Is this enough to find the age?

Probability Level 2

The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member.

Find how much younger is the new member?

The answer is 40.

2 solutions

Chew-Seong Cheong
May 31, 2016

Let the average four years ago and today be $a$ , the age of the old member and young member be $Y$ and $y$ respectively. Then the different in age $\Delta y = Y-y$ . Since the average age is the same today and four years ago. The total age today after the replacement is $10a$ . The total age before the replacement is $10(a+4)$ . Therefore, we have:

$\begin{aligned} 10(a+4) - Y + y & =10a \\ 10a + 40 - \Delta y & = 10a \\ \implies \Delta y & = \boxed{40} \end{aligned}$

The new member is $\boxed{40}$ years younger.

I am so stupid that I computed for 4 years from now

Department 8 - 5 years ago

Prince Loomba
May 31, 2016

Let the sum of nine member (total) =x and the age of old one=z so its average 4 yrs before=(x+z)/10. after 4 yrs let z be replaced by y. so now avg=(x+4*10+y)/10

now (x+z)/10=(x+40+y)/10 so after solving it found z=y+40. so old person is 40yrs older than young one.

You caught me. I forgot to add 4 years into the old member's current age

Let the ages $4$ years ago be this:

The 9 members who stayed: $\displaystyle \sum_{i=1}^9 x_i$

The old member: $a$

Let the ages today be this:

The 9 members who stayed: $\displaystyle \sum_{i=1}^9 (x_i+4) = \sum_{i=1}^9 x_i + \sum_{i=1}^9 4 = \sum_{i=1}^9 x_i + 36$

The old member: $a+4$

The new member: $b$

The average age of the members then and now are the same. Therefore, the sum of their ages then and now are also the same:

$\displaystyle\sum_{i=1}^9 x_i + a = \sum_{i=1}^9 x_i + 36 + b\\ a-b=36\\ (a+4) - b = 40$

Therefore, the old member is $\boxed{40}$ years older than the young member

Hung Woei Neoh - 5 years ago

But I suppose there were 3 tries so you should have been able to catch the mistake. Bad luck!

Prince Loomba - 5 years ago

I guessed the last two XD

Hung Woei Neoh - 5 years ago

@Hung Woei Neoh – Ok better luck next time

Prince Loomba - 5 years ago

i too did it the same way . by the do you know ayushrai

abhishek alva - 5 years ago

Yes I chat with him on slack

Prince Loomba - 5 years ago

by the way i am his cousin

abhishek alva - 5 years ago

@Abhishek Alva – Why dont you come on slack? And in which class are you?

Prince Loomba - 5 years ago

Is this enough to find the age?

The answer is 40.

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