Questions: Two machines complete a printing project in 3 h. How many machines are needed to finish the same project in 2 h?

Solution Steps

To solve this problem, we need to determine the rate at which the machines work together and then calculate how many machines are required to complete the project in a shorter time. First, find the combined rate of the two machines working together. Then, use this rate to find out how many machines are needed to complete the project in 2 hours.

The two machines complete the printing project in \( t = 3 \) hours. Therefore, the combined rate of work for the two machines is given by:

\[ \text{Rate}_{\text{two machines}} = \frac{2 \text{ machines}}{3 \text{ hours}} = \frac{2}{3} \text{ machines per hour} \]

To finish the same project in \( t = 2 \) hours, we need to find the required rate of work:

\[ \text{Rate}_{\text{required}} = \frac{2 \text{ machines}}{2 \text{ hours}} = 1 \text{ machine per hour} \]

Now, we can find the number of machines needed to achieve the required rate. Since the rate of work for one machine is \( \frac{1}{3} \) machines per hour, we can set up the equation:

\[ \text{Machines needed} = \frac{\text{Rate}_{\text{required}}}{\text{Rate}_{\text{one machine}}} = \frac{1 \text{ machine per hour}}{\frac{1}{3} \text{ machines per hour}} = 3 \text{ machines} \]

Final Answer