In The Morning, Marco Sold 12 Cups Of Lemonade For $ 3 \$3 $3 . By The End Of The Day, He Had Earned $ 9 \$9 $9 . How Many Cups Of Lemonade Did He Sell In All?Kaycee Wrote The Proportion 12 3 = 9 C \frac{12}{3}=\frac{9}{c} 3 12 ​ = C 9 ​ For This Situation.

Introduction

In this article, we will explore a problem involving the sales of lemonade by Marco. We will use a proportion to solve for the total number of cups of lemonade sold by Marco. This problem is a great example of how proportions can be used to solve real-world problems.

The Problem

Marco sold 12 cups of lemonade for $\$3$ in the morning. By the end of the day, he had earned $\$9$. We need to find out how many cups of lemonade he sold in all.

Setting Up the Proportion

Kaycee wrote the proportion $\frac{12}{3}=\frac{9}{c}$ for this situation. This proportion states that the ratio of the number of cups of lemonade sold in the morning to the amount of money earned in the morning is equal to the ratio of the total number of cups of lemonade sold to the total amount of money earned.

Understanding the Proportion

Let's break down the proportion and understand what it means. The left-hand side of the proportion, $\frac{12}{3}$, represents the ratio of the number of cups of lemonade sold in the morning to the amount of money earned in the morning. This ratio is equal to 4, since 12 divided by 3 is equal to 4.

The right-hand side of the proportion, $\frac{9}{c}$, represents the ratio of the total number of cups of lemonade sold to the total amount of money earned. We are trying to find the value of $c$, which represents the total number of cups of lemonade sold.

Solving the Proportion

To solve the proportion, we can cross-multiply and set up an equation. We can multiply the left-hand side of the proportion by $c$ and the right-hand side of the proportion by 3.

$$12c = 9 \times 3$$

Simplifying the equation, we get:

$$12c = 27$$

Dividing both sides of the equation by 12, we get:

$$c = \frac{27}{12}$$

Simplifying the fraction, we get:

$$c = 2.25$$

Interpreting the Result

The result of the proportion is that Marco sold 2.25 cups of lemonade for every dollar he earned. However, since we are trying to find the total number of cups of lemonade sold, we need to multiply the result by the total amount of money earned.

The total amount of money earned is $\$9$, so we can multiply the result by 9:

$$2.25 \times 9 = 20.25$$

Conclusion

In conclusion, Marco sold a total of 20.25 cups of lemonade. However, since we cannot sell a fraction of a cup of lemonade, we can round down to the nearest whole number.

Rounding the Result

Rounding 20.25 down to the nearest whole number, we get:

$$20$$

Therefore, Marco sold a total of 20 cups of lemonade.

Real-World Applications

This problem is a great example of how proportions can be used to solve real-world problems. In this case, we used a proportion to solve for the total number of cups of lemonade sold by Marco. This type of problem can be applied to many real-world situations, such as calculating the cost of goods sold or determining the number of units sold.

Tips and Tricks

When solving proportions, it's essential to remember to cross-multiply and set up an equation. This will help you to solve for the unknown variable.

Additionally, when interpreting the result of the proportion, make sure to consider the context of the problem. In this case, we were trying to find the total number of cups of lemonade sold, so we needed to multiply the result by the total amount of money earned.

Common Mistakes

When solving proportions, one common mistake is to forget to cross-multiply and set up an equation. This can lead to incorrect results and make it difficult to solve the problem.

Another common mistake is to misinterpret the result of the proportion. Make sure to consider the context of the problem and multiply the result by the correct value.

Conclusion

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is often written in the form $\frac{a}{b} = \frac{c}{d}$, where $a$ and $b$ are the numbers in the first ratio, and $c$ and $d$ are the numbers 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 that are equal. You can do this by looking at the problem and identifying the relationships between the different numbers. For example, in the problem we solved earlier, the ratio of the number of cups of lemonade sold in the morning to the amount of money earned in the morning was equal to the ratio of the total number of cups of lemonade sold to the total amount of money earned.

Q: How do I solve a proportion?

A: To solve a proportion, you can cross-multiply and set up an equation. This involves multiplying the left-hand side of the proportion by the denominator of the right-hand side, and multiplying the right-hand side of the proportion by the denominator of the left-hand side. For example, in the problem we solved earlier, we cross-multiplied and set up the equation $12c = 9 \times 3$.

Q: What is the difference between a proportion and an equation?

A: A proportion is a statement that two ratios are equal, while an equation is a statement that two expressions are equal. While both proportions and equations can be used to solve problems, they are used in different ways. Proportions are often used to solve problems that involve ratios, while equations are often used to solve problems that involve unknown values.

Q: Can I use proportions to solve problems that involve fractions?

A: Yes, you can use proportions to solve problems that involve fractions. In fact, proportions are often used to solve problems that involve fractions because they allow you to set up a relationship between the different fractions.

Q: Can I use proportions to solve problems that involve decimals?

A: Yes, you can use proportions to solve problems that involve decimals. In fact, proportions are often used to solve problems that involve decimals because they allow you to set up a relationship between the different decimals.

Q: How do I know when to use a proportion and when to use an equation?

A: To determine whether to use a proportion or an equation, you need to look at the problem and identify the relationships between the different numbers. If the problem involves ratios, you may want to use a proportion. If the problem involves unknown values, you may want to use an equation.

Q: Can I use proportions to solve problems that involve multiple variables?

A: Yes, you can use proportions to solve problems that involve multiple variables. In fact, proportions are often used to solve problems that involve multiple variables because they allow you to set up a relationship between the different variables.

Q: How do I check my work when using proportions?

A: To check your work when using proportions, you need to make sure that the proportion is true. You can do this by plugging in the values you found into the proportion and checking to see if it is true. If the proportion is true, then you have found the correct solution.

Q: Can I use proportions to solve problems that involve real-world applications?

A: Yes, you can use proportions to solve problems that involve real-world applications. In fact, proportions are often used to solve problems that involve real-world applications because they allow you to set up a relationship between the different variables.

Conclusion

In conclusion, proportions are a powerful tool that can be used to solve a wide range of problems. By understanding how to set up and solve proportions, you can become proficient in solving problems that involve ratios, fractions, decimals, and multiple variables. Whether you are working on a math problem or a real-world application, proportions can help you to find the solution.