mathcounts Question #1

Spike digs $\frac 83$ holes per hour.

Butch digs $\frac 74$ holes per hour.

Lucky digs $\frac 65$ holes per hour.

Together, they dig $\frac {337}{60}$ holes per hour, so they take $\frac {60}{337}$ hours per hole which means they dig 3 holes in $\frac {180}{337}$ hours.

$\frac {180}{337} \si{h} = \frac {10800}{337} \si{min} \approx \boxed{32 \si{min}}$ .

The rates of hole digging of Spike, Butch, and Lucky are $r_S = \dfrac 83$ , $r_B = \dfrac 74$ , and $r_L = \dfrac 65$ holes per hour.

The time for three of them to dig 3 holes is $t = \dfrac 3{r_S+r_B+r_L} = \dfrac 3{\frac 83+\frac 74+\frac 65} \times 60 \approx \boxed{32}$ minutes.