Each of the questions given below consists of a statement and / or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is / are sufficient to answer the given question. Read both the statements and give answer. The total of the present ages of $A$, $B$, $C$ and $D$ is 96 years. What is $B$'s present age? I. The average age of $A$, $B$ and $D$ is 20 years. II. The average age of $C$ and $D$ is 25 years.

### Concept & Strategy To find the specific value of a single variable in a system of equations, we generally need as many independent equations as there are variables, or a specific combination of equations that perfectly isolates the target variable. The core formula for average is: $$Sum = Average \times Number \ of \ Items$$ ### Step-by-Step Solution **Given:** * The total sum of their ages: $A + B + C + D = 96$ **Evaluating Statement I:** * Average age of $A$, $B$, and $D$ is 20 years. * Therefore, sum of their ages: $A + B + D = 20 \times 3 = 60$ * Substituting this into our main equation: $60 + C = 96 \Rightarrow C = 36$ * We still do not have $B$'s age. * Statement I alone is not sufficient. **Evaluating Statement II:** * Average age of $C$ and $D$ is 25 years. * Therefore, sum of their ages: $C + D = 25 \times 2 = 50$ * Substituting this into our main equation: $A + B + 50 = 96 \Rightarrow A + B = 46$ * We have the sum of $A$ and $B$, but cannot isolate $B$. * Statement II alone is not sufficient. **Evaluating Statements I and II Together:** * From both statements we know: 1) $A + B + D = 60$ 2) $C + D = 50$ * We deduced $C = 36$ (from statement I) and $A + B = 46$ (from statement II). * Since $C + D = 50$ and $C = 36$, we find $D = 14$. * Substituting $D = 14$ into the first equation: $A + B + 14 = 60 \Rightarrow A + B = 46$. * We are stuck with $A + B = 46$. There is no further information to separate $B$ from $A$. ### Exam Strategy & Shortcut In linear equations, count your variables and independent relationships. We want $B$. The main prompt gives us the total (1 equation, 4 variables). Combining I and II gives us the sums $(A+B+D)$ and $(C+D)$. No matter how you arrange these blocks, $A$ and $B$ are always locked together as $(A+B)$. Because no statement provides $A$ independently or a ratio between $A$ and $B$, $B$ can never be isolated. You can spot this without calculating the actual sums. ### Common Pitfall A common trap is seeing multiple equations and assuming "lots of data equals sufficient." Students often calculate $C = 36$ and $D = 14$, feel a false sense of progress, and incorrectly guess that combining statements will eventually yield the answer. Always keep your eye on the specific target variable ($B$). ### Final Answer **Therefore, the correct answer is If the data even in both Statements I and II together are not sufficient to answer the question.**