Madison Can Paint \[$\frac{3}{5}\$\] Of A Wall In 40 Minutes. Complete The Following Sentence To Describe How Fast Madison Can Paint Using A Unit Rate Of Walls Per Hour.Madison Can Paint \[$\qquad\$\] Wall(s) In \[$\qquad\$\]

Introduction

In this article, we will explore the concept of unit rates and how to apply them to real-world problems, such as calculating Madison's painting speed. We will use the given information about Madison's painting ability to determine how many walls she can paint in an hour.

Understanding Unit Rates

A unit rate is a ratio that compares two quantities, usually expressed as a rate per unit of time or distance. In the context of this problem, we want to find out how many walls Madison can paint in one hour. To do this, we need to convert the given rate from minutes to hours.

Converting Time Units

Madison can paint $\frac{3}{5}$ of a wall in 40 minutes. To convert this rate to hours, we need to divide the number of minutes by 60, since there are 60 minutes in an hour.

$$40 \text{ minutes} \div 60 = \frac{2}{3} \text{ hours}$$

Calculating the Unit Rate

Now that we have converted the time units, we can calculate the unit rate of walls per hour. To do this, we need to divide the number of walls painted by the time it takes to paint them.

$$\text{Unit Rate} = \frac{\text{Number of Walls}}{\text{Time}} = \frac{\frac{3}{5}}{\frac{2}{3}}$$

Simplifying the Unit Rate

To simplify the unit rate, we can multiply the numerator and denominator by the reciprocal of the denominator.

$$\frac{\frac{3}{5}}{\frac{2}{3}} = \frac{3}{5} \times \frac{3}{2} = \frac{9}{10}$$

Interpreting the Unit Rate

The unit rate of $\frac{9}{10}$ means that Madison can paint $\frac{9}{10}$ of a wall in one hour. To find the number of walls she can paint in an hour, we can multiply the unit rate by the total number of hours.

Calculating the Number of Walls Painted

Since we want to find the number of walls Madison can paint in one hour, we can multiply the unit rate by 1.

$$\text{Number of Walls} = \frac{9}{10} \times 1 = \frac{9}{10}$$

Conclusion

In this article, we calculated Madison's painting speed using a unit rate of walls per hour. We converted the given rate from minutes to hours, calculated the unit rate, and simplified it to find the number of walls she can paint in an hour. The result is $\frac{9}{10}$ of a wall per hour.

Discussion

• What is the significance of unit rates in real-world problems?
• How can we apply unit rates to other problems, such as calculating the cost of goods or services?
• What are some common applications of unit rates in mathematics and science?

Real-World Applications

Unit rates have many real-world applications, including:

• Calculating the cost of goods or services
• Determining the amount of time it takes to complete a task
• Finding the number of items that can be produced in a given time period
• Comparing the efficiency of different systems or processes

Mathematical Concepts

This article covers the following mathematical concepts:

• Unit rates
• Converting time units
• Simplifying fractions
• Multiplying fractions

Science and Engineering

Unit rates are used in many scientific and engineering applications, including:

• Calculating the speed of objects
• Determining the amount of energy required to complete a task
• Finding the number of items that can be produced in a given time period
• Comparing the efficiency of different systems or processes

Conclusion

Q: What is a unit rate?

A: A unit rate is a ratio that compares two quantities, usually expressed as a rate per unit of time or distance. In the context of this problem, we want to find out how many walls Madison can paint in one hour.

Q: Why do we need to convert the time units from minutes to hours?

A: We need to convert the time units from minutes to hours because the problem asks us to find the number of walls Madison can paint in one hour. If we don't convert the time units, we won't be able to compare the number of walls painted to the time it takes to paint them.

Q: How do we calculate the unit rate?

A: To calculate the unit rate, we need to divide the number of walls painted by the time it takes to paint them. In this case, we divide $\frac{3}{5}$ by $\frac{2}{3}$.

Q: Why do we multiply the numerator and denominator by the reciprocal of the denominator?

A: We multiply the numerator and denominator by the reciprocal of the denominator to simplify the fraction. This is a common technique used to simplify fractions.

Q: What does the unit rate of $\frac{9}{10}$ mean?

A: The unit rate of $\frac{9}{10}$ means that Madison can paint $\frac{9}{10}$ of a wall in one hour.

Q: How do we find the number of walls Madison can paint in one hour?

A: To find the number of walls Madison can paint in one hour, we can multiply the unit rate by the total number of hours. In this case, we multiply $\frac{9}{10}$ by 1.

Q: What are some common applications of unit rates in mathematics and science?

A: Unit rates have many real-world applications, including:

• Calculating the cost of goods or services
• Determining the amount of time it takes to complete a task
• Finding the number of items that can be produced in a given time period
• Comparing the efficiency of different systems or processes

Q: How can we apply unit rates to other problems?

A: We can apply unit rates to other problems by following these steps:

1. Identify the quantities to be compared
2. Convert the time units to a common unit
3. Calculate the unit rate
4. Simplify the unit rate
5. Apply the unit rate to the problem at hand

Q: What are some common mistakes to avoid when working with unit rates?

A: Some common mistakes to avoid when working with unit rates include:

• Failing to convert the time units to a common unit
• Not simplifying the unit rate
• Applying the unit rate incorrectly

Q: How can we check our work when working with unit rates?

A: We can check our work by:

• Verifying that the unit rate is correct
• Applying the unit rate to a different problem to see if it yields the same result
• Checking the units to ensure that they are consistent

Conclusion

In conclusion, unit rates are an important mathematical concept that has many real-world applications. By understanding how to calculate and simplify unit rates, we can apply them to a wide range of problems, from calculating the cost of goods or services to determining the amount of time it takes to complete a task.