Question

At a​ factory, two machines pack bottles into boxes for shipping. Machine A can pack 8 boxes per​ minute, and Machine B can pack 11 boxes per minute. Both machines have been packing boxes for some​ time, and the difference between the number of boxes that Machine B has packed and the number of boxes that Machine A has packed is less than 200. Write and solve an inequality to find the possible lengths of time to the nearest second that the machines have been working. Describe the possible solutions.

Answer 1

Let's let t be the time in minutes that both machines have been working. Then, the number of boxes packed by Machine A in t minutes is 8t, and the number of boxes packed by Machine B in t minutes is 11t.

The difference between the number of boxes packed by Machine B and Machine A is less than 200, so we can write the following inequality:

11t - 8t < 200

Simplifying, we get:

3t < 200

Dividing both sides by 3, we get:

t < 200/3

To the nearest second, this is approximately 66.67 seconds.

Therefore, the possible lengths of time that the machines have been working is t < 66.67 seconds.

Note that this inequality only gives us an upper bound on the time that the machines have been working, since we know that the difference between the number of boxes packed by Machine B and Machine A is less than 200. We do not have any information on the minimum time that the machines have been working, so there are infinitely many possible solutions that satisfy the given conditions.

Explanation: