The Recipe Makes Enough Stew For 6 People, And The Ingredients Cost $ 9.75 \$9.75 $9.75 . How Much Would It Cost To Serve 24 People?A Student Solved The Problem Using The Proportion 6 9.75 = C 24 \frac{6}{9.75} = \frac{c}{24} 9.75 6 = 24 C . Explain The Error In This

Mar 9, 2025 by ADMIN 272 views

The Recipe for a Costly Mistake: Understanding Proportions in Mathematics

In mathematics, proportions are used to solve problems involving ratios and rates. However, when dealing with proportions, it's essential to understand the concept and apply it correctly. In this article, we'll explore a common mistake made by a student while solving a problem involving proportions and discuss the error in their approach.

A recipe makes enough stew for 6 people, and the ingredients cost $\$9.75$ . How much would it cost to serve 24 people? A student solved the problem using the proportion $\frac{6}{9.75} = \frac{c}{24}$ , where $c$ represents the cost of serving 24 people.

The student's approach to solving the problem is based on the concept of proportions. However, the error lies in the fact that the student is trying to find the cost of serving 24 people by setting up a proportion that relates the number of people to the cost. This is incorrect because the proportion should relate the number of people to the amount of ingredients or the cost per person.

A proportion is a statement that two ratios are equal. In this case, the student set up the proportion $\frac{6}{9.75} = \frac{c}{24}$ . However, this proportion is not correct because it's trying to relate the number of people to the cost, rather than the number of people to the amount of ingredients or the cost per person.

To solve this problem, we need to find the cost per person and then multiply it by the number of people. Let's break it down step by step:

Find the cost per person: To find the cost per person, we need to divide the total cost by the number of people. In this case, the total cost is $\$9.75$ , and the number of people is 6. So, the cost per person is $\frac{9.75}{6} = \$1.625$ .
Multiply the cost per person by the number of people: Now that we have the cost per person, we can multiply it by the number of people to find the total cost. In this case, the number of people is 24, so the total cost is $24 \times \$1.625 = \$39$ .

In conclusion, the student's approach to solving the problem was incorrect because they set up a proportion that related the number of people to the cost, rather than the number of people to the amount of ingredients or the cost per person. To solve this problem, we need to find the cost per person and then multiply it by the number of people. By following this approach, we can find the correct answer, which is $\$39$ .

Understanding proportions is crucial in mathematics, especially when dealing with ratios and rates. By applying proportions correctly, we can solve problems involving proportions and rates. However, it's essential to understand the concept and apply it correctly to avoid making mistakes.

Proportions have numerous real-world applications, including finance, engineering, and science. In finance, proportions are used to calculate interest rates and investment returns. In engineering, proportions are used to design and build structures and machines. In science, proportions are used to understand and describe natural phenomena.

When dealing with proportions, there are several common mistakes to avoid:

Setting up the wrong proportion: Make sure to set up the proportion correctly by relating the correct ratios.
Not understanding the concept: Take the time to understand the concept of proportions and how to apply it correctly.
Not checking the units: Make sure to check the units of the ratios to ensure that they are consistent.

In conclusion, understanding proportions is crucial in mathematics, and it's essential to apply it correctly to avoid making mistakes. By following the correct approach and avoiding common mistakes, we can solve problems involving proportions and rates.
The Recipe for a Costly Mistake: Understanding Proportions in Mathematics

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It's a way of expressing a relationship between two quantities.

Q: How do I set up a proportion?

A: To set up a proportion, you need to identify the two ratios that are equal. For example, if you want to find the cost of serving 24 people, you would set up the proportion $\frac{6}{9.75} = \frac{c}{24}$ , where $c$ represents the cost of serving 24 people.

Q: What's the difference between a proportion and a ratio?

A: A ratio is a comparison of two quantities, while a proportion is a statement that two ratios are equal. For example, the ratio of 6 to 9.75 is a comparison of two quantities, while the proportion $\frac{6}{9.75} = \frac{c}{24}$ 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 value of the unknown quantity. In the case of the proportion $\frac{6}{9.75} = \frac{c}{24}$ , you would multiply the two ratios together to find the value of $c$ . This would give you the cost of serving 24 people.

Q: What's the importance of understanding proportions?

A: Understanding proportions is crucial in mathematics, especially when dealing with ratios and rates. By applying proportions correctly, you can solve problems involving proportions and rates. This is essential in many real-world applications, including finance, engineering, and science.

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

A: Some common mistakes to avoid when working with proportions include:

Setting up the wrong proportion: Make sure to set up the proportion correctly by relating the correct ratios.
Not understanding the concept: Take the time to understand the concept of proportions and how to apply it correctly.
Not checking the units: Make sure to check the units of the ratios to ensure that they are consistent.

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

A: To check your work when solving a proportion, you should:

Verify that the ratios are equal: Make sure that the two ratios are equal.
Check the units: Make sure that the units of the ratios are consistent.
Check the solution: Make sure that the solution is reasonable and makes sense in the context of the problem.

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

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

Finance: Proportions are used to calculate interest rates and investment returns.
Engineering: Proportions are used to design and build structures and machines.
Science: Proportions are used to understand and describe natural phenomena.

In conclusion, understanding proportions is crucial in mathematics, and it's essential to apply it correctly to avoid making mistakes. By following the correct approach and avoiding common mistakes, you can solve problems involving proportions and rates.