Understanding the Question on Age Difference

The question asks for the difference between the ages of two people, P and Q. We are given some important initial facts about their ages:

• The age of P and the age of Q are both two-digit numbers. This means they are between 10 and 99 years old.
• If you interchange the digits of P's age, you get Q's age.

Let's represent the age of P using digits. Suppose the tens digit of P's age is 'a' and the units digit is 'b'.

So, P's age can be written as \(10a + b\).

Since P's age is a two-digit number, \(10 \le 10a + b \le 99\). This implies that 'a' must be a digit from 1 to 9 (\(a \in \{1, 2, \ldots, 9\}\)) and 'b' must be a digit from 0 to 9 (\(b \in \{0, 1, \ldots, 9\}\)).

The age of Q is obtained by interchanging the digits of P's age. So, Q's age is \(10b + a\).

Since Q's age is also a two-digit number, \(10 \le 10b + a \le 99\). This implies that 'b' must be a digit from 1 to 9 (\(b \in \{1, 2, \ldots, 9\}\)) and 'a' must be a digit from 0 to 9 (\(a \in \{0, 1, \ldots, 9\}\)).

Combining the conditions for both P and Q being two-digit ages (greater than 10 and less than 100), both the tens digit and the units digit must be between 1 and 9. So, \(a \in \{1, 2, \ldots, 9\}\) and \(b \in \{1, 2, \ldots, 9\}\).

The difference between their ages is the absolute value of the difference: \(|(10a + b) - (10b + a)| = |9a - 9b| = 9|a - b|\).

To answer the question "What is the difference of their ages?", we need to find a unique value for \(9|a - b|\). This requires finding a unique value for \(|a - b|\) or determining the specific values of 'a' and 'b'.

Analyzing Statement-I on Age Comparison

Statement-I says: The age of P is greater than the age of Q.

Mathematically, this is \(10a + b > 10b + a\).

Subtracting \(10b + a\) from both sides gives:

\(10a - a + b - 10b > 0\)

\(9a - 9b > 0\)

\(9(a - b) > 0\)

\(a - b > 0\)

\(a > b\)

So, Statement-I tells us that the tens digit of P's age ('a') is greater than the units digit of P's age ('b'). Since P's digits are swapped to get Q's age, this also means the tens digit of Q's age ('b') is less than the units digit of Q's age ('a'). Crucially, it tells us $a > b$.

With $a > b$, the difference in their ages is $9(a - b)$.

We know \(a, b \in \{1, 2, \ldots, 9\}\) and \(a > b\). Let's list some possible pairs of (a, b) and the corresponding age difference:

As you can see from the table, Statement-I narrows down the possibilities for 'a' and 'b' (specifically, \(a > b\)), but it does not give a single unique value for the difference in their ages. The difference can be 9, 18, 27, etc., depending on the values of 'a' and 'b'.

Therefore, Statement-I alone is not sufficient to answer the question about the age difference.

Analyzing Statement-II on Age Sum and Difference

Statement-II says: The sum of their ages is 11/6 times their difference.

Let the age of P be \(A_P = 10a + b\) and the age of Q be \(A_Q = 10b + a\). We know \(a, b \in \{1, 2, \ldots, 9\}\).

Sum of ages: \(A_P + A_Q = (10a + b) + (10b + a) = 11a + 11b = 11(a + b)\).

Difference of ages: \(|A_P - A_Q| = |(10a + b) - (10b + a)| = |9a - 9b| = 9|a - b|\).

According to Statement-II:

\(A_P + A_Q = \frac{11}{6} |A_P - A_Q|\)

Substitute the expressions for sum and difference:

\(11(a + b) = \frac{11}{6} \times 9|a - b|\)

\(11(a + b) = \frac{99}{6} |a - b|\)

\(11(a + b) = \frac{33}{2} |a - b|\)

Multiply both sides by 2:

\(22(a + b) = 33 |a - b|\)

Divide both sides by 11:

\(2(a + b) = 3 |a - b|\)

Since 'a' and 'b' are digits from 1 to 9, \(a+b\) is always positive. The absolute difference \(|a - b|\) is also non-negative.

We need to consider two cases for \(|a - b|\):

Case 1: \(a \ge b\)

If \(a \ge b\), then \(|a - b| = a - b\). The equation becomes:

\(2(a + b) = 3(a - b)\)

\(2a + 2b = 3a - 3b\)

Adding \(3b\) to both sides and subtracting \(2a\) from both sides:

\(2b + 3b = 3a - 2a\)

\(5b = a\)

We are looking for digits \(a, b \in \{1, 2, \ldots, 9\}\) that satisfy \(a = 5b\).

