The ages of the people at a gathering were forty-nine, twenty-seven, thirty-seven, forty-four, thirty-four, thirty-six, nineteen, twenty-four, twenty-three, forty, twenty, twenty-one, and forty-three. When one more person joined the gathering, the mean age became thirty-one. How old was the person who joined?
Well first, let’s just think about the word mean. What does mean mean? Well in this case, we’re talking about mean age. So the mean age, really, it means if I shared the ages out equally between everyone, how old would each person be? And that enables each person to make a sort of comparison between themselves and the rest of the group. Do I have more than my equal share of age? Or do I have less than my equal share of age? They are the sorts of questions you can start to answer by looking at the mean.
Well how do we calculate the mean age then? Well the process of sharing things out equally means that we gather in all the ages, or we add up all the ages. And to share them out between everyone, we’re just dividing them amongst that number of people. So we add up all the ages and divide by the number of people that we’ve got.
Now the next complication is that, we know the answer to that question, the mean is thirty-one, what we don’t know is the age of the person who joined the group. So let’s give that a letter; let’s call that, say, 𝑥. So let’s write that. Let 𝑥 equal the age of the person who joined. So the answer to our question is: They were 𝑥 years old! Well sadly, it’s not quite as simple as that. I think they’re probably looking for an actual number in this question.
So in order to get our marks, we’d better work out what that number is. Now remember, the calculation we’re gonna do is that, the mean age is equal to the sum of all the ages divided by how many people there were. Right. Well we’ve got some of the information we need. We know that the mean age became thirty-one, so we can fill in that box. Now we’ve gotta count up how many people there were. Well that’s one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, plus the person who joined, that’s fourteen. And now we need to add up all the ages and put that in the other box. And we didn’t quite have enough room for that, so I just extended the line so forty-nine plus twenty-seven plus thirty-seven, et cetera, plus 𝑥, is the sum of the ages. So I’ll just evaluate those numbers added together. And the sum of all the ages that we know is four hundred and seventeen. So I can rewrite that top line, and that give us, thirty-one is equal to four hundred and seventeen plus 𝑥 over fourteen. So if I multiply both sides of my equation by fourteen, fourteen times thirty-one on the left-hand side gives me four hundred and thirty-four, and multiplying the right-hand side by fourteen means I can cancel those out, which just leave me without a fraction.
So four hundred and thirty-four is equal to four hundred and seventeen plus 𝑥. Well if I now subtract four hundred and seventeen from each side of my equation, four hundred and thirty-four minus four hundred and seventeen just leaves me with seventeen, and subtracting four hundred and seventeen from the right-hand side just leaves me with 𝑥. So 𝑥 is equal to seventeen.
So our answer is: The joiner had an age of seventeen.