Question

The average ages in years) of persons in groups A, B and C are 37.23 and 41 years, respectively. If the average age of
persons in groups A and B combined is 29. that of B and C is 33 and that of A and C is 39.5, then the average age of all
the persons combined in these three groups is:​

Answer 1

The average age of all the persons combined in these three groups is:​ 34

Step-by-step explanation:

Let say number of People in group A , B & C  are  A , B & C respectively

Total age of group A = 37A

Total age of Group B = 23B

Total age of Group C  = 41C

A + B Group total age  = 29A + 29B

B + C group total age  =  33B + 33C

A + C Group Total age  = 39.5A  + 39.5C

Adding these  

2( A + B + C Group total age)  = 29A + 29B + 33B + 33C + 39.5A  + 39.5C

=> 74A + 46B + 82C = 68.5A  + 62B  + 72.5C

=> 5.5A + 9.5C  = 16B

Total age of Group B =  A + B + C Group total age - A + C Group Total age

23B  = 37A + 23B + 41C - 39.5A  - 39.5C

=> 2.5A = 1.5C

=> 0.5A  = 0.3C

=> A = 0.6C

5.5A + 9.5C  = 16B

=> 5.5*0.6C + 9.5C = 16B

=> 3.3C +  9.5C = 16B

=> 12.8C = 16B

=> B = 0.8C

Total age =  37A + 23B + 41C  = 37(0.6C) + 23(0.8C) + 41C

= 22.2C + 18.4C + 41C

= 81.6C

Persons A + B + C = 0.6C + 0.8C + C = 2.4C

Average = 81.6C/ 2.4C = 34

the average age of all the persons combined in these three groups is:​ 34

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Answer 2

34 years.

Step-by-step explanation:

Let there are x people in group A, y people in group B and z people in group C.

So, the total age in group A is 37x, in group B is 23y and in the group, C is 41z.

Now, the average age of  persons in groups A and B combined is 29.

So, $\frac{37x + 23y}{x + y} = 29$

⇒ 37x + 23y = 29x + 29y

⇒ 8x = 6y

⇒ 4x = 3y

⇒ $\frac{x}{y} = \frac{3}{4}$ ............ (1)

Again, the average age of  persons in groups B and C combined is 33.

So, 23y + 41z = 33(y + z)

⇒ 10y = 8z

⇒ $\frac{y}{z} = \frac{4}{5}$ .............. (2)

Hence, from equations (1) and (2) we get,

x : y : z = 3 : 4 : 5

Let, x = 3k, y = 4k and z = 5k

Hence, the average age of all the persons combined in the three group A, B, and C will be

= $\frac{37x + 23y + 41z}{x + y + z}$

= $\frac{37(3k) + 23(4k) + 41(5k)}{3k + 4k + 5k}$

= 34 years. (Answer)