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Pat's watch gains an extra 10 seconds every 2 hours. Kim's watch loses 5 seconds every 3 hours. If both watches are set to the correct time at 8 o'clock in the morning and run without interruption, after 72 hours, what will be the difference in time between Pat's watch and Kim's watch?
Let's understand what's happening with each watch in plain English:
Pat's watch: It runs fast, gaining 10 seconds every 2 hours. This means that when 2 real hours pass, Pat's watch shows 2 hours and 10 seconds.
Kim's watch: It runs slow, losing 5 seconds every 3 hours. This means that when 3 real hours pass, Kim's watch only shows 2 hours, 59 minutes, and 55 seconds.
What we need to find: After 72 hours of real time have passed, how far apart will these two watches be from each other?
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding of rates
For Pat's watch (gains 10 seconds every 2 hours):
In 72 hours, there are \(72 \div 2 = 36\) periods of 2 hours each.
Since the watch gains 10 seconds in each 2-hour period:
Total gain = \(36 \times 10 = 360\) seconds
For Kim's watch (loses 5 seconds every 3 hours):
In 72 hours, there are \(72 \div 3 = 24\) periods of 3 hours each.
Since the watch loses 5 seconds in each 3-hour period:
Total loss = \(24 \times 5 = 120\) seconds
Now let's think about where each watch will be after 72 hours:
Pat's watch: Shows the correct time PLUS 360 seconds (because it's running fast)
Kim's watch: Shows the correct time MINUS 120 seconds (because it's running slow)
The difference between Pat's watch and Kim's watch is:
\(360 + 120 = 480\) seconds
We add these because Pat's watch is ahead of the correct time while Kim's watch is behind the correct time, so they're moving in opposite directions from the true time.
Process Skill: INFER - Recognizing that when one watch runs fast and another runs slow, we add both drifts to find the total separation
We have 480 seconds total difference. Let's convert this to minutes:
\(480 \div 60 = 8\) minutes exactly
Since this converts to a whole number of minutes with no remaining seconds, our answer is 8 minutes.
The difference in time between Pat's watch and Kim's watch after 72 hours will be 8 minutes.
This matches answer choice (E) 8 min.
1. Misinterpreting the direction of watch errors: Students may confuse which watch gains time and which loses time. The problem states Pat's watch "gains" 10 seconds (runs fast) and Kim's watch "loses" 5 seconds (runs slow). Some students might reverse these concepts, thinking that "gaining" means the watch is behind or "losing" means the watch is ahead.
2. Misunderstanding what "difference between the watches" means: Students might think they need to find how far each watch is from the correct time, rather than understanding that they need to find the total separation between Pat's watch reading and Kim's watch reading after 72 hours.
3. Incorrectly handling the time periods: Students may struggle with the fact that Pat's watch has a 2-hour cycle while Kim's watch has a 3-hour cycle. They might try to use the same time period for both watches or attempt to find a common time period unnecessarily, making the problem more complex than needed.
1. Calculation errors when finding the number of periods: Students may make arithmetic mistakes when dividing 72 by 2 (for Pat's watch) or 72 by 3 (for Kim's watch). These basic division errors would cascade through the entire solution.
2. Incorrectly combining the watch drifts: When finding the total difference between the watches, students might subtract the drifts (\(360 - 120 = 240\) seconds) instead of adding them (\(360 + 120 = 480\) seconds). This happens because they don't properly visualize that the watches are drifting in opposite directions from the true time.
3. Unit conversion errors: Students may make mistakes when converting 480 seconds to minutes, either forgetting to divide by 60, or making computational errors during the conversion process.
1. Selecting an answer with incorrect units: After calculating 480 seconds correctly and converting to 8 minutes, students might select an answer choice that expresses the same time in different units (like choosing "6 min 40 sec" if they miscalculated, rather than recognizing their answer should be exactly 8 minutes with no seconds).
2. Second-guessing the clean result: Since 480 seconds converts to exactly 8 minutes with no remaining seconds, some students might doubt this "too clean" result and select a more complex answer choice with both minutes and seconds, thinking their solution must be wrong because it's too neat.