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Math party

Math party

The average age of people at a party coincidentally coincides with the number of people attending. At a certain time of the night a young man of 29 years arrives at the party and (oh, surprise) the average age of the party continues to coincide with the number of people attending.

How many people were initially at the party?

Solution

Let's suppose X It's the amount of people at the party at the beginning.

Since we know that it coincides with the average age, we will need to know what the average age is, which is calculated by adding the ages of all and dividing by the total number of people at the party.

Let's call S to the sum of the ages of all of the party. It is true that X = S / X. Or, put another way, S = X * X = X2.

When the 29-year-old comes to the party, we have to S increases by 29 units and X increases by one unit, but equality is maintained between these two quantities, that is, S + 29 = (X + 1)2 = X2 + 2X + 1.

Using now the two equations, we substitute the S in the second and we get that X2 + 29 = X2 + 2X + 1 and simplifying this equation we have to 28 = 2X, that is X = 14.

That is there were 14 people at the party They had an average age of 14 years (the sum of their ages must be 196). However, the arrival of the new guest, rose the
average age at 15 years, while increasing the number of participants by one (now the sum of their ages amounts to 225).