In a standard analog clock, when the hour hand and the minute hand are in exactly opposite directions (that is, they form a straight line of 180 degrees), how many minute spaces apart are the two hands on the dial?

Introduction / Context:
Questions on clocks often test your understanding of how the circular dial is divided and how angles between the hour and minute hands are related to these divisions. In this problem, you are asked to find how many 'minute spaces' separate the hands when they are in exactly opposite directions, forming a straight line (180 degrees). Knowing the relationship between minute spaces and degrees makes this type of question very quick to solve once the concept is clear.

Given Data / Assumptions:

• The clock dial is a full circle of 360 degrees.
• The dial is divided into 60 equal minute spaces around the circumference.
• When the hands are in opposite directions, the angle between them is 180 degrees.
• Each minute space corresponds to a fixed angle in degrees.

Concept / Approach:
The key idea is that the clock face is divided into 60 equal minute spaces. Since a full circle is 360 degrees, each minute space represents 360 / 60 degrees. Once we know the angle represented by one minute space, we can determine how many such spaces add up to 180 degrees, which corresponds to the hands being opposite each other in a straight line.

Step-by-Step Solution:
Step 1: A clock has 60 minute spaces over a full angle of 360 degrees. Step 2: Angle per minute space = 360 / 60 = 6 degrees. Step 3: When the hands are in opposite directions, they make an angle of 180 degrees. Step 4: Number of minute spaces corresponding to 180 degrees = 180 / 6. Step 5: 180 / 6 = 30 minute spaces. Step 6: Therefore, when the hands are opposite each other, they are 30 minute spaces apart on the dial.

Verification / Alternative check:
Another way to think about it is to note that 30 minute spaces is exactly half of the 60 spaces on the dial. Being opposite each other means the hands are separated by half the circle, which is consistent with a straight angle of 180 degrees. Since half of 60 is 30, this again confirms that the hands must be 30 minute spaces apart when they are opposite.

Why Other Options Are Wrong:
• 15 minute spaces: 15 * 6 = 90 degrees, which is a right angle, not opposite directions.
• 20 minute spaces: 20 * 6 = 120 degrees, which is an obtuse angle but not a straight line.
• 25 minute spaces: 25 * 6 = 150 degrees, still less than 180 degrees, so the hands are not exactly opposite.

Common Pitfalls:
Students sometimes confuse right angles (90 degrees) with opposite directions (180 degrees). Another common mistake is to forget that each minute space represents 6 degrees, and instead assume some other value or mix up hour divisions with minute divisions. Carefully remembering that 360 degrees divided by 60 minute marks gives 6 degrees per minute space helps avoid these errors. Also, mixing up the idea of overlap (0 degrees) and opposite directions (180 degrees) can lead to incorrect answers.

Final Answer:
When the two hands of a clock are in opposite directions, they are separated by 30 minute spaces on the dial.