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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
37% (01:46) correct
63%
(01:58)
wrong
based on 86
sessions
History
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Time
Result
Not Attempted Yet
Following every 11 consecutive days of swimming practice, Danny has 3 days off. On the 183rd day from today, will Danny practice swimming or have a day off ?
(1) Exactly 13 days ago Danny was on a day off
(2) Tomorrow Danny will start his three days off
(1) Exactly 13 days ago Danny was on a day off
(2) Tomorrow Danny will start his three days off
09 Jan 2023, 00:00
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Bunuel
We can start with statement 2, as Statement 2 is relatively simpler to work with than Statement 1.
Statement 2
Tomorrow Danny will start his three days off
It is given that tomorrow is the first day of this three days off, so we can precisely find out whether Danny will be swimming on the 183rd day.
This statement is sufficient. Hence we can eliminate A, C and E.
Statement 1
Exactly 13 days ago Danny was on a day off
The statements tells us that Danny was on a off-day 13 days ago, however the information doesn't provide any further details with respect to that day. The day could be the first day of his three off days, or it could have been the second or the third day. Also, at this stage we don't know if knowing that information will change what Danny would be doing on the 183rd day from today.
To find the information out, we can consider a streak of 14 days in which Danny swims for 11 days and has 3 days off. As we know the information on the 13th day prior to today, we can start the analysis by considering creating a streak as shown below -
Attachment:
Screenshot 2023-01-09 121705.jpg [ 54.13 KiB | Viewed 1446 times ]
If 13 days ago was the first day of this off day , the 183rd day will also be the first day off.
This is because 182 is divisible by 14, so there will be \(\frac{182}{14}\) streaks between the 13 days ago and the 183rd day.
Therefore we can conclude that whatever activity Danny was doing 13 days ago, will be the same Danny will be doing 183 days from now.
We can also visualize this using. For example, if we shift one cell to the left, the 183rd is still an "off-day".
Attachment:
Screenshot 2023-01-09 122449.jpg [ 48.03 KiB | Viewed 1432 times ]
In the same manner, if we shift two cells to the left, the 183rd is still an "off-day".
Attachment:
Screenshot 2023-01-09 122927.jpg [ 52.77 KiB | Viewed 1429 times ]
Hence this statement is also sufficient.
Option D
04 Mar 2025, 02:36
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Danny follows a repeating cycle where he swims for 11 consecutive days, followed by 3 days off, creating a cycle of 14 days (11 practice days + 3 off days). Therefore, every 14 days, the schedule repeats.
\(\\
\textbf{Step 1: Analyzing Statement (1)}\\
\\
\)
\(\\
\textbf{Statement (1): "Exactly 13 days ago Danny was on a day off." } \\
\\
\)
From today, there are 196 days between the 183rd day and the day 13 days ago. Since:
\(196 \mod 14 = 0\\
\)
This means that after every 14-day cycle, the same days repeat.
\(Thus, \textbf{Statement (1) alone is sufficient}\).
\(\\
\textbf{Step 2: Analyzing Statement (2)}\\
\\
\)
\(\\
\textbf{Statement (2): "Tomorrow Danny will start his three days off." }\\
\\
\)
Given that Danny starts his break tomorrow. Now, we can calculate the 183rd day from today.
\(\\
Thus, \textbf{Statement (2) alone is sufficient}.\\
\\
\)
\(\\
\textbf{Final Conclusion}\\
\\
\)
Both statements provide sufficient information to determine that Danny will be swimming on the 183rd day. Therefore, the answer is:
\(\boxed{\text{D (Each statement alone is sufficient)}}\\
\)
\(\\
\textbf{Step 1: Analyzing Statement (1)}\\
\\
\)
\(\\
\textbf{Statement (1): "Exactly 13 days ago Danny was on a day off." } \\
\\
\)
From today, there are 196 days between the 183rd day and the day 13 days ago. Since:
\(196 \mod 14 = 0\\
\)
This means that after every 14-day cycle, the same days repeat.
\(Thus, \textbf{Statement (1) alone is sufficient}\).
\(\\
\textbf{Step 2: Analyzing Statement (2)}\\
\\
\)
\(\\
\textbf{Statement (2): "Tomorrow Danny will start his three days off." }\\
\\
\)
Given that Danny starts his break tomorrow. Now, we can calculate the 183rd day from today.
\(\\
Thus, \textbf{Statement (2) alone is sufficient}.\\
\\
\)
\(\\
\textbf{Final Conclusion}\\
\\
\)
Both statements provide sufficient information to determine that Danny will be swimming on the 183rd day. Therefore, the answer is:
\(\boxed{\text{D (Each statement alone is sufficient)}}\\
\)
