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. Find out the value of the eleventh number in a set of eleven numbers. I. The average of the first ten numbers in the set is $x$. II. The average of all the eleven numbers is $y$.
Aptitude
Average
Difficulty: Easy
Choose an option
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AIf the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question
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BIf the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question
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CIf the data either in Statement I or in Statement II alone are sufficient to answer the question
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DIf the data even in both Statements I and II together are not sufficient to answer the question
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EIf the data in both Statements I and II together are necessary to answer the question
Answer
Correct Answer: If the data in both Statements I and II together are necessary to answer the question
Explanation
### Concept & Logic
To find the value of a specific term added to a sequence, you subtract the sum of the initial terms from the sum of all terms combined.
$$N_{11} = (Sum \ of \ 11 \ terms) - (Sum \ of \ 10 \ terms)$$
### Step-by-Step Solution
**Target:** Find the exact value or definitive expression for the 11th number.
**Evaluating Statement I:**
* The average of the first 10 numbers is $x$.
* Sum of the first 10 numbers = $10x$.
* We have no information about the 11th number or the total set.
* Statement I alone is not sufficient.
**Evaluating Statement II:**
* The average of all 11 numbers is $y$.
* Sum of all 11 numbers = $11y$.
* We have no information to separate the first 10 from the 11th.
* Statement II alone is not sufficient.
**Evaluating Statements I and II Together:**
* We know the sum of all 11 numbers is $11y$.
* We know the sum of the first 10 numbers is $10x$.
* The 11th number is simply the difference between these two sums:
$$11th \ number = 11y - 10x$$
* Since $x$ and $y$ are given parameters representing known values in the context of the statements, this deterministic algebraic expression uniquely defines the 11th number. Both statements are required.
### Exam Strategy & Shortcut
Don't panic if a Data Sufficiency question yields an algebraic expression instead of a flat number (like 42). If variables are presented as the "given" data in the statements (e.g., "the average is $x$"), expressing the final answer purely in terms of those given variables constitutes a sufficient answer.
### Common Pitfall
A common trap is marking "not sufficient" because the final answer is $11y - 10x$ rather than a constant numerical integer. In data sufficiency, if you can define the exact relationship based entirely on the provided constants/variables, you have solved it.
### Final Answer
**Therefore, the correct answer is If the data in both Statements I and II together are necessary to answer the question.**