Problem 45 - Algebra and Trigonometry

Question

Sharing a Job Candy and Tim share a paper route. It takes Candy 70 min to deliver all the papers, and it takes Tim 80 min. How long does it take the two when they work together?

Step-by-Step Solution

Answer

When working together, Candy and Tim take about 37.33 minutes.

Step 1: Determine Candy and Tim's Rates

To find out how fast Candy and Tim can deliver papers, calculate their work rates. Candy completes 1 job in 70 minutes, so her rate is \( \frac{1}{70} \) jobs per minute. Similarly, Tim's rate is \( \frac{1}{80} \) jobs per minute, since he completes 1 job in 80 minutes.

Step 2: Add Their Rates

Since Candy and Tim are working together, their rates combine. Add Candy's rate \( \frac{1}{70} \) and Tim's rate \( \frac{1}{80} \) to find their combined rate: \[ \frac{1}{70} + \frac{1}{80} = \frac{8}{560} + \frac{7}{560} = \frac{15}{560} = \frac{3}{112} \] This fraction represents the part of the job they complete together in one minute.

Step 3: Find the Time Taken Together

To determine how long it takes for both Candy and Tim to complete the paper route together, calculate the reciprocal of their combined rate:\[ \text{Time} = \frac{112}{3} \approx 37.33 \text{ minutes} \]Therefore, it takes Candy and Tim approximately 37.33 minutes to deliver all the papers when working together.

Key Concepts

Collaborative Work

Collaborative work involves two or more individuals joining forces to complete a task more efficiently than if they worked alone. In math problems that involve collaborative work, the goal is to determine how combining the efforts of all parties speeds up task completion. In some instances, each person or entity may have a different rate of work. For example, in our problem, Candy and Tim deliver newspapers individually at different speeds. When they work together, their combined efforts should decrease the total time needed to complete the task.

This requires us to think about how individual work rates can be added together to obtain a combined work rate. If each person's work rate is expressed as a fraction representing the part of the job they complete per unit of time, these can be combined to find how quickly the group finishes the task collectively. This concept is key in solving work rate problems involving collaborative efforts.

Calculating Work Rates

Calculating work rates involves expressing the amount of work completed by a person in a specific amount of time. In work problems, this quantity is typically represented as a fraction of a job completed per unit of time (e.g., per minute or per hour). For Candy and Tim, calculating their individual work rates is the first step towards understanding how they perform work together.

Candy completes her paper route in 70 minutes, which translates to a work rate of \( \frac{1}{70} \) jobs per minute. Similarly, Tim completes his job in 80 minutes, giving a work rate of \( \frac{1}{80} \) jobs per minute. By evaluating these rates, we have a clear picture of how much work each can accomplish alone. The next part of the problem involves combining these rates to find out how fast they work together, which leads us to the importance of adding these fractions effectively.

Fraction Addition

Adding fractions is a critical skill in solving work rate problems, especially when combining work rates of multiple people or machines to find an overall rate. When you have two fractions, like Candy’s \( \frac{1}{70} \) and Tim’s \( \frac{1}{80} \), you'll generally need a common denominator to add them.

A common technique is to convert each rate so they share a denominator. In this example, the least common multiple of 70 and 80 is 560. Thus, \( \frac{1}{70} \) is converted to \( \frac{8}{560} \) and \( \frac{1}{80} \) to \( \frac{7}{560} \). Adding these fractions gives \( \frac{15}{560} \). Simplifying further, \( \frac{15}{560} \) becomes \( \frac{3}{112} \), representing the fraction of the job they complete together per minute.

Mastering fraction addition helps calculate the combined work rate, offering a practical understanding of how individual contributions blend in collaborative settings.