Understanding the Machine Rate

The problem states that 7 machines take 7 minutes to produce 7 identical toys. This implies a direct relationship between machines, toys, and time.

From the given information, we can deduce the rate of production:

• If 7 machines make 7 toys in 7 minutes, then, on average, each machine makes 1 toy in those 7 minutes.
• This means the time required for a single machine to make a single toy is 7 minutes.

Calculating Time for 100 Machines

We need to find the time it takes for 100 machines to make 100 toys, assuming they work at the same rate.

Since each machine takes 7 minutes to make one toy, and we have 100 machines working simultaneously:

• Each of the 100 machines can work on one toy.
• Each machine will complete its toy in 7 minutes.
• Therefore, all 100 machines will finish their respective 100 toys in the same amount of time it takes one machine to finish one toy.

The calculation is as follows:

Rate of one machine = $\frac{1 \text{ toy}}{7 \text{ minutes}}$

Total rate of 100 machines = $100 \times \frac{1 \text{ toy}}{7 \text{ minutes}} = \frac{100}{7} \frac{\text{toys}}{\text{minute}}$

Time = $\frac{\text{Total Toys}}{\text{Total Rate}} = \frac{100 \text{ toys}}{\frac{100}{7} \frac{\text{toys}}{\text{minute}}} = 100 \times \frac{7}{100} \text{ minutes} = 7 \text{ minutes}$

Thus, it would take 7 minutes for 100 machines to make 100 toys.