How many times between 6: 00 am and 6: 00 pm, do the hands of the clock make a straight line.

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answer is C.

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Detailed Solution

Concept: Find pairs of integers that are parallel to one another and make a straight line with a small and large needle in this question using the clock's diagram. Then, utilize this knowledge to approach the question's solution.
Let's build a clock diagram to determine how many straight lines were formed between 6:00 am and 6:00 pm.
Question Image

As seen in the above diagram, a clock has two needles. The larger needle stands for the minutes, while the smaller needle stands for the hour. As we all know, there are 24 hours between 6:00 am and 6:00 pm. In the given diagram, from six in the morning to six in the evening, count how many numerals are parallel to one another. As we can see from the above diagram, the numbers 12 and 6 are parallel to one another, with the larger needle at 12 and the smaller needle at 6. When the larger needle is at 12 and the smaller needle is at 6, it creates a straight line, just as happens when the needles switch places. According to the preceding assertion, since the numbers 12 and 6 are parallel to one another, we can infer that these numbers constitute straight lines.
Since the larger needle and smaller needle will be placed at each number from six o'clock in the morning until six o'clock in the evening, the parallel rows of numbers will form two straight lines: the first when the larger needle is at the first number and the smaller needle is at the smaller number, and the second when the needles switch places. The parallel sets of numbers from 6 o'clock in the morning to 6 o'clock in the evening are therefore (1, 7), (2, 8), (3, 9), (4, 10), (5, 11) , (6, 12) , (7, 1) , (8, 2) , (9, 3) , and (10, 4) . (11, 5) Thus, each set here results in a straight line. Since there are 11 sets of integers running parallel to one another from 6:00 am to 6:00 pm, Therefore, between 6:00 am and 6:00 pm, 11 straight lines were created in total.
Hence, the correct option is 3.

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