Phoebe runs a T-shirt printing business. She ships the T-shirts to retailers in boxes of 200. She has two automated printing machines that can print designs on T-shirts. One machine can print 200 T-shirts in 50 minutes. The other machine can print 200 T-s

Phoebe's T-shirt printing business utilizes two machines with different printing speeds. Here’s a breakdown of how they perform: - **Machine A**: Can print 200 T-shirts in 50 minutes. - This means the rate of Machine A is: \[ \frac{200 \text{ T-shirts}}{50 \text{ minutes}} = 4 \text{ T-shirts per minute} \] - **Machine B**: Can print 200 T-shirts in 150 minutes. - The rate of Machine B is: \[ \frac{200 \text{ T-shirts}}{150 \text{ minutes}} \approx 1.33 \text{ T-shirts per minute} \] ### Combined Printing Rate If both machines are working together, their combined printing rate is: \[ 4 \text{ T-shirts per minute} + 1.33 \text{ T-shirts per minute} \approx 5.33 \text{ T-shirts per minute} \] ### Printing Time for 200 T-shirts To find out how long it takes both machines together to print one box (200 T-shirts): \[ \frac{200 \text{ T-shirts}}{5.33 \text{ T-shirts per minute}} \approx 37.5 \text{ minutes} \] ### Summary - **Machine A** prints 200 T-shirts in 50 minutes at a rate of 4 T-shirts per minute. - **Machine B** prints 200 T-shirts in 150 minutes at a rate of approximately 1.33 T-shirts per minute. - Together, both machines can print a box of 200 T-shirts in about 37.5 minutes. Update (2025-04-29): Let's analyze the printing capabilities of Phoebe's two machines. 1. **Machine 1**: - Prints 200 T-shirts in 50 minutes. - Therefore, the printing rate is: \[ \text{Rate of Machine 1} = \frac{200 \text{ T-shirts}}{50 \text{ minutes}} = 4 \text{ T-shirts per minute} \] 2. **Machine 2**: - Prints 200 T-shirts in 150 minutes. - Therefore, the printing rate is: \[ \text{Rate of Machine 2} = \frac{200 \text{ T-shirts}}{150 \text{ minutes}} = \frac{4}{3} \text{ T-shirts per minute} \approx 1.33 \text{ T-shirts per minute} \] Now, let's find the combined printing rate of both machines: \[ \text{Combined Rate} = \text{Rate of Machine 1} + \text{Rate of Machine 2} = 4 \text{ T-shirts per minute} + \frac{4}{3} \text{ T-shirts per minute} \] To add these rates together, we can convert \(4\) into a fraction: \[ 4 = \frac{12}{3} \] Now we can add: \[ \text{Combined Rate} = \frac{12}{3} + \frac{4}{3} = \frac{16}{3} \text{ T-shirts per minute} \] Thus, combining the production rates yields a total rate of: \[ \text{Combined Rate} \approx 5.33 \text{ T-shirts per minute} \] Next, if Phoebe wants to calculate the time it takes to print a full box of 200 T-shirts using both machines, we can use the combined rate: \[ \text{Time to print 200 T-shirts} = \frac{200 \text{ T-shirts}}{\frac{16}{3} \text{ T-shirts per minute}} = 200 \times \frac{3}{16} = \frac{600}{16} = 37.5 \text{ minutes} \] **Summary**: The two machines together can print 200 T-shirts in 37.5 minutes.