Working independently, Machine X can fill an order in 15 hours. Working independently, Machine Y can fill the same order in 10 hours. If Machine X works independently for 12 hours to fill an order and then Machine Y works independently to complete the order, how many hours does it take Machine Y to complete the order?

To solve this problem, we need to determine how much of the order Machine X completes in 12 hours and then find out how much time Machine Y needs to finish the remaining part of the order. 1. **Calculate the work rate of each machine:** - Machine X can fill the order in 15 hours, so its work rate is: \[ R_X = \frac{1}{15} \text{ orders per hour} \] - Machine Y can fill the order in 10 hours, so its work rate is: \[ R_Y = \frac{1}{10} \text{ orders per hour} \] 2. **Calculate the amount of work done by Machine X in 12 hours:** - The work done by Machine X in 12 hours is: \[ W_X = R_X \times \text{time} = \frac{1}{15} \times 12 = \frac{12}{15} = \frac{4}{5} \text{ of the order} \] 3. **Determine the remaining work to be done:** - The remaining work after Machine X has worked for 12 hours is: \[ W_{\text{remaining}} = 1 - W_X = 1 - \frac{4}{5} = \frac{1}{5} \text{ of the order} \] 4. **Calculate the time it takes for Machine Y to complete the remaining work:** - Let \( t \) be the time taken by Machine Y to complete the remaining work. We can set up the equation: \[ R_Y \times t = W_{\text{remaining}} \] - Substituting the values we have: \[ \frac{1}{10} \times t = \frac{1}{5} \] 5. **Solve for \( t \):** - Rearranging the equation gives: \[ t = \frac{1}{5} \div \frac{1}{10} = \frac{1}{5} \times \frac{10}{1} = 2 \text{ hours} \] Thus, it takes Machine Y **2 hours** to complete the order after Machine X has worked for 12 hours.