From 2:11 to 2:18, the rotation angle of the minute hand of the clock is 42 degrees. How many minutes has the minute hand of the clock coincided with the hour hand for the first time from 4 o'clock?

From 2:11 to 2:18, the rotation angle of the minute hand of the clock is 42 degrees. How many minutes has the minute hand of the clock coincided with the hour hand for the first time from 4 o'clock?

Minute hand turns 360 / 60 = 6 ° per minute
360 / 12 = 30 ° per hour
30 / 60 = 0.5 ° per minute
4 o'clock the needle points to 12 o'clock
The angle between the hour hand and the branch is 30 ° * 4 = 120 °
The minute hand moves 6-0.5 ° more per minute = 5.5 °
120 / 5.5 = 21.818 minutes = 21 minutes 49.091 seconds

The clock starts from 12 o'clock. How long does it take for the first minute hand to coincide with the hour hand
Next time the minute hand coincides with the hour hand, it must be at 1:00
So let's set 1 point X minutes, and the minute hand coincides with the hour hand
(1-1/12)x=5
11/12x=5
x=60/11
So the next time the minute hand coincides with the hour hand is 1:60 / 11
After 60 + 60 / 11 = 65 and 5 / 11 minutes, the first minute hand coincides with the hour hand
Who can help me to analyze the solution of this formula

My method
X-60=X/12
X = 65 5 / 11
In an hour after 5 and 5 / 11
My train of thought: in the same time, the minute hand walked a full circle more than the hour hand, it would coincide
Hour hand: 5 / 60 grids per minute, that is 1 / 12 grids
Minute hand: one block per minute
One hour minus the square of minute hand, that is, 60 square is equal to the square of hour hand
The final number x is the number of squares and minutes of the minute hand or pointer
Note: I said a few grid, understand as a few minutes, the smallest grid on the dial

How many minutes has it taken for the minute hand of the clock to coincide with the hour hand for the first time
The best equation

The problem of distance on clock face
Minute hand, rotation per minute: 360 △ 60 = 6 degrees
Clockwise, turn every minute: 360 △ 12 △ 60 = 0.5 degrees
4 o'clock, minute hand behind hour hand: 4 / 12 × 360 = 120 degrees
For the first coincidence of the two needles, it takes 120 ÷ (6-0.5) = 240 / 11 minutes