Solving Work Rate Time Problems

Work rate time problems are a common type of math problem that involves figuring out how long it takes to complete a task, given the rate at which work is being done. These problems often involve multiple people or machines working together, and they can be a bit tricky to solve if you're not familiar with the concepts involved. But don't worry, with a little practice, you'll be able to solve these problems like a pro!

Understanding the Basics

Before we dive into solving work rate time problems, let's first understand the key concepts involved:

• Work: The amount of task completed.
• Rate: The amount of work done per unit of time.
• Time: The duration it takes to complete the work.

The relationship between these three concepts can be expressed by the following formula:

Work = Rate x Time

Solving Work Rate Time Problems

Here's a step-by-step guide to solving work rate time problems:

1. Identify the knowns: Determine the values for work, rate, and time that are given in the problem.
2. Set up an equation: Use the formula Work = Rate x Time to set up an equation that represents the problem.
3. Solve for the unknown: Solve the equation to find the value of the unknown variable (work, rate, or time).

Example Problem

Let's say John can paint a house in 6 hours, and Mary can paint the same house in 4 hours. How long will it take them to paint the house together?

Solution

1. Identify the knowns:
   • John's rate: 1/6 of the house per hour
   • Mary's rate: 1/4 of the house per hour
   • Work: 1 (whole house)
2. Set up an equation:

   Let 't' be the time it takes them to paint the house together. The combined rate is the sum of their individual rates:

   (1/6)t + (1/4)t = 1

3. Solve for the unknown:
   1. Find a common denominator for the fractions (12):

      (2/12)t + (3/12)t = 1

   2. Combine the fractions:

      (5/12)t = 1

   3. Multiply both sides by 12/5 to isolate 't':

      t = 12/5 hours = 2.4 hours

Therefore, it will take John and Mary 2.4 hours to paint the house together.

Tips for Solving Work Rate Time Problems

• Pay attention to units: Make sure the units of work, rate, and time are consistent. For example, if the rate is given in hours, the time should also be in hours.
• Think about the whole: The total work completed is always 1 (representing the entire task).
• Practice, practice, practice: The more problems you solve, the more comfortable you'll become with these concepts.

Work rate time problems can be a bit challenging at first, but with a clear understanding of the concepts and a step-by-step approach, you'll be able to solve them with ease. Remember to always check your work and make sure your answer makes sense in the context of the problem.