If machine A and B can produce 800 units in 3 hours and 8 hours, respectively, in how many hours can machine A and B working together at these constant rates produce 800 units?
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.
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