We are given the average age of three people A, B, and C is 30. Let their integer ages be $x, y,$ and $z$ respectively, with the condition $x \le y \le z$.

Average Age Calculation

The formula for the average age is:

Multiplying both sides by 3, we get the sum of their ages:

Age Relationship

We are told that B's age ($y$) is 5 more than A's age ($x$):

Substituting Age Relationship

Substitute $y = x + 5$ into the sum of ages equation:

Simplify the equation:

We can express $z$ in terms of $x$:

Applying Constraints

The ages must satisfy the condition $x \le y \le z$.

• The condition $x \le y$ becomes $x \le x + 5$, which is always true for any $x$.
• The condition $y \le z$ becomes $x + 5 \le z$.

Substitute $z = 85 - 2x$ into the inequality $x + 5 \le z$:

Add $2x$ to both sides:

Subtract 5 from both sides:

Divide by 3:

Since $x$ must be an integer, the maximum possible value for $x$ is 26.

Finding Minimum Value of z

To find the minimum possible value of $z$, we need to use the maximum possible value of $x$, because $z = 85 - 2x$. A larger $x$ results in a smaller $z$.

Using the maximum integer value for $x$, which is $x = 26$:

Verification

Let's verify if these ages satisfy all conditions:

• If $x = 26$, then $y = x + 5 = 26 + 5 = 31$.
• We found $z = 33$.
• The ages are $x=26, y=31, z=33$.
• Are they integers? Yes.
• Is $x \le y \le z$? $26 \le 31 \le 33$. Yes.
• Is the average age 30? $\frac{26 + 31 + 33}{3} = \frac{90}{3} = 30$. Yes.

All conditions are met. Therefore, the minimum possible value of $z$ is 33.