To solve this problem, let's define the variables and set up an equation based on the information provided.

1. Define Variables:

   • Let $s$ represent the time (in minutes) it takes Machine A to fill a case by itself.
   • Machine B takes $s + 10$ minutes to fill a case by itself.

2. Determine Rates:

   • Machine A's filling rate is $\frac{1}{s}$ cases per minute.
   • Machine B's filling rate is $\frac{1}{s + 10}$ cases per minute.

3. Combined Rate:

   • When both machines work together, their combined rate is the sum of their individual rates: $\frac{1}{s} + \frac{1}{s + 10}$
   • Together, they take 15 minutes to fill one case, so their combined rate is also $\frac{1}{15}$ cases per minute.

4. Set Up the Equation: $\frac{1}{s} + \frac{1}{s + 10} = \frac{1}{15}$

5. Solve the Equation: Multiply both sides by $15s(s + 10)$ to eliminate the denominators: $15(s + 10) + 15s = s(s + 10)$ Expanding and simplifying this equation will allow us to solve for $s$.

Let me solve this equation for you.The solutions to the equation are approximately:

• $s \approx -5.81$ (which we discard since time cannot be negative)
• $s \approx 25.81$

Thus, Machine A takes approximately 25.81 minutes to fill a case, and Machine B takes $25.81 + 10 = 35.81$ minutes to fill a case.

Would you like further details on this solution or have any other questions?

Here are some related questions for practice:

1. What would happen to the combined filling time if Machine A's rate doubled?
2. How would the equation change if Machine B took $s + 5$ minutes instead?
3. If Machine A were twice as fast, how would that affect the time taken to fill the case together?
4. How long would it take if a third machine, Machine C, filled cases in $s + 15$ minutes and worked alongside A and B?
5. What if the two machines had to fill two cases together; how long would that take?

Tip: When dealing with combined rates, add individual rates to find the combined rate.