Problem 55 - Intermediate Algebra

Question

Fireplaces. A mason and his assistant work together for 6 hours on a brick fireplace before the mason has to leave the job. The assistant finishes the job alone in 10 hours. If the mason can construct a fireplace in 18 hours working alone, how long does it take his assistant working alone to construct a fireplace?

Step-by-Step Solution

Answer

The assistant takes 24 hours to construct a fireplace alone.

Step 1: Define Work Rates

First, define the work rates of the mason and the assistant. The mason's work rate is \( \frac{1}{18} \) of a fireplace per hour because he can complete one fireplace in 18 hours. Let \( A \) be the time it takes for the assistant to complete one fireplace alone; then the assistant's work rate is \( \frac{1}{A} \) fireplaces per hour.

Step 2: Calculate Joint Work for 6 Hours

When the mason and the assistant work together for 6 hours, their combined work rate is \( \frac{1}{18} + \frac{1}{A} \). Thus, in 6 hours, they complete \( 6 \times \left(\frac{1}{18} + \frac{1}{A}\right) \) of the fireplace.

Step 3: Assistant's Remaining Work

After 6 hours of joint work, the assistant continues to work alone for 10 hours. Thus, the assistant completes \( 10 \times \frac{1}{A} \) of the fireplace in these hours.

Step 4: Total Work Equals One Fireplace

The entire job, i.e., one full fireplace, is completed when the sum of work by both phases, together and assistant alone, totals to 1. Therefore, setup the equation: \[ 6 \left(\frac{1}{18} + \frac{1}{A}\right) + 10 \left(\frac{1}{A}\right) = 1 \]

Step 5: Simplify the Equation

Simplify the equation. First, distribute the 6: \[ \frac{6}{18} + \frac{6}{A} + \frac{10}{A} = 1 \] Simplify \( \frac{6}{18} \) to \( \frac{1}{3} \) and combine terms: \[ \frac{1}{3} + \frac{16}{A} = 1 \]

Step 6: Solve for A

Isolate \( \frac{16}{A} \) by subtracting \( \frac{1}{3} \) from both sides: \[ \frac{16}{A} = 1 - \frac{1}{3} = \frac{2}{3} \] Now solve for \( A \) by multiplying both sides by \( A \) and dividing by \( \frac{2}{3} \): \[ A = \frac{16 \times 3}{2} = 24 \]

Step 7: Conclusion

The assistant, thus working alone, takes 24 hours to construct a fireplace.

Key Concepts

Algebraic Equations

Algebraic equations are powerful tools that help break down real-world scenarios into solvable mathematical expressions. In the context of work rate problems, algebraic equations help us calculate the time it takes for different individuals working at varying speeds to complete a task together or alone.
To create an algebraic equation for a work problem, we must first establish each person's work rate, which is the amount of work they can complete per unit of time. For the mason and his assistant, we defined their work rates as fractions of a fireplace per hour:

• For the mason: \( \frac{1}{18} \) fireplaces/hour
• For the assistant: \( \frac{1}{A} \) fireplaces/hour, where \( A \) is unknown

By establishing these work rates, we can set up equations that combine their respective contributions toward completing a full fireplace. This lays the groundwork for solving how long each worker takes individually and together on the project.

Problem Solving Steps

When approaching any work rate problem, using a clear set of sequential steps can guide us to the correct answer efficiently. Here are the problem-solving steps applied to the fireplace scenario:

**Step 1: Define Work Rates**
We started by defining the work rates of both the mason and the assistant.
**Step 2: Calculate Joint Work**
We assessed the joint work rate over the time both worked together and used it to figure out how much of the fireplace they completed in those crucial hours.
**Step 3: Evaluate Remaining Work**
We considered how much work the assistant finished alone after the mason left.
**Step 4: Total the Work**
We combined work from both the joint effort and the assistant’s solo work to ensure it equalled one complete fireplace.
**Step 5: Simplify Equation and Solve**
In this step, we manipulated the equation to express it in its simplest form and solve for the assistant's rate.

• Combine like terms
• Simplify fractions
• Isolate \( A \) to find the standalone time the assistant needs

These structured steps highlight how organizing one’s approach can demystify complex problems and make them far more manageable.

Joint Work Problems

Joint work problems explore scenarios where multiple figures collaborate to complete a task, each contributing at their own pace. Understanding these relationships involves:

• Identifying each person’s work rate
• Setting a combined work rate when working together
• Evaluating contributions over specified time periods

In this fireplace-building example, we have both a mason and his assistant involved. Initially, they contribute their efforts together for a portion of the job, and later, one person finishes it. By conducting this split analysis, we gain insight into how to resolve issues of shared work and find individual timings. Joint work problems require a foundational comprehension of algebra to establish accurate equations reflecting contributions. These problems frequently arise in project planning and resource management scenarios, making them critical to tackle efficiently in real-life applications.