Arnav's Present Age: Data Sufficiency Analysis

This question requires us to determine Arnav's present age by evaluating the sufficiency of three provided statements. We need to identify which combination of these statements is essential to uniquely calculate Arnav's current age.

Defining Variables for Age Calculations

To solve this age problem, let's assign variables to the present ages:

• Let Arnav's present age be $\text{A}$ years.
• Let Arnav's son's present age be $\text{S}$ years.

Converting Statements into Algebraic Equations

Each statement provides a specific relationship between Arnav's age and his son's age, which we can translate into an algebraic equation:

Statement I: Five years ago, Arnav's age was double that of his son's age at that time.

• Arnav's age five years ago was $\text{A} - 5$.
• Arnav's son's age five years ago was $\text{S} - 5$.
• From Statement I, the equation is:

  $\text{A} - 5 = 2(\text{S} - 5)$

  $\text{A} - 5 = 2\text{S} - 10$

  $\text{A} = 2\text{S} - 10 + 5$

  $\text{A} = 2\text{S} - 5$ (Equation 1)

Statement II: Present ages of Arnav and his son are in the ratio of 11 ∶ 6.

• From Statement II, the ratio of their present ages gives us:

  $\frac{\text{A}}{\text{S}} = \frac{11}{6}$

  $6\text{A} = 11\text{S}$ (Equation 2)

Statement III: Five years hence, the ratio of Arnav's age and his son's age will become 12 ∶ 7.

• Arnav's age five years from now will be $\text{A} + 5$.
• Arnav's son's age five years from now will be $\text{S} + 5$.
• From Statement III, the equation is:

  $\frac{\text{A} + 5}{\text{S} + 5} = \frac{12}{7}$

  $7(\text{A} + 5) = 12(\text{S} + 5)$

  $7\text{A} + 35 = 12\text{S} + 60$

  $7\text{A} - 12\text{S} = 60 - 35$

  $7\text{A} - 12\text{S} = 25$ (Equation 3)

Checking Sufficiency of Statement Combinations

To find the unique value of Arnav's present age ($\text{A}$), we need at least two independent linear equations involving $\text{A}$ and $\text{S}$. Let's examine each possible pair of statements:

Sufficiency with Statements I and II

• We have:

  1. $\text{A} = 2\text{S} - 5$ (from Statement I)

  2. $6\text{A} = 11\text{S}$ (from Statement II)

• Substitute the expression for $\text{A}$ from Equation 1 into Equation 2:

  $6(2\text{S} - 5) = 11\text{S}$

  $12\text{S} - 30 = 11\text{S}$

  $12\text{S} - 11\text{S} = 30$

  $\text{S} = 30$

• Now, substitute the value of $\text{S}$ back into Equation 1 to find $\text{A}$:

  $\text{A} = 2(30) - 5$

  $\text{A} = 60 - 5$

  $\text{A} = 55$

• Conclusion: Using Statements I and II, we can determine Arnav's present age as 55 years. Therefore, these two statements are sufficient.

Sufficiency with Statements I and III

• We have:

  1. $\text{A} = 2\text{S} - 5$ (from Statement I)

  2. $7\text{A} - 12\text{S} = 25$ (from Statement III)

• Substitute the expression for $\text{A}$ from Equation 1 into Equation 3:

  $7(2\text{S} - 5) - 12\text{S} = 25$

  $14\text{S} - 35 - 12\text{S} = 25$

  $2\text{S} - 35 = 25$

  $2\text{S} = 25 + 35$

  $2\text{S} = 60$

  $\text{S} = 30$

• Now, substitute the value of $\text{S}$ back into Equation 1 to find $\text{A}$:

  $\text{A} = 2(30) - 5$

  $\text{A} = 60 - 5$

  $\text{A} = 55$

• Conclusion: Using Statements I and III, we can determine Arnav's present age as 55 years. Therefore, these two statements are sufficient.

Sufficiency with Statements II and III

• We have:

  1. $6\text{A} = 11\text{S} \implies \text{S} = \frac{6\text{A}}{11}$ (from Statement II)

  2. $7\text{A} - 12\text{S} = 25$ (from Statement III)

• Substitute the expression for $\text{S}$ from Equation 2 into Equation 3:

  $7\text{A} - 12\left(\frac{6\text{A}}{11}\right) = 25$

  $7\text{A} - \frac{72\text{A}}{11} = 25$

• To eliminate the fraction, multiply the entire equation by 11:

  $11(7\text{A}) - 11\left(\frac{72\text{A}}{11}\right) = 11(25)$

  $77\text{A} - 72\text{A} = 275$

  $5\text{A} = 275$

  $\text{A} = \frac{275}{5}$

  $\text{A} = 55$

• Conclusion: Using Statements II and III, we can determine Arnav's present age as 55 years. Therefore, these two statements are sufficient.

Final Decision on Sufficiency

Since any combination of two statements (I & II, I & III, or II & III) is sufficient to find Arnav's present age, the answer to the question is that any two of the three statements are necessary.