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Tutorials that teach
Work, Rate, and Time in a System of Equations

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- Work, Rate, and Time Relationship
- Combined Rates
- Solving a Work, Rate, and Time Problem with a System of Equations

**Work, Rate, and Time**

The relationship between work, rate, and time can be modeled using a similar equation to distace, rate, and time:

We multiply the rate at which someone completes work or a particular job, by the amount of time they spent working.

**Combined Rates**

If two people are working together, we can add their rates together and reflect this in the formula for work, rate and time. For example if a professor can grade 10 papers in one hour, and her teaching assistant can grade 7 papers in one hour, we know that their combined rate is 17 papers per hour.

In general, if two people are working together, we use: as the rate in the formula, to show that Rate 1 and Rate 2 are added together

**Solving a Work, Rate, and Time Problem using a System of Equations**

Let's return to the professor and teaching assistant scenario:

A professor can grade 80 papers in the same amount of time it takes for her teaching assistant to grade 60 papers. The teaching assistant grades 1 less paper per hour than the professor. How many papers can the two grade in one hour?

Let's use what we know about work, rate, and time (and combined rate) to create some equations based off of what we know:

Where r_{1} is the hourly rate of the professor, and r_{2}_{ }is the hourly rate of the teaching assistant.

We also know that the teaching assistant grades one less paper per hour, so we can subtract 1 from the professors rate to represent the teaching assistant's rate relative to the professor. This is shown below:

We can use this equivalent expression for r_{2} in one of the other equations in our system. Instead of writing r_{2}, we will write r_{1}–1. This will allow us to make further substitutions and eventually solve for an unknown variable:

Now that we know the value for t, we can use this in another equation in our system to solve for other variables. We eventually want to know how many papers the professor and assistant can grade in one hour. We already have an equation for this in our system; it is the equation we found by adding two equations together to show combined rate:

We can simply plug in 20 for t, and solve for (r_{1} + r_{2}). Notice that we don't necessarily need to solve for the two rates individually, because we are interested in what their combined rate is: