Problem 47 - Algebra and Trigonometry

Question

Sharing a Job \(\quad\) Betty and Karen have been hired to paint the houses in a new development. Working together the women can paint a house in two-thirds the time that it takes Karen working alone. Betty takes 6 \(\mathrm{h}\) to paint a house alone. How long does it take Karen to paint a house working alone?

Step-by-Step Solution

Answer

Karen takes 3 hours to paint a house alone.

Step 1: Define the Variables

Let Karen's time to paint a house alone be \( x \) hours. We know that Betty takes 6 hours to paint a house alone.

Step 2: Establish the Combined Work Rate

Betty's work rate is \( \frac{1}{6} \) of a house per hour. Karen's work rate is \( \frac{1}{x} \) of a house per hour. Together, they have a combined work rate of \( \frac{1}{x} + \frac{1}{6} \) per hour.

Step 3: Relate the Combined Rate to Time

Since they can paint a house together in two-thirds the time it takes Karen alone, this means their combined rate equals \( \frac{1}{\frac{2}{3}x} = \frac{3}{2x} \) of a house per hour.

Step 4: Set Up the Equation

Now, set up the equation for the combined work rate: \( \frac{1}{x} + \frac{1}{6} = \frac{3}{2x} \).

Step 5: Solve the Equation

To solve \( \frac{1}{x} + \frac{1}{6} = \frac{3}{2x} \), multiply through by \( 6x \) to clear the fractions: \( 6 + x = 9 \).

Step 6: Solve for x

Subtract 6 from both sides: \( x = 3 \). This means Karen takes 3 hours to paint a house working alone.

Key Concepts

Understanding Combined Work Rate

In work rate problems, understanding the combined work rate is crucial. This concept helps you determine how quickly two people can complete a task together, compared to working alone. The combined work rate is simply the sum of each individual's work rate.

To find an individual's work rate, you calculate what portion of the job they can complete in one unit of time. For example, if Betty can paint a house in 6 hours, her work rate is \( \frac{1}{6} \) of a house per hour. Similarly, if Karen's work rate is \( \frac{1}{x} \) of a house per hour, when they work together, their combined work rate is:

• Betty's work rate: \( \frac{1}{6} \)
• Karen's work rate: \( \frac{1}{x} \)

So, their combined work rate is \( \frac{1}{x} + \frac{1}{6} \). This combined rate tells us how much of the house they can paint together in one hour.

Time and Work Formula: The Key to Solving Problems

The time and work formula is a fundamental tool in solving work rate problems. It connects the work rate with the time taken to complete a job. The principle is: the total work is one complete unit (like one house), and time multiplied by work rate equals this total work.

The formula can be expressed as:

\[ \text{Time} \times \text{Work Rate} = \text{Total Work} \]

In the given problem, since Betty and Karen can jointly paint a house in two-thirds the time it takes Karen alone, we can say their joint work rate must complete the house in \( \frac{2}{3}x \) time. This allows us to write another expression for their joint work rate:

\[ \text{Combined Rate} = \frac{1}{\frac{2}{3}x} = \frac{3}{2x} \]

This indicates that their combined efficiency is greater when working together, hence reducing the total time needed.

Algebraic Equations in Time and Work Problems

Using algebraic equations is essential for solving time and work problems, as they allow you to set up a relationship between different rates and times. In this context, equations help find unknown variables representing how long each person takes to complete a job alone.

For the exercise, we know:

• Betty's rate: \( \frac{1}{6} \)
• Karen's rate: \( \frac{1}{x} \)
• Combined rate matching two-thirds of Karen's time alone: \( \frac{3}{2x} \)

The algebraic equation representing their combined work can be set up as:

\[ \frac{1}{x} + \frac{1}{6} = \frac{3}{2x} \]

Solving this involves clearing the fractions by multiplying through by a common denominator (in this case, \(6x\)), and solving for \(x\):

\[ 6 + x = 9 \]
\[ x = 3 \]

This shows that it takes Karen 3 hours to paint a house alone. By manipulating these equations, you can effectively solve work rate problems and find the time different workers take to complete tasks separately.