Calculating Machine Production Time Together

This problem involves calculating the combined rate at which two machines, A and B, produce units and then determining the time it takes for them to produce a specific quantity (800 units) when working together.

Understanding Individual Machine Rates

First, let's determine the production rate of each machine individually. The rate is calculated as the number of units produced divided by the time taken.

• Machine A's Rate: Produces 800 units in 3 hours. Rate_A = $ \frac{800 \text{ units}}{3 \text{ hours}} $
• Machine B's Rate: Produces 800 units in 8 hours. Rate_B = $ \frac{800 \text{ units}}{8 \text{ hours}} $

Calculating Combined Production Rate

When machines A and B work together, their rates add up. To find the combined rate, we sum their individual rates:

Combined Rate = Rate_A + Rate_B

Combined Rate = $ \frac{800}{3} + \frac{800}{8} $ units per hour

To add these fractions, we find a common denominator, which is 24:

Combined Rate = $ \left( \frac{800 \times 8}{3 \times 8} \right) + \left( \frac{800 \times 3}{8 \times 3} \right) $

Combined Rate = $ \frac{6400}{24} + \frac{2400}{24} $

Combined Rate = $ \frac{6400 + 2400}{24} $

Combined Rate = $ \frac{8800}{24} $ units per hour

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (which is 8):

Combined Rate = $ \frac{8800 \div 8}{24 \div 8} = \frac{1100}{3} $ units per hour

Determining Time for Combined Production

Now, we need to find out how many hours it takes for both machines working together (at the combined rate of $ \frac{1100}{3} $ units per hour) to produce 800 units. The formula for time is:

Time = $ \frac{\text{Total Units}}{\text{Combined Rate}} $

Time = $ \frac{800 \text{ units}}{ \frac{1100}{3} \text{ units per hour}} $

To divide by a fraction, we multiply by its reciprocal:

Time = $ 800 \times \frac{3}{1100} $ hours

Time = $ \frac{800 \times 3}{1100} $ hours

Time = $ \frac{2400}{1100} $ hours

Finally, we simplify this fraction by dividing both the numerator and the denominator by 100:

Time = $ \frac{24}{11} $ hours

Therefore, machines A and B working together can produce 800 units in $ \frac{24}{11} $ hours.