Back To Course

Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

It's amazing how much time can be saved when people work together. In this lesson, we'll practice figuring out just how much time we're talking about as we solve problems involving time.

*Many hands make light work.* This is an old saying that means big jobs get easier when more people help. Imagine you're painting a house. That's a big job for one person. But what if you had 5 helpers? It'll go much faster.

There's another saying: *Many hands make light pizza.* This one notes how those same people who help you paint the house also eat the pizza that would've lasted you for days.

If we don't worry about the pizza, we can focus on how working together saves time. Let's consider a simple example. Allison is working on a jigsaw puzzle. Alone, it would take her 3 hours to complete. If her friend Beth did the same puzzle, it would take her 6 hours to complete. What if they worked together?

We wouldn't say it's 3 + 6. Those numbers represent the total time each person takes when working alone. And, obviously, it wouldn't take Allison and Beth nine hours to do the puzzle together. Well, it would if they're really bad at teamwork.

We need a formula. The formula needs to be built on the rate each person takes.

Allison's rate is 1/3. It takes her 3 hours to complete the puzzle, so she completes 1/3 of the puzzle in an hour. Beth's rate is 1/6. Let's call *x* the time it would take them to work together. By the same logic, their rate when working together is 1/*x*.

So we can state that 1/3 + 1/6 = 1/*x*. In other words, this is our **work rate formula**:

In this formula, T1 is the time it takes person 1, T2 is the time it takes person 2, and Tt is the time it takes them when working together.

This formula works no matter how many people are involved. You can just add a 1 / T3, T4, etc. And the numerator is always 1 to represent the one task.

So we have our equation for Allison and Beth: *1/3 + 1/6 = 1/x*. How do we solve it? All those pesky denominators make it a little tricky. We can get rid of the fractions by finding the least common multiple. Here, it's 6*x*. So let's multiply both sides by 6*x*.

That gets us 6*x*/3 + 6*x*/6 = 6*x*/*x*. This simplifies to 2*x* + *x* = 6. 2*x* + *x* is 3*x*. And 6/3 is 2. So *x* = 2

That means that if Allison and Beth join forces, this dynamic duo of puzzle building can complete the job in just 2 hours, as opposed to 3 hours for Allison or 6 hours for Beth.

Let's try one with multiple people. Actually, let's do one that doesn't even involve people. *A garden hose can fill a swimming pool with water in 15 hours. A larger hose can do the job in 10 hours. A fire hose can do the job in 6 hours. If we use all three hoses together, how long will it take to fill the pool?*

Sometimes, you just need to go for a swim and you aren't willing to wait around for 15 hours. Besides, if you happen to have a fire hose at the ready, why not use it?

This problem doesn't involve people, but the concept of how we solve it is the same. The garden hose fills the pool at a rate of 1/15. The larger hose does it at 1/10. And rate of the fire hose is 1/6. So our equation looks like this: *1/15 + 1/10 + 1/6 = 1/x*.

There are three hoses working together, but we still just find the least common multiple. Here, it's 30*x*. So let's multiply both sides by 30*x*.

We get 30*x*/15 + 30*x*/10 + 30*x*/6 = 30*x*/*x*. That simplifies to 2*x* + 3*x* + 5*x* = 30. 2*x* + 3*x* + 5*x* is 10*x*. And 30 divided by 10 is 3. So *x* = 3. If we use all three hoses, we'll get the job done in just 3 hours! That's hose-based teamwork!

Sometimes we know the time it takes when people work together, but not the time it takes when they work alone. Here's another problem: *Kyle can mow a lawn 1.25 times faster than Steve. If they work together, the job takes 5 hours. How long would it take Steve to mow the lawn by himself?*

First of all, that must be one big lawn. 5 hours? Are they cutting it with nail clippers? Anyway, we start this one by defining our variables. Let's say Kyle takes *x* hours to mow the lawn by himself. Steve is 1.25 times slower than Kyle, so it takes him 1.25*x*.

Now we can set up our equation like this: *1/*x* + 1/1.25*x* = 1/5*. So it's the same equation, we're just using the variables differently. To solve this, we again find the least common multiple. That's 5*x*. Let's multiply both sides by 5*x*.

