Question

The minute hand of a clock overtakes the hour hand at intervals of 64 minutes of correct time. How much a day does the clock gain or lose?

Answer 1

First calculate the interval of time it should take the minute hand to overtake the hour hand.

Let
x = distance traveled by hour hand
1 + x = distance traveled by minute hand

Time = Distance/Rate
x/(1/12) = (1 + x)/1
x = (1/12)(1 + x) = 1/12 + x/12
(11/12)x = 1/12
x = 1/11

Calculate time for minute hand to travel a distance 1 + x.

Time = Distance/Rate
(1 + x)/1 = 1 + x = 1 + 1/11 = 12/11 hour

Actual time to overtake on clock is 64 minutes.

64 min = 64/60 hours = 16/15 hour

Correct Clock Speed/Actual Clock Speed
(12/11) / (16/15) = 12*15 / (11*16) = 180/176 = 45/44

The clock gains:
45/44 - 1 = 1/44 day = 24/44 hour = 6/11 hour = 360/11 min

Answer 2

Step-by-step explanation:

65 5/11 -64*(24*60min/64)

=720/11-64=16/11

=16/11*(24*60/64)

=1440/44

by changing into mixed fraction

=32 32/44

the clock gains 32 32/44