A Company Sold 3,000 Computers In One Month, But 12 Were Returned. If 3,500 Were Sold The Next Month, The Company Would Expect 14 To Be Returned.What Is A Valid Proportion To Represent The Problem?A. { \frac{3,000}{12} = \frac{14}{3,500}$}$B.

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Understanding Proportional Relationships in Business

In the world of business, understanding proportional relationships is crucial for making informed decisions and predicting future outcomes. A company's sales and returns can be a great example of this concept. In this article, we will explore a problem involving a company's sales and returns, and how to represent it as a valid proportion.

The Problem

A company sold 3,000 computers in one month, but 12 were returned. If 3,500 were sold the next month, the company would expect 14 to be returned. What is a valid proportion to represent the problem?

Analyzing the Situation

To solve this problem, we need to understand the relationship between the number of computers sold and the number of returns. Let's start by analyzing the given information:

  • 3,000 computers were sold in the first month, and 12 were returned.
  • 3,500 computers were sold in the second month, and 14 are expected to be returned.

We can see that the number of returns is proportional to the number of computers sold. This means that the ratio of returns to sales is constant.

Representing the Problem as a Proportion

To represent this problem as a proportion, we need to set up a ratio that shows the relationship between the number of returns and the number of computers sold. Let's use the following variables:

  • x = number of computers sold
  • y = number of returns

We can set up the proportion as follows:

yx=123,000\frac{y}{x} = \frac{12}{3,000}

This proportion shows that the ratio of returns to sales is constant, and it can be used to predict the number of returns for a given number of sales.

Comparing the Proportions

Now, let's compare the proportion we just set up with the one given in the problem:

3,00012=143,500\frac{3,000}{12} = \frac{14}{3,500}

At first glance, this proportion may seem valid, but it's actually not. The problem is that the ratio of returns to sales is not constant in this proportion. The number of returns is not proportional to the number of computers sold.

A Valid Proportion

To find a valid proportion, we need to set up a ratio that shows the relationship between the number of returns and the number of computers sold. Let's use the following variables:

  • x = number of computers sold
  • y = number of returns

We can set up the proportion as follows:

yx=123,000\frac{y}{x} = \frac{12}{3,000}

This proportion shows that the ratio of returns to sales is constant, and it can be used to predict the number of returns for a given number of sales.

Conclusion

In conclusion, a valid proportion to represent the problem is:

yx=123,000\frac{y}{x} = \frac{12}{3,000}

This proportion shows that the ratio of returns to sales is constant, and it can be used to predict the number of returns for a given number of sales.

Discussion

This problem is a great example of how proportional relationships can be used in business to make informed decisions. By understanding the relationship between the number of computers sold and the number of returns, the company can predict future outcomes and make adjustments to their sales strategy.

Real-World Applications

Proportional relationships are used in many real-world applications, including:

  • Business: Understanding proportional relationships can help businesses make informed decisions about pricing, production, and sales.
  • Science: Proportional relationships are used to describe the relationships between variables in scientific experiments.
  • Engineering: Proportional relationships are used to design and optimize systems, such as electrical circuits and mechanical systems.

Final Thoughts

In conclusion, proportional relationships are an important concept in mathematics and have many real-world applications. By understanding proportional relationships, we can make informed decisions and predict future outcomes. The problem presented in this article is a great example of how proportional relationships can be used in business to make informed decisions.

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Q&A: Understanding Proportional Relationships in Business

In our previous article, we explored a problem involving a company's sales and returns, and how to represent it as a valid proportion. In this article, we will answer some frequently asked questions about proportional relationships in business.

Q: What is a proportional relationship?

A: A proportional relationship is a relationship between two variables where the ratio of the variables is constant. In other words, if you multiply or divide one variable by a certain factor, the other variable will also be multiplied or divided by the same factor.

Q: How do I determine if a relationship is proportional?

A: To determine if a relationship is proportional, you need to check if the ratio of the variables is constant. You can do this by dividing the two variables and checking if the result is the same for all values of the variables.

Q: What is the difference between a proportional relationship and a non-proportional relationship?

A: A proportional relationship is a relationship where the ratio of the variables is constant, while a non-proportional relationship is a relationship where the ratio of the variables is not constant.

Q: How do I represent a proportional relationship as a proportion?

A: To represent a proportional relationship as a proportion, you need to set up a ratio that shows the relationship between the two variables. The proportion should be in the form of:

yx=ab\frac{y}{x} = \frac{a}{b}

where y is the dependent variable, x is the independent variable, a is the constant ratio, and b is the constant multiplier.

Q: What is the significance of proportional relationships in business?

A: Proportional relationships are significant in business because they help companies make informed decisions about pricing, production, and sales. By understanding proportional relationships, companies can predict future outcomes and make adjustments to their sales strategy.

Q: Can you give an example of a proportional relationship in business?

A: Yes, here's an example:

A company sells 100 units of a product per day at a price of $10 per unit. If the company wants to increase sales by 20%, how many units will it need to sell per day at the same price?

To solve this problem, we need to set up a proportional relationship between the number of units sold and the price per unit. Let's say the number of units sold is x and the price per unit is y. We can set up the proportion as follows:

xy=10010\frac{x}{y} = \frac{100}{10}

Since the company wants to increase sales by 20%, we need to multiply the number of units sold by 1.2 (which is 20% more than 1). So, the new number of units sold per day will be:

x=100×1.2=120x = 100 \times 1.2 = 120

Therefore, the company will need to sell 120 units per day at the same price to increase sales by 20%.

Q: Can you give another example of a proportional relationship in business?

A: Yes, here's another example:

A company has a production cost of $50 per unit and sells the product at a price of $100 per unit. If the company wants to increase production costs by 15%, how much will it need to increase the price per unit?

To solve this problem, we need to set up a proportional relationship between the production cost and the price per unit. Let's say the production cost is x and the price per unit is y. We can set up the proportion as follows:

xy=50100\frac{x}{y} = \frac{50}{100}

Since the company wants to increase production costs by 15%, we need to multiply the production cost by 1.15 (which is 15% more than 1). So, the new production cost per unit will be:

x=50×1.15=57.50x = 50 \times 1.15 = 57.50

Therefore, the company will need to increase the price per unit by $7.50 to increase production costs by 15%.

Conclusion

In conclusion, proportional relationships are an important concept in business that can help companies make informed decisions about pricing, production, and sales. By understanding proportional relationships, companies can predict future outcomes and make adjustments to their sales strategy. We hope this Q&A article has helped you understand proportional relationships in business better.

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Additional Resources