A certain clock marks every hour by striking a number of times equal to the hour, and the time required...

1. Translate the problem requirements: Understand that when a clock strikes \(\mathrm{N}\) times, there are \(\mathrm{N}\) strokes and \((\mathrm{N}-1)\) intervals between strokes, plus the duration of each stroke equals the duration of each interval
2. Visualize the striking pattern: Map out exactly what happens during the striking sequence to identify all time components
3. Set up the timing relationship: Use the 6:00 scenario to find the duration of individual strokes and intervals
4. Apply the pattern to 12:00: Use the same timing relationship to calculate the total duration for 12 strikes

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what happens when the clock strikes. When it's 6:00, the clock strikes 6 times. But here's the key insight we need to grasp: the problem tells us that "the time required for a stroke is exactly equal to the time interval between strokes."

What does this mean in everyday terms? Imagine the clock going "BONG... BONG... BONG... BONG... BONG... BONG" at 6:00. Each "BONG" takes some amount of time (let's call this \(\mathrm{t}\) seconds), and each pause between the BONGs also takes the same amount of time \(\mathrm{t}\) seconds.

So when the clock strikes 6 times:

• We have 6 strokes (each taking \(\mathrm{t}\) seconds)
• We have 5 intervals between the strokes (each taking \(\mathrm{t}\) seconds)
• Total time = \(6\mathrm{t} + 5\mathrm{t} = 11\mathrm{t}\) seconds

The problem states this total time is 22 seconds at 6:00.
Process Skill: TRANSLATE - Converting the problem's description into mathematical understanding

2. Visualize the striking pattern

Let's draw out exactly what happens at 6:00 to make this crystal clear:

Time →
Stroke 1: [\(\mathrm{t}\) seconds] → Gap → Stroke 2: [\(\mathrm{t}\) seconds] → Gap → Stroke 3: [\(\mathrm{t}\) seconds] → Gap → Stroke 4: [\(\mathrm{t}\) seconds] → Gap → Stroke 5: [\(\mathrm{t}\) seconds] → Gap → Stroke 6: [\(\mathrm{t}\) seconds]

Counting the components:

• Number of strokes = 6 (each lasting \(\mathrm{t}\) seconds)
• Number of gaps = 5 (each lasting \(\mathrm{t}\) seconds)
• Total time = \(6\mathrm{t} + 5\mathrm{t} = 11\mathrm{t}\) seconds

This visualization helps us see the pattern: for \(\mathrm{N}\) strikes, we get \(\mathrm{N}\) strokes plus \((\mathrm{N}-1)\) intervals.
Process Skill: VISUALIZE - Mapping out the sequence to identify all time components

3. Set up the timing relationship

Now we can use the information from 6:00 to find the value of \(\mathrm{t}\) (the duration of each stroke and each interval).

We know that at 6:00:

• Total time = \(11\mathrm{t} = 22\) seconds
• Therefore: \(\mathrm{t} = 22 ÷ 11 = 2\) seconds

So each individual stroke takes 2 seconds, and each interval between strokes also takes 2 seconds.

4. Apply the pattern to 12:00

Now let's apply this same pattern to 12:00, when the clock strikes 12 times.

Using our pattern from step 2:

• Number of strokes = 12 (each lasting 2 seconds)
• Number of intervals = 11 (each lasting 2 seconds)
• Total time = \((12 × 2) + (11 × 2) = 24 + 22 = 46\) seconds

Let's double-check this makes sense: We have 23 total "events" (12 strokes + 11 intervals), each taking 2 seconds, giving us \(23 × 2 = 46\) seconds.

4. Final Answer

At 12:00, the time that elapses between the beginning of the first stroke and the end of the last stroke is 46 seconds.

This matches answer choice (D) 46.

Common Faltering Points

Errors while devising the approach

Students often fail to recognize that BOTH the strokes and the intervals between strokes take time. They may think only the actual striking sounds count, forgetting that the problem states "the time required for a stroke is exactly equal to the time interval between strokes." This leads them to set up equations considering only the stroke times or only the interval times.

A critical error is miscounting the number of intervals. For 6 strokes, students might think there are 6 intervals instead of 5. The pattern is: for \(\mathrm{N}\) strokes, there are \((\mathrm{N}-1)\) intervals between them. Missing this relationship leads to completely wrong equations like \(12\mathrm{t} = 22\) instead of \(11\mathrm{t} = 22\).

Some students may think the 22 seconds refers to just the time for the strokes themselves, or just the intervals, rather than the total time from the beginning of the first stroke to the end of the last stroke. This misinterpretation leads to setting up the wrong equation from the start.

Errors while executing the approach

even with the correct setup of \(11\mathrm{t} = 22\), students may make simple division errors and calculate \(\mathrm{t}\) incorrectly (such as getting \(\mathrm{t} = 3\) or \(\mathrm{t} = 1\) instead of \(\mathrm{t} = 2\)). This error propagates through the entire solution.

Students may correctly find \(\mathrm{t} = 2\) but then make errors when applying it to 12:00. Common mistakes include using 12 intervals instead of 11, or forgetting to multiply both components by 2, leading to calculations like \(12 + 11 = 23\) instead of \((12 × 2) + (11 × 2) = 46\).

Errors while selecting the answer

Students might arrive at intermediate values during their calculation (such as 24 for just the stroke time, or 22 for just the interval time) and mistakenly select these as their final answer instead of the total time of 46 seconds. This often happens when students lose track of what the question is actually asking for.