Calculating combined work rates is a valuable skill, especially when dealing with tasks that involve multiple workers or machines. This concept is widely used in various fields such as project management, engineering, and everyday problem-solving.
Understanding Individual Work Rates
First, let’s understand what a work rate is. A work rate is the amount of work done per unit of time. For example, if a worker can complete a task in 5 hours, their work rate is $frac{1}{5}$ of the task per hour.
Combining Work Rates
When multiple workers are involved, their combined work rate is the sum of their individual work rates. Suppose Worker A can complete a task in 5 hours, and Worker B can complete the same task in 3 hours. Their individual work rates are:
- Worker A: $frac{1}{5}$ of the task per hour
- Worker B: $frac{1}{3}$ of the task per hour
To find their combined work rate, you simply add these rates together:
$text{Combined Work Rate} = frac{1}{5} + frac{1}{3}$
To add these fractions, find a common denominator. In this case, the common denominator is 15:
$frac{1}{5} = frac{3}{15}$
$frac{1}{3} = frac{5}{15}$
So, the combined work rate is:
$frac{3}{15} + frac{5}{15} = frac{8}{15}$
This means that together, Worker A and Worker B can complete $frac{8}{15}$ of the task per hour.
Calculating Total Time
To find out how long it will take for both workers to complete the task together, take the reciprocal of the combined work rate:
$text{Total Time} = frac{1}{text{Combined Work Rate}} = frac{1}{frac{8}{15}} = frac{15}{8} text{ hours}$
So, it will take them $frac{15}{8}$ hours, or 1.875 hours, to complete the task together.
Practical Example
Imagine you have two machines working on assembling gadgets. Machine X can assemble a gadget in 4 hours, and Machine Y can do it in 6 hours. What is their combined work rate, and how long will it take them to assemble one gadget together?
- Machine X: $frac{1}{4}$ of the gadget per hour
- Machine Y: $frac{1}{6}$ of the gadget per hour
Combined work rate:
$frac{1}{4} + frac{1}{6}$
Find the common denominator (12):
$frac{1}{4} = frac{3}{12}$
$frac{1}{6} = frac{2}{12}$
So, the combined work rate is:
$frac{3}{12} + frac{2}{12} = frac{5}{12}$
To find the total time:
$frac{1}{frac{5}{12}} = frac{12}{5} text{ hours}$
Therefore, it will take them $frac{12}{5}$ hours, or 2.4 hours, to assemble one gadget together.
Conclusion
Understanding how to calculate combined work rates can simplify complex tasks and optimize efficiency. Whether it’s workers on a construction site or machines in a factory, knowing how to add individual work rates and find the total time required can be incredibly useful.