Question

If three masons can build \(245 \mathrm{ft}\) of wall in 4.50 days, how many feet of wall can one mason build in a day? Assume that each mason works at the same rate, and that the same length of wall is built each day.

Short answer

One mason can build 54.44 ft of wall in a day.

Step-by-step solution

Determine the Total Work Done

First, calculate the total amount of work done by all three masons in 4.50 days. This is equal to the total length of the wall built by the three masons.

Calculate the Daily Work Rate for All Masons

Divide the total work done by the number of days to find the daily work rate for all three masons.

Find the Daily Work Rate for One Mason

Since each mason works at the same rate, divide the daily work rate for all masons by the number of masons to determine the daily work rate for one mason.

Key concepts

Proportional Reasoning

Proportional reasoning is a cornerstone of mathematical understanding, particularly when it comes to solving rate problems. It involves recognizing and working with the multiplicative relationship between quantities. In simpler terms, if one quantity increases or decreases, the other does so in a way that this ratio remains constant.

In our exercise, the concept of proportional reasoning allows us to deduce that if three masons are able to build a certain length of wall in a given timeframe, one mason would build a smaller proportion of that wall in the same timeframe. This kind of reasoning helps in scaling the work done up or down according to the number of workers or the time available, maintaining the ratio of work done to the corresponding quantity (number of workers or time).

Work Rate

Work rate in mathematics refers to the amount of work done per unit time. It's commonly used in problems dealing with productivity or efficiency. To solve work rate problems, one needs to understand how to apportion work according to different rates and times.

The exercise showcases a practical application of work rate. By calculating how much wall three masons build collectively in 4.5 days, then breaking it down to a daily rate, we use the concept of work rate to find an individual mason’s daily output. This involves understanding that work rate is a measure of one's capacity to accomplish a task within a given time, and is thus an important concept for figuring out individual contributions to a collective effort.

Mathematical Problem-Solving

Mathematical problem-solving is a sequential process that involves understanding the problem, devising a plan, carrying out that plan, and finally reviewing the solution. This process is central to tackling mathematical questions effectively and efficiently.

In solving the given rate problem, first, the total amount of work (the length of the wall built by all masons) is established. Next, this total is divided by the time taken to determine a daily rate for the group. Lastly, dividing the group's daily rate by the number of masons gives the daily rate for a single mason. This step-by-step approach showcases how to systematically solve problems by breaking them down into more manageable parts.

Unit Rate Calculations

Unit rate calculations involve determining the amount of quantity per one unit of another quantity. This type of calculation is often used to compare different rates and make informed decisions based on these comparisons.

In our context, the unit rate calculation is employed to determine how much wall a single mason builds per day. By first calculating the total length of the wall built by all masons and then dividing by the quantity of masons and the number of days, we isolate the quantity per one mason per day. This is an illustrative example of how unit rate calculations provide a standardized method for comparing productivity or efficiency on an individual basis.