟹ x = 4.9 g

2. Find the mean proportional of 3 and 27.

Solution:

The mean proportional of 3 and 27 = +√3×27 = +√81 = 9.

 

3. Find the mean between 6 and 0.54.

Solution:

The mean proportional of 6 and 0.54 = +√6×0.54 = +√3.24 = 1.8

 

4. If two extreme terms of three continued proportional numbers be pqr, prq; what is the mean proportional?

Solution:

Let the middle term be x

Therefore, pqrx = xprq

⟹ x2 = pqr × prq = p2r2

⟹ x = √p2r2 = pr

Therefore, the mean proportional is pr.

 

5. Find the third proportional of 36 and 12.

Solution:

If x is the third proportional then 36, 12 and x are continued proportion.

Therefore, 3612 = 12x

⟹ 36x = 12 × 12

⟹ 36x = 144

⟹ x = 14436

⟹ x = 4.