If $64$ people can complete a piece of work in $8$ days by working $12$ hours a day, then how many hours a day should $32$ people work to complete the same work in $15$ days?

We can use the standard work equivalence formula (Chain Rule):
$$M_1\times D_1\times H_1=M_2\times D_2\times H_2$$

Where:

* $M_1=64$ people, $D_1=8$ days, $H_1=12$ hours/day
* $M_2=32$ people, $D_2=15$ days, $H_2=$ number of hours required per day

Substitute the given parameters into the equation:

$$64\times 8\times 12=32\times 15\times H_2$$

Now, solve for $H_2$:

$$H_2=\frac{64\times 8\times 12}{32\times 15}$$

Since $\frac{64}{32}=2$:

$$H_2=\frac{2\times 8\times 12}{15}$$

$$H_2=\frac{16\times 12}{15}$$

$$H_2=\frac{192}{15}$$

$$H_2=12.8\text{hours}$$

Thus, they should work $12.8\text{hours}$ a day.