Solving the Problem with a Proportion

Oct 13, 2025 by recoveryshake.com 19 views

Introduction

Proportions are a fundamental concept in mathematics that help us solve real-world problems involving ratios and percentages. In this article, we will explore how to use proportions to solve a common problem: finding the total earnings of an individual based on their hourly wage and the number of hours worked.

Understanding the Problem

Let's consider the problem presented in the question: Rosa earns $\$12$ per hour and works for 7.5 hours. We need to find the total amount she will earn in dollars, denoted by $x$ . To solve this problem, we can use a proportion, which is a statement that two ratios are equal.

What is a Proportion?

A proportion is a mathematical statement that two ratios are equal. It is written in the form:

\frac{a}{b} = \frac{c}{d}

where $a$ and $b$ are the two values in the first ratio, and $c$ and $d$ are the two values in the second ratio.

Solving the Problem with a Proportion

To solve the problem, we can set up a proportion using the information given. We know that Rosa earns $\$12$ per hour, so the ratio of her earnings to the number of hours she works is $\frac{12}{7.5}$ . We want to find the total amount she will earn, denoted by $x$ , so the ratio of her earnings to the number of hours she works is also $\frac{x}{7.5}$ .

We can set up the proportion as follows:

\frac{12}{7.5} = \frac{x}{7.5}

However, this is not the only possible proportion that can be used to solve the problem. We can also set up the proportion as follows:

\frac{1}{12} = \frac{7.5}{x}

\frac{1}{7.5} = \frac{12}{x}

Evaluating the Options

Now that we have set up the proportion, let's evaluate the options given in the question:

A. $\frac{x}{12} = \frac{1}{7.5}$

This option is not a valid proportion because it does not match the proportion we set up.

B. $\frac{1}{12} = \frac{x}{7.5}$

This option is a valid proportion because it matches the proportion we set up.

C. $\frac{1}{7.5} = \frac{12}{x}$

This option is also a valid proportion because it matches the proportion we set up.

Conclusion

In conclusion, we have seen how to use proportions to solve a real-world problem involving hourly wages and the number of hours worked. We have also evaluated the options given in the question and determined which ones are valid proportions. By using proportions, we can solve a wide range of problems involving ratios and percentages.

Tips and Tricks

When setting up a proportion, make sure to use the correct ratios and values.
Use the proportion to solve for the unknown value.
Check your work by plugging in the values and making sure the proportion is true.

Common Mistakes

Not using the correct ratios and values when setting up the proportion.
Not solving for the unknown value.
Not checking the work by plugging in the values and making sure the proportion is true.

Real-World Applications

Proportions have many real-world applications, including:

Finding the total cost of an item based on its price and the number of items purchased.
Determining the amount of time it will take to complete a task based on the rate of work and the amount of work to be done.
Calculating the interest on a loan based on the principal amount, interest rate, and time period.

Final Thoughts

Q: What is a proportion?

A: A proportion is a mathematical statement that two ratios are equal. It is written in the form:

\frac{a}{b} = \frac{c}{d}

where $a$ and $b$ are the two values in the first ratio, and $c$ and $d$ are the two values in the second ratio.

Q: How do I set up a proportion?

A: To set up a proportion, you need to identify the two ratios and the unknown value. Then, you can write the proportion in the form:

\frac{a}{b} = \frac{c}{d}

where $a$ and $b$ are the two values in the first ratio, and $c$ and $d$ are the two values in the second ratio.

Q: What is the difference between a proportion and a ratio?

A: A ratio is a comparison of two values, while a proportion is a statement that two ratios are equal. For example, the ratio of 2:3 is a comparison of two values, while the proportion $\frac{2}{3} = \frac{4}{6}$ is a statement that two ratios are equal.

Q: How do I solve a proportion?

A: To solve a proportion, you need to find the unknown value. You can do this by cross-multiplying and solving for the unknown value.

Q: What are some real-world applications of proportions?

A: Proportions have many real-world applications, including:

Finding the total cost of an item based on its price and the number of items purchased.
Determining the amount of time it will take to complete a task based on the rate of work and the amount of work to be done.
Calculating the interest on a loan based on the principal amount, interest rate, and time period.

Q: How do I check my work when solving a proportion?

A: To check your work, you can plug in the values and make sure the proportion is true. You can also use a calculator to check your work.

Q: What are some common mistakes to avoid when solving proportions?

A: Some common mistakes to avoid when solving proportions include:

Not using the correct ratios and values when setting up the proportion.
Not solving for the unknown value.
Not checking the work by plugging in the values and making sure the proportion is true.

Q: How do I use proportions to solve problems involving percentages?

A: To use proportions to solve problems involving percentages, you can set up a proportion using the percentage and the total value. For example, if you want to find 20% of 100, you can set up the proportion:

\frac{20}{100} = \frac{x}{100}

where $x$ is the unknown value.

Q: How do I use proportions to solve problems involving rates?

A: To use proportions to solve problems involving rates, you can set up a proportion using the rate and the time period. For example, if you want to find the distance traveled based on a rate of 60 miles per hour and a time period of 2 hours, you can set up the proportion:

\frac{60}{2} = \frac{x}{2}

where $x$ is the unknown value.

Q: What are some advanced applications of proportions?

A: Some advanced applications of proportions include:

Using proportions to solve problems involving exponential growth and decay.
Using proportions to solve problems involving trigonometry and geometry.
Using proportions to solve problems involving calculus and differential equations.

Conclusion

In conclusion, proportions are a powerful tool for solving real-world problems involving ratios and percentages. By using proportions, we can solve a wide range of problems and make informed decisions. Remember to use the correct ratios and values when setting up the proportion, solve for the unknown value, and check your work by plugging in the values and making sure the proportion is true.