Answered

Lea and Marie are washing all the windows in their house. It would take Lea 80 minutes to wash the windows by herself, and it would take Marie 60 minutes to wash them by herself. Lea creates a table so she can find out how long it will take to wash the windows if they work together. What is the missing value in the table that represents the rate for Marie?




Rate (part/min)Time (min)Part of Windows Washed

Lea		$\frac{1}{80}$$t$$\frac{1}{80} t$

Marie		?$t$$\frac{1}{60} t$





A$\frac{1}{60}$
B$60$
C$1$

Verified Answer :

  • Define the work rate as the fraction of a job completed per unit time, given by \(\text{Rate} = \frac{1}{\text{Time}}\).
  • Using the information that Marie takes \(60\) minutes to finish the job, her work rate is \(\frac{1}{60}\) part/min.
  • Verify this with the table, where the amount of work done is \(\frac{1}{60}t\), confirming the rate is \(\frac{1}{60}\).
  • Select the matching option from the list, which is \(\boxed{\frac{1}{60}}\).

Explanation

  1. Understanding the Relationship Between Rate and Time

The problem asks us to determine the rate at which Marie washes windows, given the time it takes her to complete the task alone. Work rate is defined as the portion of a task completed per unit of time. For a task that takes a total time \(T\) to complete, the rate \(R\) is given by the formula \(R = \frac{1}{T}\). We are given that it takes Marie \(60\) minutes to wash all the windows by herself. Therefore, her work rate is \(R = \frac{1}{60}\) part of the windows per minute.

  1. Identifying the Missing Rate Value

We are provided with a table that shows the work done by Lea and Marie over a time period \(t\). For Marie, the table shows the time as \(t\) and the part of the windows washed as \(\frac{1}{60}t\). Since the "Part of Windows Washed" is calculated as \(\text{Rate} \times \text{Time}\), we have \(\text{Rate} \times t = \frac{1}{60}t\). Dividing both sides by \(t\) (assuming \(t > 0\)), we find that the rate for Marie is \(\frac{1}{60}\). This matches our calculation from the first step. Comparing this result with the given choices (\(\frac{1}{60}\), \(60\), \(1\)), the correct value is \(\frac{1}{60}\).

Examples

Understanding work rates is essential for project management and planning. For instance, if you and a friend are tasked with painting a room or assembling furniture, knowing your individual rates helps you predict how much time you will need when working together. By adding your individual rates (e.g., \(\frac{1}{A} + \frac{1}{B}\)), you can determine your combined work rate, allowing you to estimate precisely when a collaborative project will be finished. This logic applies to any scenario involving resource allocation and task scheduling.