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# Word Problems: Work and Workers

In our daily lives, we're all too familiar with the concept of "work." Whether it's studying for an exam, practicing math problems, or even doing household chores, work is a constant. However, in mathematics, "work" carries a unique significance. Let's dive into the world of math and explore some interesting word problems centered around work, enhancing our understanding and seeing how this mathematical principle applies in everyday scenarios.

## What is work?

Work has a few different definitions in math and science. In physics, work is the amount of force required to move an object. We express this with the formula Work = Force x Distance, or simply. $W=(F)(D)$. In math, we may also see word problems involving work in a more familiar context. For example, our teacher might ask us how long it will take for two people to paint a house if both are working together at different rates. It is this second definition that we will be dealing with today.

## The concept of shared work

A "shared work" problem involves numerous "workers" operating at different rates. We might also say that these problems involve workers who work at varying levels of efficiency.

The interesting thing about these problems is that they can be expressed with linear equations. In other words, we can use our known values as coefficients while assigning unknowns to variables.

The trickiest part is using our knowledge of linear equations to "translate" word problems into linear equations.

## Solving a shared work problem

Consider this shared work problem: Tom and Robert work at a toy factory. Tom can build one toy in 2 hours, while Robert takes 3 hours to build the same toy. How long would it take if they both work together?

To solve this, we can construct a linear equation using the given information. Let's denote t as the time it takes for them to build a toy together.

Tom can build 1 toy in 2 hours, so in 1 hour, he builds $\frac{1}{2}$ of a toy. Similarly, Robert, being a little slower, builds $\frac{1}{3}$ of a toy in 1 hour. Since 't' is the time it takes for them to build 1 toy together, Tom's contribution in this time would be $\frac{t}{2}$, and Robert's would be $\frac{t}{3}$.

In time 't', the sum of their contributions should equal 1 whole toy, so we can write the equation as: $\frac{t}{2}+\frac{t}{3}=1$

Solving for 't' involves dealing with fractions. To make this easier, we can multiply each term by 6 (the least common multiple of 2 and 3) to get rid of the denominators: $6\times \frac{t}{2}+6\times \frac{t}{3}=6\times 1$

Simplify this to: $3\times t+2\times t=6$

Combine like terms: $5\times t=6$

Then, solve for 't' by dividing both sides by 5: $t=\frac{6}{5}$ hours

This translates to 1 hour and 1/5 of an hour. To convert this fraction of an hour into minutes, we multiply by 60 (since 1 hour equals 60 minutes): $\frac{1}{5}\times 60=12$ minutes

Hence, it should take Tom and Robert 1 hour and 12 minutes to build a toy together. This means that they are more productive working as a team than individually, despite Tom being the faster worker. Keep in mind that if more than one job is involved, the equation will need to be adjusted accordingly.

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## Pair your student with a suitable tutor who understands problems involving work

If you've been searching for a math tutor who can help your student cover word problems involving work, look no further than Varsity Tutors. We'll pair your student with a tutor who can deliver personalized instruction in an encouraging, patient 1-on-1 environment. Tutors can cater to your student's ability level, learning style, and even their hobbies. Students can also ask as many questions as they like during these tutoring sessions. Speak with our Educational Directors for more information.

## Pair your student with a suitable tutor who understands problems involving work

If you've been searching for a math tutor who can help your student cover word problems involving work, look no further than Varsity Tutors. We'll pair your student with a tutor who can deliver personalized instruction in an encouraging, patient 1-on-1 environment. Tutors can cater to your student's ability level, learning style, and even their hobbies. Students can also ask as many questions as they like during these tutoring sessions. Speak with our Educational Directors for more information.

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