Problem 45 - Elementary Algebra

Question

For Problems \(33-50\), set up an equation and solve the problem. (Objective 2 ) Mark can overhaul an engine in 20 hours, and Phil can do the same job by himself in 30 hours. If they both work together for a time, and then Mark finishes the job by himself in 5 hours, how long did they work together?

Step-by-Step Solution

Answer

Mark and Phil worked together for 9 hours.

Step 1: Understanding the Problem

We need to find out how long Mark and Phil worked together to finish an engine overhaul. Individually, Mark takes 20 hours and Phil takes 30 hours to complete the task. After working together for some time, Mark finishes the job by himself, taking an additional 5 hours.

Step 2: Equation Setup

Let the time they worked together be denoted by \( t \) hours. In \( t \) hours, Mark completes \( \frac{t}{20} \) of the job, and Phil completes \( \frac{t}{30} \) of the job. When Mark works alone for the last 5 hours, he completes \( \frac{5}{20} \) of the job. The entire job done is equal to 1.

Step 3: Combine and Solve for Total Work

The equation for their combined work is: \[\frac{t}{20} + \frac{t}{30} + \frac{5}{20} = 1\]Combine the fractions to solve:\[\frac{3t}{60} + \frac{2t}{60} + \frac{15}{60} = 1\]Simplify and solve for \( t \):\[\frac{5t + 15}{60} = 1\]

Step 4: Solving the Equation

Multiply both sides by 60 to clear the fractions:\[5t + 15 = 60\]Then subtract 15 from both sides:\[5t = 45\]Divide by 5:\[t = 9\]So, the time they worked together is 9 hours.

Key Concepts

Understanding Work and Time Problems

Work and time problems are common in algebra and involve scenarios where multiple agents complete a task at different rates. These problems can be tricky because you need to find out how long it will take for these agents to complete the task when they work together or separately. Often, the task is standardized to one complete job, which helps to simplify the equation setup.

In these types of problems, imagine each worker contributing a portion of the work according to their speed. For instance, if one person can finish a job in 20 hours, they complete \(\frac{1}{20}\) of the job per hour. Similarly, someone who completes the same job in 30 hours does \(\frac{1}{30}\) per hour. The goal is to set up and solve an equation where all contributions add to one whole job.

Equation Setup for Work Problems

Setting up the equation correctly is crucial in solving work and time problems. The key is to express the job as fractions of the whole task completed per hour by each individual.

For Mark and Phil:

• Mark completes \(\frac{1}{20}\) of the job in one hour.
• Phil completes \(\frac{1}{30}\) of the job in one hour.

If they both work together for \(t\) hours, their combined contribution is described by \(\frac{t}{20} + \frac{t}{30}\). Afterward, if Mark works alone for an additional 5 hours, we add \(\frac{5}{20}\) from his solo effort.

The entire job must add up to 1. Therefore, the equation to solve is: \[\frac{t}{20} + \frac{t}{30} + \frac{5}{20} = 1.\] This equation represents the sum of work done by Mark and Phil together plus Mark alone, equating to the whole job.

Problem Solving Strategies

Solving these equations requires careful manipulation of fractions. Let's break it down:

• Combine fractions: Convert them to have a common denominator. In this case, the least common denominator is 60.
• Rewrite the equation: \(\frac{t}{20}\), \(\frac{t}{30}\), and \(\frac{5}{20}\) become \(\frac{3t}{60}\), \(\frac{2t}{60}\), and \(\frac{15}{60}\) respectively.
• Add and solve: The equation simplifies to \(\frac{5t + 15}{60} = 1\). Multiply both sides by 60 to remove the denominators.

Perform the math: Multiply 5t + 15 by 60 to get 60, then subtract 15 from both sides, resulting in \(5t = 45\). Divide by 5 to find \(t = 9\).

These steps demonstrate how systematic approaches—such as combining and simplifying fractions—help tackle algebraic problems efficiently.