Solving Work Rate Time Problems

Work rate time problems are a common type of word problem in algebra. These problems involve figuring out how long it takes to complete a task, given the rate at which people or machines can work. To solve these problems, you need to understand the relationship between work, rate, and time.

Understanding the Relationship

The key formula to remember is:

Work = Rate x Time

Here’s what each part represents:

• Work: The amount of the task completed (e.g., painting a house, filling a pool).
• Rate: How much work is done per unit of time (e.g., painting 1/2 a house per hour, filling 10 gallons per minute).
• Time: The duration of the work (e.g., 2 hours, 15 minutes).

Solving Work Rate Time Problems

Here’s a step-by-step approach to solving work rate time problems:

1. Identify the knowns: Determine the work, rate, and time values that are given in the problem.
2. Identify the unknown: Figure out what you’re trying to find (usually time or rate).
3. Set up an equation: Use the formula Work = Rate x Time to set up an equation that relates the knowns and the unknown.
4. Solve for the unknown: Use algebraic techniques to solve the equation for the unknown variable.

Example Problem

Let’s say you have two painters, A and B. Painter A can paint a house in 6 hours, while painter B can paint the same house in 4 hours. How long would it take them to paint the house together?

Solution

1. Knowns:
   • Painter A’s rate: 1/6 of the house per hour
   • Painter B’s rate: 1/4 of the house per hour
   • Work: 1 whole house
2. Unknown: Time it takes them to paint together (let’s call it ‘t’).
3. Equation:

   Work = Rate x Time

   (1/6)t + (1/4)t = 1 (Since they’re working together, their individual rates add up)

4. Solve:
   1. Find a common denominator for the fractions: (2/12)t + (3/12)t = 1
   2. Combine like terms: (5/12)t = 1
   3. Multiply both sides by 12/5 to isolate ‘t’: t = 12/5 hours = 2.4 hours

Therefore, it would take painters A and B 2.4 hours to paint the house together.

Key Points to Remember

• Always remember the formula: Work = Rate x Time
• When people or machines work together, their individual rates add up.
• If a task is completed in ‘x’ time, the rate is 1/x.

Practice solving work rate time problems, and you’ll become more comfortable with them. Remember to organize your information, set up the equation carefully, and use your algebra skills to solve for the unknown.