Question

The number n is of the form 9.94l6, where l denotes the thousandths digit. What digit is l?
(1) If n were rounded to the nearest hundredth, the result would be 9.95.
(2) If n were rounded to the nearest thousandth, the result would be 9.946.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

Be precise about rounding rules. When rounding a certain decimal place, you only look at the single digit immediately to its right. Here, for thousandths, we look at ten-thousandths. For hundredths, we look at thousandths.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the value of the digit `l` in the number `n = 9.94l6`. We need to use the rules of rounding decimals to determine the value of `l`.
- To round to the nearest hundredth, we look at the thousandths digit (`l`). If `l` is 5 or greater, we round the hundredths digit up. Otherwise, we keep it as it is.
- To round to the nearest thousandth, we look at the ten-thousandths digit (6). If it is 5 or greater, we round the thousandths digit (`l`) up. Otherwise, we keep it as it is.
Step 2: Detailed Explanation:
Analyzing Statement (1):
The number `n = 9.94l6` is rounded to the nearest hundredth, and the result is 9.95.
The hundredths digit in `n` is 4. For it to be rounded up to 5, the next digit (the thousandths digit, `l`) must be 5, 6, 7, 8, or 9.
\[ l \in \{5, 6, 7, 8, 9\} \]
Since `l` can be any of these five digits, this statement does not give a unique value for `l`. Therefore, statement (1) alone is not sufficient.
Analyzing Statement (2):
The number `n = 9.94l6` is rounded to the nearest thousandth, and the result is 9.946.
To round to the nearest thousandth, we look at the ten-thousandths digit, which is 6.
Since 6 is greater than or equal to 5, we must round the thousandths digit, `l`, up.
This means that after rounding `l` up by 1, the result is 6.
So, \(l + 1 = 6\).
This implies that \(l = 5\).
Let's verify this. If \(l=5\), the number is 9.9456. Rounding this to the nearest thousandth gives 9.946. This matches the statement.
If \(l=6\), the number is 9.9466. Rounding this would give 9.947, which does not match.
Therefore, the only possible value for `l` is 5. This statement is sufficient.
Step 3: Final Answer:
Statement (2) alone is sufficient to find a unique value for `l`, while statement (1) is not. Therefore, the correct option is (B).