when the minute hand goes between 0 and 12, the hour hand goes between 9 and 10.
if we let x = the number of minutes that have elapsed from when the hour hand is on 9 and the minute hand is on 0 (same as 12), then we get the following formulas:
m = 12x/60
h = x/60 + 9
these formulas tell you the time on the clock where the hour hand the minute hand reside after x minutes.
you get the following table of values that confirm that the formulas are correct.
when 0 minutes have elapsed, the little hand is at the 9 and the big hand is at the 12 position on the clock.
when 30 minutes have elapsed, the little hand is at the 9.3 and the big hand is at 6 position on the clock.
when 60 minutes have elapsed, the little hand is at the 10 and the big hand is at the 12 potion on the clock.
the equations track the little hand (h) and the big hand (m) positions on the clock exactly as far as the accuracy of the calculator used allow.
you want to find when the distance between 12 and the hour hand is the same as the distance between 12 and the minute hand.
the distance between 12 and the minute hand can be expressed as 12 - m which becomes 12 - 12x/60.
the distance between 12 and the hour hand can be expressed as 12 - h which becomes 12 - (x/60 + 9)
12 - 12x/60 can be simplified to 720 - 12x.
12 - (x/60 + 9) can be simplified to 180 - x.
720 - 12x is the distance between 12 and the minute hand.
180 - x is the distance between 12 and the hour hand.
you want to know when these distances are the same, so you want to know when the absolute value of the expressions are equal to each other.
since it's possible for the minute hand to be on one side of 12 and the hour hand on the other, then the absolute value is required because the same distance can be expressed as plus or minus whatever it is.
you get abs(720-12x) = abs(180-x)
abs(...) means absolute value of ...
this absolute value equation can be solved by breaking these equations up into 2 pieces.
first piece is (720-12x) = (180-x)
second piece is (720-12x) = -(180-x)
solving the first piece gets you x = 49.09090909
solving the second piece gets you x = 69.23076923
when x = 49.09090909:
the minute hand is at m = 12x/60 = 9.818181818 and the hour hand is at h = x/60 + 9 = 9.818181818
since they are both at the same position on the clock, then they are both the same distance from 12.
when x = 69.23076923 is invalid because more than 60 minutes have elapsed and the hour hand is now past 10 position on the clock.
your solution is that both h and m are at 9.818181818 position on the clock when 49.09090909 minutes have elapsed.
9.818181818 position on the clock equates to 9 hours and 49 minutes and 5 seconds rounded to the nearest second.
round this to the nearest minutes and you get 9 hours and 49 minutes.
This can be expressed at 9:49 am or 9:49 pm.