• If \(b = 1\), then \(a = 5 \times 1 = 5\). This gives the pair (a, b) = (5, 1). Here \(a=5\) and \(b=1\), so \(a \ge b\) condition is met. P's age is 51, Q's age is 15. Both are valid two-digit ages. The difference is \(51 - 15 = 36\). Let's check the original statement: Sum = 66, Difference = 36. Is \(66 = \frac{11}{6} \times 36\)? Yes, \(66 = 11 \times 6\), which is true.
• If \(b = 2\), then \(a = 5 \times 2 = 10\). This is not a single digit between 1 and 9. So this pair is not possible.
• Any larger value of 'b' will result in an 'a' that is not a single digit.

So, under the condition \(a \ge b\), the only possible pair of digits is (5, 1), resulting in ages 51 and 15, with a difference of 36.

Case 2: \(a < b\)

If \(a < b\), then \(|a - b| = -(a - b) = b - a\). The equation becomes:

\(2(a + b) = 3(b - a)\)

\(2a + 2b = 3b - 3a\)

Adding \(3a\) to both sides and subtracting \(2b\) from both sides:

\(2a + 3a = 3b - 2b\)

\(5a = b\)

We are looking for digits \(a, b \in \{1, 2, \ldots, 9\}\) that satisfy \(b = 5a\).

• If \(a = 1\), then \(b = 5 \times 1 = 5\). This gives the pair (a, b) = (1, 5). Here \(a=1\) and \(b=5\), so \(a < b\) condition is met. P's age is 15, Q's age is 51. Both are valid two-digit ages. The difference is \(|15 - 51| = |-36| = 36\). Let's check the original statement: Sum = 66, Difference = 36. Is \(66 = \frac{11}{6} \times 36\)? Yes, \(66 = 11 \times 6\), which is true.
• If \(a = 2\), then \(b = 5 \times 2 = 10\). This is not a single digit between 1 and 9. So this pair is not possible.
• Any larger value of 'a' will result in a 'b' that is not a single digit.

So, under the condition \(a < b\), the only possible pair of digits is (1, 5), resulting in ages 15 and 51, with a difference of 36.

In both possible cases derived from Statement-II (\(a \ge b\) leading to (5,1) and \(a < b\) leading to (1,5)), the calculated difference in ages is uniquely 36.

Therefore, Statement-II alone is sufficient to answer the question about the age difference.

Evaluating Data Sufficiency Options

Based on our analysis:

• Statement-I alone is not sufficient because it leads to multiple possible age differences (9, 18, 27, etc.).
• Statement-II alone is sufficient because it uniquely determines the age difference to be 36.

Now let's look at the given options:

1. The Question cannot be answered even by using both the Statements together.

   This is incorrect, as Statement-II alone is sufficient.

2. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

   This matches our findings. Statement-II alone can answer the question, but Statement-I alone cannot.

3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

   This is incorrect, as Statement-II alone is sufficient.

4. The Question can be answered by using either Statement alone.

   This is incorrect, as Statement-I alone is not sufficient.

Thus, option (b) correctly describes the situation.

Conclusion on Age Difference Sufficiency

Statement-I provides a condition relating the digits (\(a > b\)) but does not narrow down the possible ages or their difference to a single value. Statement-II provides a relationship between the sum and difference of the ages, which, when translated into an equation involving the digits, leads to only one possible value for the difference, regardless of which age is greater.

Therefore, Statement-II alone is sufficient to determine the difference in ages, while Statement-I alone is not sufficient.

Revision Table: Analyzing Statement Sufficiency

Additional Information on Data Sufficiency Questions

Data sufficiency questions test your ability to determine if the given statements provide enough information to answer the main question. You don't necessarily have to solve the problem completely, just decide if the information is sufficient.

Typical approach involves:

• Understanding the core question being asked.
• Analyzing Statement-I alone: Assume Statement-I is true and see if you can uniquely answer the question. Ignore Statement-II for now.
• Analyzing Statement-II alone: Assume Statement-II is true and see if you can uniquely answer the question. Ignore Statement-I for now.
• Analyzing both Statements together: If neither statement alone is sufficient, assume both statements are true simultaneously and see if you can uniquely answer the question.

The options in data sufficiency questions usually revolve around which statement(s) provide enough information.

In this specific problem, the age digits being interchanged is a key constraint that sets up the variables and relationships before even considering the statements.