Question

Let a, b and c be the ages of three persons P, Q and R respectively where \(a≤b≤c\) are natural numbers. If the average age of P, Q, R is 32 years and if the age of Q is exactly 6 years more than that of P, then what is the minimum possible value of c?

• A. 34 years
• B. 36 years
• C. 38 years
• D. 32 years

Answer 1

The Correct Option is A

To solve this problem, we need to find the minimum possible value of \( c \), where \( a \), \( b \), and \( c \) are the ages of persons P, Q, and R respectively. The given conditions are:

1. The average age of P, Q, and R is 32 years.
2. The age of Q (b) is exactly 6 years more than the age of P (a).
3. The ages follow \( a \le b \le c \).

Let's solve the problem step-by-step.

First, using the condition on average age:

\[ \frac{a + b + c}{3} = 32 \]

Multiplying both sides by 3 gives:

\[ a + b + c = 96 \]

Next, using the condition \( b = a + 6 \):

Substituting \( b = a + 6 \) into the sum equation, we get:

\[ a + (a + 6) + c = 96 \]

Simplifying this gives:

\[ 2a + 6 + c = 96 \]

Subtracting 6 from both sides, we obtain:

\[ 2a + c = 90 \]

Since \( a \le b = a + 6 \le c \), substitute \( b \) and \( c \) to check the minimum value:

Assume \( a \) takes the smallest integer value possible. Start from logical assumptions:

• When \( a = 1 \):
  • \( c = 90 - 2 \times 1 = 88 \) \(\rightarrow\) Does not satisfy \( b \le c \).
• When \( a = 2 \):
  • \( c = 90 - 2 \times 2 = 86 \) \(\rightarrow\) Does not satisfy \( b \le c \).
• ...and so on.
• When \( a = 28 \):
  • Then \( b = a + 6 = 34 \)
  • And \( c = 90 - 2 \times 28 = 34 \)
  • Here, all conditions \( a \le b \le c \) and \( b = a + 6 \) are satisfied.

Thus, the minimum possible value of \( c \) is 34 years.

Therefore, the correct answer is 34 years.