Average [औसत] - 01

Question: The average age of A and B is 20 years. If C were to replace A, the average would be 19 and if C were to replace B, the average would be 21. What are the ages of A, B and C? [A तथा B की औसत आयु 20 वर्ष है. यदि A को हटा कर C आ जाता है, तो औसत 19 वर्ष हो जाती है और यदि B को हटा कर C आ जाता है, तो यह औसत 21 वर्ष हो जाती है. बताएं कि A, B तथा C की आयु क्या है?]  [Ans. (a) 22, 18, 20  ]

(a) 22, 18, 20   (b) 18, 22, 20   (c) 22, 20, 18   (d) 18, 20, 22   (e) none of these

Detailed Explanation:

First Method: Given that the average of A and B is 20 years. If C were to replace A, the average would be 19; that means the age of C is 2 less than A. Also, if C were to replace B, the average would be 21; that means the age of C is 2 more than A.

So, C = A – 2 and C = B + 2

That is, A = C + 2 and B = C – 2

Since the average of A and B is 20 years, so A + B = 2 × 20 = 40

C + 2 + C – 2 = 40, which implies that 2C = 40, or, C = 20

A = 20 + 2 = 22 and, B = 20 – 2 = 18

Therefore, the ages of A, B and C are respectively 22 years, 18 years and 20 years. Ans. 

Second Method: Given that the average of A and B is 20 years; C and B is 19 years and A and C is 21 years. So,

A + B = 2 × 20   ……… (1)

C + B = 2 × 19   ……… (2)

A + C = 2 × 21   ……… (3)

2 × (A + B + C) = 2 × (20 + 19 + 21)   [Adding (1), (2) and (3), we get]

(A + B + C) = 60    ………. (4)

A = 22     [From equations (2) and (4)]

B = 18      [From equations (3) and (4)]

And, C = 20    [From equations (1) and (4)]

Therefore, the ages of A, B and C are respectively 22 years, 18 years and 20 years. Ans.