Runnning at the same constant rate, $6$ identical machines can produce a total of $180$ bottles per hour. How many bottles could $15$ such machines produce in $30$ minutes?

### Concept & Work Equivalence Formula Production output ($W$) is proportional to the number of machines ($M$) and the operating time ($T$). $$ \frac{M_1 \times T_1}{W_1} = \frac{M_2 \times T_2}{W_2} $$ ### Step-by-Step Solution * **Given**: - $M_1 = 6$ machines, $T_1 = 60$ minutes ($1$ hour), $W_1 = 180$ bottles. - $M_2 = 15$ machines, $T_2 = 30$ minutes. * **Calculation**: Plug values into the formula to find $W_2$: $$ \frac{6 \times 60}{180} = \frac{15 \times 30}{W_2} $$ Simplify the left side: $$ \frac{360}{180} = \frac{450}{W_2} $$ $$ 2 = \frac{450}{W_2} $$ $$ W_2 = \frac{450}{2} = 225 $$ ### Exam Strategy & Shortcut First, find the rate of one machine. $6$ machines make $180$ bottles in an hour, so $1$ machine makes $30$ bottles in an hour, which means $15$ bottles in $30$ minutes. For $15$ machines, the production in $30$ minutes is $15 \times 15 = 225$ bottles. ### Common Pitfall Failing to unify the time units (mixing hours and minutes). Always convert "per hour" and "in 30 minutes" to the same unit (e.g., minutes) before calculating. ### Final Answer Therefore, the correct answer is **225**.