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Please solve it with explanation -

The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member.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

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    $\begingroup$ In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. $\endgroup$
    – Martin R
    Apr 5, 2019 at 13:12
  • $\begingroup$ We like to see your thoughts, not merely a statement of a problem. You have all sorts of strange tags, but not (algebra-precalculus). Does this mean you do not know algebra? Algebra is what I would use to solve it. $\endgroup$
    – GEdgar
    Apr 5, 2019 at 13:12
  • $\begingroup$ What's with the tags? I.. er.. don't see the relevance of sage, constructive math and computational maths there. It doesn't have anything to do with mathematicians either. $\endgroup$
    – Trebor
    Apr 5, 2019 at 13:13
  • $\begingroup$ It is not a contest level question. Regarding solution, the sum of ages should not change, appears that simple. Use it to set up an equation and solve. $\endgroup$
    – Narasimham
    Apr 5, 2019 at 13:17

2 Answers 2

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Notice that while there is one member being replaced, the sum of the ages of the other nine members only changes by 36. So, let us consider their ages four years ago to be x1, ..., x10, and naturally, we can order them by increasing age, making x10 either the oldest or one of the oldest, and the person who will be replaced by the younger member, whose age, four years ago, we shall call x11. Also, denote x1 + ... + x9 by S.

Now, look at what has been given to you. The average of their ages four years ago is (S + x10)/10. That equates to the average of their ages now. Write down what that is, in terms of S and x11. Now, you need to find x11 - x10, because, of course, if you and I differ by ten years now, we will differ by 10 years four years from now.

P.S. I believe, from what little I have seen of this site, that it is traditional on this site to make an attempt at the problem you have asked and present ideas you have thought about, rather than to simply ask for a solution. You should probably do that the next time you have a question.

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    $\begingroup$ Please use MathJax to format math expressions. For example, $x_{11}$ prints as $x_{11}$ $\endgroup$
    – saulspatz
    Apr 5, 2019 at 13:35
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Let's solve it using algebra. Let's use variables to represent the ages today.

Let $A$ be the age (today) of the old member leaving the committee. His age 4 years ago was $A-4$.

Let $B$ be the age (today) of the new member replacing him.

We want to know $Y = A-B$, that is, how much younger is the new member.

Let $S$ be the sum of the ages (today) of the other nine members. Then the sum of the ages of these other members 4 years ago was $S-36$.

What was the average age 4 years ago?

$$ \frac{1}{10}\big(A-4+S-36\big)=\frac{1}{10}\big(A+S-40\big) $$

What is the average age today?

$$ \frac{1}{10}\big(B+S\big) = \frac{1}{10}\big(A-Y+S\big) $$

These must be equal: $$ \frac{1}{10}\big(A+S-40\big) = \frac{1}{10}\big(A+S-Y\big) $$ So we conclude $40=Y$, which is what we wanted to know.

hints
It is best to write everything out in words like this. Do not just write a bunch of equations.

We could have solved instead using variables to represent the ages 4 years ago. But using some variables for the ages today and other variables for the ages 4 years ago would likely be confusing, so should be avoided.

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