Introduction / Context:
This question is a classic example of a clock angle problem from quantitative aptitude. It asks how many degrees the hour hand has rotated from noon to ten minutes past five. Understanding the angular speed of the hour hand and how long it has been moving allows you to find the exact angle. Such questions are common in competitive exams that test numerical reasoning and time and work with clocks.
Given Data / Assumptions:
- The clock is started at noon, which is 12:00.
- The time in question is 5:10, that is five hours and ten minutes after noon.
- Only the rotation of the hour hand from its starting position at 12:00 is considered.
- We assume the clock is ideal, and both hands move uniformly at their standard rates.
Concept / Approach:
The key concept is the speed of the hour hand. A full circle is 360 degrees, and the hour hand completes one full circle in twelve hours. Therefore, the hour hand moves 360 divided by 12 which equals 30 degrees per hour. Equivalently, since each hour is 60 minutes, the hour hand moves 0.5 degrees per minute. By calculating how many minutes have passed between noon and 5:10, you can multiply by 0.5 degrees per minute to find the total angle turned.
Step-by-Step Solution:
Step 1: Determine the time interval from noon to 5:10. From 12:00 to 5:00 is 5 hours, and from 5:00 to 5:10 is an additional 10 minutes.
Step 2: Convert the total time to minutes. Five hours is 5 multiplied by 60 which equals 300 minutes. Add 10 minutes to get 310 minutes in total.
Step 3: Recall that the hour hand moves 0.5 degrees each minute.
Step 4: Multiply the minutes by the angular speed: angle turned equals 310 multiplied by 0.5 degrees per minute.
Step 5: Compute 310 multiplied by 0.5 which equals 155 degrees.
Step 6: Compare this result with the answer choices 145, 150, 155, and 160 and identify 155 as the matching value.
Verification / Alternative check:
Another way to check is to break the motion into hours and minutes separately. For five full hours, the hour hand moves 5 multiplied by 30 which equals 150 degrees. In ten additional minutes, it moves 10 multiplied by 0.5 which equals 5 degrees. Adding these gives 150 plus 5 which equals 155 degrees. This alternative calculation confirms the earlier result and ensures that there is no arithmetic mistake.
Why Other Options Are Wrong:
145: This value could come from miscalculating the hours or minutes, but it does not match the correct total of 155 degrees.
150: This corresponds only to the movement in a full five hours and ignores the extra ten minutes.
160: This overestimates the angle, perhaps by incorrectly using the minute hand speed instead of the hour hand speed.
Common Pitfalls:
A common mistake is to forget that the hour hand moves continuously and not just in jumps each hour. Students sometimes calculate only the angle for whole hours and ignore the additional minutes, leading to 150 degrees instead of 155 degrees. Another error is mixing up the angular speeds of the minute and hour hands. Memorizing that the hour hand moves 0.5 degrees per minute and 30 degrees per hour can prevent these mistakes.
Final Answer:
By 10 minutes past five, the hour hand has turned through
155 degrees from its position at noon.
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