Solving Work Rate Problems: A Carpenter's Challenge

In this article, we will explore how to solve a common work rate problem through a real-world example involving a carpenter and his chair-making skills. We will break down the problem using mathematical reasoning and apply it to similar scenarios, ensuring clarity and understanding for readers.

The Problem: Chair Production

The problem states: 'A carpenter can make 52 chairs in 13 days. How many days will it take to make 28 chairs?' This is a typical work rate problem, where we need to determine the time required to complete a specific number of tasks given the worker's productivity.

Calculating the Carpenter's Rate

To solve this problem, we first need to find the carpenter's rate of work. This can be calculated as follows:

Step 1: Calculate the rate of chair production

The carpenter makes 52 chairs in 13 days. To find the carpenter's rate of production, we use the formula:

Rate Total chairs / Total days

This gives us:

Rate frac{52 text{ chairs}}{13 text{ days}} 4 text{ chairs per day}

This means the carpenter can produce 4 chairs in one day.

Calculating the Time Needed for 28 Chairs

Now that we know the carpenter's rate of production, we can determine how many days it will take to produce 28 chairs:

Step 2: Calculate the time to make 28 chairs

We use the formula:

Time Total chairs / Rate of production

This gives us:

Time frac{28 text{ chairs}}{4 text{ chairs per day}} 7 text{ days}

Therefore, it will take the carpenter 7 days to make 28 chairs.

Alternative Method: Proportional Reasoning

We can also solve this problem using proportional reasoning, which involves setting up and solving a proportion:

Method 3: Using Proportional Reasoning

We start with the given information:

52 chairs in 13 days

To find the number of days required for 28 chairs, we set up the proportion:

frac{52 text{ chairs}}{13 text{ days}} frac{28 text{ chairs}}{x text{ days}}

We solve for ( x ) by cross-multiplying:

52x 28 times 13

x frac{28 times 13}{52} 7 text{ days}

Understanding the Concept

The problem can also be broken down as follows:

52 chairs take 13 days, so 1 chair is made in (frac{13}{52}) days.

For 28 chairs, the time required is:

28 times frac{13}{52} 7 text{ days}

Summary of Key Points

The carpenter's rate of production is 4 chairs per day. To produce 28 chairs, it takes 7 days. The problem can be solved using either the rate of production or proportional reasoning.

Conclusion

In this article, we explored how to solve a work rate problem using a practical example involving a carpenter and his chair-making skills. We demonstrated how to calculate the rate of production and how to determine the time required to complete a specific number of tasks. This method can be applied to various scenarios in real-life situations, making it a valuable tool for solving work rate problems.