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 average of three quotations for a particular item is ₹ 120. Is the highest quotation less than or equal to ₹ 139? I. The lowest quotation is of ₹ 90. II. One of the quotations is ₹ 125.
Aptitude
Average
Difficulty: Hard
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 & Strategy
In Data Sufficiency, a "Yes/No" question is considered answered if the data provides a definitive "Yes" OR a definitive "No". If the data allows for both possibilities, it is insufficient. To test extremes (highest/lowest limits), we use the fixed total sum. Let the quotations be $x \le y \le z$.
$$Total = 120 \times 3 = 360$$
### Step-by-Step Solution
**Given:**
* Total sum of quotations = $120 \times 3 = 360$.
* Question asks: Is the maximum value ($z$) $\le 139$?
**Evaluating Statement I:**
* Lowest quotation ($x$) = 90.
* Remaining sum for the other two: $y + z = 360 - 90 = 270$.
* We know $y \ge 90$ (since 90 is the lowest). To find the maximum possible $z$, we minimize $y$.
* If $y = 90$, then $z = 180$. (Is $180 \le 139$? **No**)
* To find the minimum possible $z$, we set $y = z$. $2z = 270 \Rightarrow z = 135$. (Is $135 \le 139$? **Yes**)
* Since $z$ could be 180 or 135, we do not have a definitive answer. Statement I alone is not sufficient.
**Evaluating Statement II:**
* One quotation is 125.
* Remaining sum = $360 - 125 = 235$.
* We don't know if 125 is the lowest, middle, or highest. The highest could easily be over 139 (e.g., quotations are 10, 125, 225) or under 139 (e.g., quotations are 117, 118, 125).
* Statement II alone is not sufficient.
**Evaluating Statements I and II Together:**
* Lowest is 90 ($x = 90$).
* One is 125. Since it's higher than 90, let's assign it to $y$ ($y = 125$).
* The remaining quotation (highest, $z$) is calculated as:
$$z = 360 - (90 + 125)$$
$$z = 360 - 215 = 145$$
* The highest quotation is exactly 145.
* Now answer the prompt's question: Is $145 \le 139$? The answer is a definitive **No**.
* Because we have a definitive, unshakeable answer to the question, the data is sufficient. Both statements are needed.
### Exam Strategy & Shortcut
A definitive "No" is a perfectly sufficient answer! Many students calculate that the highest is 145, see that the prompt asked if it was $\le 139$, get confused by the mismatch, and incorrectly mark "not sufficient." Remember, sufficiency just means you have enough data to close the case, regardless of whether the verdict is true or false.
### Common Pitfall
Failing to test the boundaries in Statement I. Students might arbitrarily pick $y = 135, z = 135$ and incorrectly assume "Yes" is the only outcome, thus mistakenly thinking Statement I is sufficient alone. Always test the extreme edge cases (minimizing $y$ to maximize $z$).
### Final Answer
**Therefore, the correct answer is If the data in both Statements I and II together are necessary to answer the question.**