In a standard analog clock, when the hour hand and the minute hand are at right angles to each other (forming a 90 degree angle), how many minute spaces apart are the two hands on the dial?

Introduction / Context:
This problem tests a basic but important concept about the division of a clock face into minute spaces and how these correspond to angles. When the hour and minute hands are at right angles to each other, they form a 90 degree angle. You are asked to express this separation as a number of minute spaces on the dial. Understanding how degrees relate to minute spaces simplifies many clock problems involving angles.

Given Data / Assumptions:

• A clock face is a full circle of 360 degrees.

• The dial is divided into 60 equal minute spaces.

• Each minute space corresponds to an equal angle at the center of the clock.

• A right angle is 90 degrees.

Concept / Approach:
Since the entire 360 degree circle is divided into 60 minute spaces, each minute space represents 360 / 60 degrees. Knowing the degree measure of a right angle, we simply divide that angle by the degrees per minute space to find the number of minute spaces that correspond to a 90 degree separation between the hands.

Step-by-Step Solution:
Step 1: Total angle in a full circle of the clock = 360 degrees. Step 2: Total number of minute spaces on the dial = 60. Step 3: Angle per minute space = 360 / 60 = 6 degrees. Step 4: For a right angle, we need an angular separation of 90 degrees between the hands. Step 5: Number of minute spaces for a right angle = 90 / 6. Step 6: 90 / 6 = 15 minute spaces. Step 7: Therefore, when the hands are at right angles, they are 15 minute spaces apart on the clock dial.

Verification / Alternative check:
You can also reason that half a circle (180 degrees) corresponds to 30 minute spaces (as each space is 6 degrees), so a right angle, which is half of 180 degrees, would correspond to half of 30 minute spaces. Half of 30 is 15, confirming that 15 minute spaces represent a right angle between the hands.

Why Other Options Are Wrong:
• 13 minute spaces: 13 * 6 = 78 degrees, which is less than 90 degrees, so the hands are not at a right angle.
• 14 minute spaces: 14 * 6 = 84 degrees, still an acute angle and short of 90 degrees.
• 16 minute spaces: 16 * 6 = 96 degrees, which is slightly greater than 90 degrees, so the hands are more than a right angle apart.

Common Pitfalls:
A typical mistake is to mix up hour divisions (12 large divisions) with minute spaces (60 divisions), leading to incorrect angle calculations. Some students also forget that each minute space is 6 degrees, not 5 or any other number. Remembering the simple relation that 360 degrees divided by 60 spaces equals 6 degrees per space helps quickly handle such problems.

Final Answer:
When the two hands of a clock are at right angles, they are separated by 15 minute spaces on the dial.