**Ques - The ratio of third proportional to 21 and 42 and mean proportional to 16 and 49 is ?**

**(a) 3 : 1**

**(b) 2 : 3**

**(c) 1 : 3**

**(d) 4 : 3**

**Solution -**

Let,

The third proportion = x.

so,

21, 42 and x be in continued proportion.

Therefore,

21 : 42 :: 42 : x

21/42 = 42/x

1/2 = 42/x

**x = 84**

Thus,

**The third proportion is 84.**

Now,

Let,

Mean proportion = y

so,

16, y and 49 be in continued proportion.

Therefore,

16 : y :: y : 49

16/y = y/49

16 × 49 = y²

y² = 16 × 49

y = squareroot of 16 × 49

y = (16 × 49)½

y = 4 × 7

**y = 28**

Thus,

**The Mean proportion is 28.**

Therefore,

Requried ratio = 84 : 28

**= 3 : 1**

The ratio of third proportional to 21 and 42 and mean proportional to 16 and 49 is 3 : 1.

Hence,

The correct answer is option **(a) 3 : 1.**

Thank You