We get 5*x*/*x* + 5*x*/1.25*x* = 5*x*/5. That simplifies to 5 + 5/1.25 = *x*. 5/1.25 simplifies to 4. So 5 + 4 = *x*. So *x* = 9. What is *x* again? That's how long Kyle takes to mow the lawn. We want to know about Steve. Steve is 1.25 slower. So what's 1.25 * 9? 11.25. So it takes Steve 11 and a quarter hours to do the job by himself. Somebody get this guy a riding mower.

To summarize, we learned about teamwork. Specifically, we focused on the work rate formula. This formula helps us understand the relationship between the time it takes different individuals to perform a task and the time it takes if they work together.

We can write the formula like this:

We can have as many 1 / T-whatevers we need, depending on how many people are involved in the problem.

After you've completed this lesson, you should be able to:

- Identify the work rate formula
- Use this formula to solve problems involving the time it takes individuals to perform a task when working together

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
9 in chapter 3 of the course:

Back To Course

Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- Ratios & Rates: Definitions & Examples 6:37
- How to Solve Problems with Money 8:29
- Proportion: Definition, Application & Examples 6:05
- Calculations with Ratios and Proportions 5:35
- Percents: Definition, Application & Examples 6:20
- How to Solve Word Problems That Use Percents 6:30
- Math Combinations: Formula and Example Problems 7:14
- How to Calculate a Permutation 6:58
- How to Solve Problems with Time 6:18
- Go to High School Algebra: Calculations, Ratios, Percent & Proportions

- Computer Science 336: Network Forensics
- Computer Science 220: Fundamentals of Routing and Switching
- Global Competency Fundamentals & Applications
- Introduction to the Principles of Project Management
- Praxis Elementary Education: Reading & Language Arts - Applied CKT (7902): Study Guide & Practice
- Practical Applications for Business Ethics
- Practical Applications for Marketing
- Practical Applications for HR Management
- Practical Applications for Organizational Behavior
- Analyzing Texts Using Writing Structures
- MBLEx Prep Product Comparison
- AEPA Prep Product Comparison
- ASCP Prep Product Comparison
- NCE Prep Product Comparison
- TASC Test Score Information
- What is the TASC Test?
- Praxis Prep Product Comparison

- Diclofenac vs. Ibuprofen
- Developing & Managing a High-Quality Library Collection
- Library Space Planning
- Literacy Strategies for Teachers
- Arithmetic Operations in R Programming
- Practical Application: Understanding Employee Behavior
- Positive Global Outcomes of Global Competence
- Practical Application: Color Wheel Infographic
- Quiz & Worksheet - Developing a Learner-Centered Classroom
- Quiz & Worksheet - Technology for Teaching Reading
- Quiz & Worksheet - Pectoralis Major Anatomy
- Quiz & Worksheet - Oral & Written Communication Skills
- Quiz & Worksheet - How to Teach Reading to ELL Students
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Glencoe Algebra 1: Online Textbook Help
- UExcel Introduction to Macroeconomics: Study Guide & Test Prep
- Business 107: Organizational Behavior
- College Chemistry: Certificate Program
- High School World History: Help and Review
- Leadership Styles in Organizational Behavior: Help and Review
- FTCE: Word Choice
- Quiz & Worksheet - Light and Relativity
- Quiz & Worksheet - Circuit, Path & Sum of Degrees Theorems
- Quiz & Worksheet - Issues that Affect Population Size
- Quiz & Worksheet - Factors that Impact Air Masses
- Quiz & Worksheet - Evaluating Environmental Health Threats

- Assessing Weighted & Complete Graphs for Hamilton Circuits
- Alice Stebbins Wells: Biography
- Equivalent Fractions Lesson Plan
- Dred Scott Lesson Plan
- How Many Questions Are on the TExES PPR?
- How Long Should I Study For the GMAT?
- Homeschooling in Massachusetts
- Still Life Drawing Lesson Plan
- Homeschooling in Delaware
- Listening Activities for Kids
- Dolch Sight Words for Third Grade
- What is on the TABE Test?

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject