Understanding the Average Age Problem

This problem involves calculating the present age of a person, referred to as 'A', using information about the average ages of a group of people (P, Q, R, and S) at different times.

Calculating Past Total Age

We are given that the average age of four people (P, Q, R, S) two years ago was 30 years. To find the sum of their ages two years ago, we multiply the average age by the number of people:

Sum of ages of P, Q, R, S (two years ago) = Average age $\times$ Number of people

Sum of ages (two years ago) = $30 \times 4 = 120$ years.

Calculating Present Total Age of P, Q, R, S

Since the initial average was calculated two years ago, each of the four individuals (P, Q, R, S) is now 2 years older. Therefore, the total increase in their combined age is:

Increase in total age = Number of people $\times$ Number of years passed

Increase = $4 \times 2 = 8$ years.

Now, we can find the sum of the present ages of P, Q, R, and S:

Sum of present ages (P, Q, R, S) = Sum of ages (two years ago) + Increase in total age

Sum of present ages (P, Q, R, S) = $120 + 8 = 128$ years.

Calculating Present Total Age of P, Q, R, S, and A

When person A is included, the group size becomes 5 people. The problem states that the average present age of these five people (P, Q, R, S, A) is 35 years. The sum of their present ages is:

Sum of ages (P, Q, R, S, A, present) = Average present age $\times$ Number of people

Sum of ages (P, Q, R, S, A, present) = $35 \times 5 = 175$ years.

Determining the Present Age of A

To find the present age of person A, we subtract the sum of the present ages of P, Q, R, and S from the sum of the present ages of all five people (P, Q, R, S, A):

Present age of A = Sum of ages (P, Q, R, S, A, present) - Sum of present ages (P, Q, R, S)

Present age of A = $175 - 128 = 47$ years.

Conclusion

The present age of person A is 47 years.