Ideal speed of minute hand = 60 minute distance per hr
Ideal speed of hour hand = 5 minute distance per hr
So, ideally collision happens when
$60 \times x = 5 \times x + 60$
i.e., the minute hand completes one full revolution extra to the hour hand.
Solving, $x = \frac{12}{11}\; hr = \frac{720}{11}$ minutes $=65\frac{5}{11}$ minutes.
Given collision time of the clock = 64 minutes.
So, clock is losing $1 \frac{5}{11}$ minutes every $64$ minutes.
So, minutes lost in a day ($60 \times 24$ minutes)
$ = 1 \frac{5}{11} \times \frac{60 \times 24}{64}
\\= \frac{16}{11} \times \frac{15 \times 24}{16}
\\= \frac{360}{11} = 32 \frac{8}{11}.$