Five years ago the mean age of a family was 25 years. What is the present mean age of the family?

Hint: Mean is a statistical value which is the average of the given data, or can be also defined as the ratio of the sum of the number of the given statistical values and the total number of values. Mean is simply called as the average of the collected samples in the given data values. In statistics the mean and average are the same, which is the ratio of summation of the values observations to the total number of observations.

Complete step-by-step solution:
The mean formula can be mathematically represented as:
$ \Rightarrow {x_n} = \dfrac{{\sum\limits_i {{x_i}} }}{n}$, where ${x_n}$ is the arithmetic mean, $\sum\limits_i {{x_i}} $is the sum of the observations and $n$ is the total number of observations.
$ \Rightarrow \dfrac{{\sum\limits_i {{x_i}} }}{n} = \dfrac{{{x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n}}}{n}$
So here in the problem given the mean age or the average age of the family which is 25 years. Let there are n members in the family of ages of each are ${x_1},{x_2},{x_3}, \cdot \cdot \cdot \cdot ,{x_n}$.
$ \Rightarrow {x_n} = 25$;
$\therefore \dfrac{{\sum\limits_i {{x_i}} }}{n} = 25$
$ \Rightarrow \dfrac{{{x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n}}}{n} = 25$
$ \Rightarrow {x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n} = 25n$
$ \Rightarrow \sum\limits_i {{x_i}} = 25n$
Now after 5 years, each person is older by 5 years i.e, each person’s age is added with 5 years. Now let the new mean be ${X_n}$.
$\therefore {X_n} = \dfrac{{({x_1} + 5) + ({x_2} + 5) + ({x_3} + 5) + \cdot \cdot \cdot \cdot + ({x_n} + 5)}}{n}$
$ \Rightarrow {X_n} = \dfrac{{{x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n} + (n \times 5)}}{n}$
$ \Rightarrow {X_n} = \dfrac{{{x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n} + 5n}}{n}$
But here we found that ${x_1} + {x_2} + {x_3} + \cdot \cdot \cdot \cdot + {x_n} = 25n$, substituting this in the new mean expression:
$ \Rightarrow {X_n} = \dfrac{{25n + 5n}}{n}$
$ \Rightarrow {X_n} = \dfrac{{30n}}{n}$
$ \Rightarrow {X_n} = 30$
$\therefore $The present mean age of the family is 30 years.

The present mean age of the family is 30 years.

Note: While solving for the mean using arithmetic mean formula, here it is solved carefully that already given the mean age of 5 years ago, so the no. of observations doesn’t change here only the value of the observations are altered in order to find the new mean.