Problem 73 - Algebra 2

Question

PAINING Two employees of a painting company paint houses together. One painter can paint a house alone in 3 days, and the other painter can paint the same size house alone in 4 days. How long will it take them to paint one house if they work together?

Step-by-Step Solution

Answer

It will take them approximately 1.71 days to paint the house together.

Step 1: Determine Each Painter's Rate

Let's first calculate how much of the house each painter can paint per day. The first painter can complete the house in 3 days, so each day they paint \( \frac{1}{3} \) of the house. Similarly, the second painter can complete the house in 4 days, so each day they paint \( \frac{1}{4} \) of the house.

Step 2: Add the Rates Together

When the two painters work together, their combined rate is the sum of their individual rates. So together, they paint \( \frac{1}{3} + \frac{1}{4} \) of the house per day. To add these fractions, find a common denominator (12), and combine: \( \frac{1}{3} = \frac{4}{12} \) and \( \frac{1}{4} = \frac{3}{12} \), so \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \).

Step 3: Calculate Time to Complete the House

The combined rate tells us that together they paint \( \frac{7}{12} \) of the house in one day. To find how many days it takes to complete one whole house, set up the equation \( \frac{7}{12}x = 1 \). Solve for \( x \) by multiplying both sides by \( \frac{12}{7} \): \( x = \frac{12}{7} \).

Step 4: Convert the Fraction to a Decimal or Mixed Number

The answer \( x = \frac{12}{7} \) is approximately equal to 1.714 days. This means it will take them a little over 1 2/3 days to paint the house together.

Key Concepts

Work Rate Formula

The work rate formula is a powerful tool used in solving problems related to simultaneous work by different entities. In essence, the formula helps you determine how fast a task can be completed when multiple people or machines collaborate. Here, we're dealing with two painters working together to finish the painting of a house.

The basic idea is to express each worker's ability to complete a task within a given time frame as their *work rate*. For instance, if Painter A can complete the house on their own in 3 days, their work rate is the fraction of the house they can paint in one day, denoted as \(\frac{1}{3}\) per day. Similarly, if Painter B can finish the house in 4 days, their work rate is \(\frac{1}{4}\) per day.

To find the combined work rate when both painters work together, we simply add their rates together. This addition reflects the speed at which they can jointly paint the house. Thus, the combined work rate of both painters is \(\frac{1}{3} + \frac{1}{4}\) of the house per day. The resulting rate is the foundation for determining how long it will take them to finish the task when collaborating.

Fraction Addition

Fraction addition is critical in work rate problems, especially when calculating the combined rate of work performed by multiple workers. When two fractions have different denominators, like \(\frac{1}{3}\) and \(\frac{1}{4}\) in our example, you cannot directly add them. Instead, you must find a common denominator.

The smallest common multiple of 3 and 4 is 12, so we convert each fraction to have this common denominator. We express \(\frac{1}{3}\) as \(\frac{4}{12}\) and \(\frac{1}{4}\) as \(\frac{3}{12}\). Now, these fractions have the same denominator, allowing for straightforward addition:

• \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)

This final fraction, \(\frac{7}{12}\), represents the combined portion of the house painted per day by both painters. Understanding this concept ensures you can handle similar problems where combining rates is necessary for finding a solution.

Problem Solving in Algebra

Solving work rate problems often involves logical steps and algebraic reasoning. The goal is to find out how long a task takes when workers combine efforts. Here's how algebra helps in our example:

After obtaining the combined work rate of \(\frac{7}{12}\) house per day, the next step is to determine the total time to complete the house. We set this up with the equation:

• \(\frac{7}{12}x = 1\)

Where \(x\) represents the total time in days. Solve this equation by isolating \(x\). Multiply both sides by \(\frac{12}{7}\) to get:

• \(x = \frac{12}{7} = 1.714\)

Converting the fraction \(\frac{12}{7}\) yields approximately 1.714 days. This tells us that both painters together will take just over 1.7 days to paint the house. Algebraic techniques are essential here to turn a situation into a solvable equation, enabling clear and accurate problem solving.