NIMCET 2007 — Mathematics PYQ

To find the average age of the group, we can utilize a key characteristic property of Arithmetic Progressions (A.P.).

Step 1: Understand the given variables

Let the total number of students in the group be $n$.

• The age of the youngest student (first term, $a$) $=5$

• The age of the oldest student (last term, $l$) $=15$

Step 2: Formula for the Sum of an A.P.

The sum of all terms ($S_n$) in an Arithmetic Progression when the first and last terms are known is given by:

$$S_n=\frac{n}{2}(a+l)$$

Substituting our known values into the equation:

$$S_n=\frac{n}{2}(5+15)$$

$$S_n=\frac{n}{2}(20)=10n$$

Step 3: Calculate the Average Age

By definition, the average value is computed by dividing the total sum of the observations by the number of observations ($n$):

$$\text{Average Age}=\frac{\text{Sum of ages }(S_n)}{\text{Total number of students }(n)}$$

Substitute the expression for $S_n$ into the average formula:

$$\text{Average Age}=\frac{10n}{n}$$

The variable $n$ cancels out from both the numerator and the denominator:

$$\text{Average Age}=10$$

Shortcut Rule: For any sequence in an Arithmetic Progression, the average (arithmetic mean) of the series is always equal to the average of its first and last terms:

$$\text{Average}=\frac{a+l}{2}=\frac{5+15}{2}=10$$

Correct Answer

The correct option is (c) 10.