This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — To determine how many times the hands of a clock coincide between 7 a.m. and 7 p.m., we need to understand the general behavior of clock hands. Step 1: Understand the relative speed of the clock hands. The minute hand completes a full circle (360^) in 60 minutes, so its speed is 6^ per minute. The hour hand completes a full circle (360^) in 12 hours (720 minutes), so its speed is 0.5^ per minute. The relative speed at which the minute hand gains on the hour hand is 6^ - 0.5^ = 5.5^ per minute. Step 2: Calculate the time between coincidences. For the hands to coincide, the minute hand must gain 360^ on the hour hand. The time taken for the hands to coincide is (360^)/(5.5^/minute) = (720)/(11) minutes. This is approximately 65.45 minutes. Step 3: Determine the number of coincidences in a 12-hour period. A 12-hour period consists of 12 × 60 = 720 minutes. The number of times the hands coincide in 12 hours is 720 minutes(720)/(11) minutes/coincidence = 11 times. Step 4: Identify the specific coincidences within the given time frame. The hands coincide 11 times in any 12-hour period. The only exception to a simple count of 12 coincidences (one per hour) is that the hands coincide exactly at 12:00, which counts as one coincidence for both the "11-12" hour and the "12-1" hour. This means there is no separate coincidence between 11 and 12, and between 12 and 1, other than at 12:00 itself. The period from 7 a.m. to 7 p.m. is exactly 12 hours. This period includes the 12:00 p.m. coincidence. The coincidences occur approximately at: Between 7 a.m. and 8 a.m. (approx. 7:38 a.m.) Between 8 a.m. and 9 a.m. (approx. 8:44 a.m.) Between 9 a.m. and 10 a.m. (approx. 9:49 a.m.) Between 10 a.m. and 11 a.m. (approx. 10:55 a.m.) At 12:00 p.m. (noon) Between 1 p.m. and 2 p.m. (approx. 1:05 p.m.) Between 2 p.m. and 3 p.m. (approx. 2:11 p.m.) Between 3 p.m. and 4 p.m. (approx. 3:16 p.m.) Between 4 p.m. and 5 p.m. (approx. 4:22 p.m.) Between 5 p.m. and 6 p.m. (approx. 5:27 p.m.) Between 6 p.m. and 7 p.m. (approx. 6:33 p.m.) Counting these, there are 11 coincidences. The final answer is B. 11. Two more to go! Send the next one 📸