Identify the correct equation

Oct 13, 2025 by windailys.com 135 views

Which Best Describes the Error That Sameer Made in Solving the Equation?

Understanding the Concept of Proportional Relationships

In mathematics, proportional relationships are a fundamental concept that helps us understand how different quantities are related to each other. When we say that two quantities are proportional, it means that one quantity is a constant multiple of the other. This concept is often represented using ratios or fractions, where the ratio of the two quantities remains constant.

The Given Equation and Sameer's Solution

The given equation is $\frac{3}{4} = \frac{x}{12}$ . This equation represents a proportional relationship between the two quantities, where the ratio of $\frac{3}{4}$ is equal to the ratio of $\frac{x}{12}$ . To solve for $x$ , Sameer multiplied both sides of the equation by 12, resulting in $9 = x$ .

Analyzing Sameer's Solution

At first glance, Sameer's solution may seem correct. However, upon closer inspection, we can see that there is an error in his solution. The error lies in the fact that Sameer multiplied both sides of the equation by 12, but he did not multiply the numerator of the fraction on the left-hand side by 12.

The Correct Solution

To solve for $x$ , we need to multiply both sides of the equation by 12, but we must also multiply the numerator of the fraction on the left-hand side by 12. This results in:

$\frac{3}{4} \cdot 12 = \frac{x}{12} \cdot 12$

Simplifying the left-hand side, we get:

$9 = x$

However, this is not the correct solution. The correct solution is:

$\frac{3}{4} \cdot 12 = \frac{x}{12} \cdot 12$

Simplifying the left-hand side, we get:

$9 = \frac{x}{12} \cdot 12$

Multiplying both sides by 12, we get:

$108 = x$

The Error in Sameer's Solution

The error in Sameer's solution is that he did not multiply the numerator of the fraction on the left-hand side by 12. This resulted in an incorrect solution, where $x$ was equal to 9 instead of 108.

Conclusion

Understanding the Importance of Proportional Relationships

Proportional relationships are an essential concept in mathematics, and understanding them is crucial for solving problems in various fields, such as science, engineering, and economics. When we encounter a problem that involves proportional relationships, we need to be able to identify the correct equation and solve for the unknown quantity.

Real-World Applications of Proportional Relationships

Proportional relationships have numerous real-world applications, such as:

Finance: When calculating interest rates or investment returns, proportional relationships are used to determine the amount of money that will be earned or lost.
Science: When studying the behavior of physical systems, proportional relationships are used to model the relationships between different variables.
Engineering: When designing systems or structures, proportional relationships are used to ensure that the system or structure is stable and efficient.

Tips for Solving Problems Involving Proportional Relationships

When solving problems involving proportional relationships, here are some tips to keep in mind:

Identify the correct equation: Make sure that you have identified the correct equation that models the proportional relationship between the two quantities.
Multiply both sides by the same value: When multiplying both sides of the equation by a value, make sure that you multiply both sides by the same value.
Simplify the equation: Simplify the equation by combining like terms and canceling out any common factors.

Conclusion

In conclusion, the error that Sameer made in solving the equation was that he did not use the correct equation to model the proportional relationship between the two quantities. He multiplied both sides of the equation by 12, but he did not multiply the numerator of the fraction on the left-hand side by 12. This resulted in an incorrect solution, where $x$ was equal to 9 instead of 108. By understanding the concept of proportional relationships and following the tips outlined above, we can ensure that we solve problems correctly and accurately.
Q&A: Understanding Proportional Relationships and Solving Equations

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about proportional relationships and solving equations.

Q: What is a proportional relationship?

A: A proportional relationship is a relationship between two quantities where one quantity is a constant multiple of the other. This means that if we multiply one quantity by a certain value, the other quantity will also be multiplied by the same value.

Q: How do I identify a proportional relationship?

A: To identify a proportional relationship, look for a ratio or fraction that remains constant between the two quantities. For example, if we have the equation $\frac{3}{4} = \frac{x}{12}$ , we can see that the ratio of $\frac{3}{4}$ is equal to the ratio of $\frac{x}{12}$ .

Q: How do I solve an equation involving a proportional relationship?

A: To solve an equation involving a proportional relationship, multiply both sides of the equation by the same value. For example, if we have the equation $\frac{3}{4} = \frac{x}{12}$ , we can multiply both sides by 12 to get $9 = x$ .

Q: What is the difference between a proportional relationship and a direct variation?

A: A proportional relationship and a direct variation are similar concepts, but they are not exactly the same thing. A direct variation is a specific type of proportional relationship where the ratio of the two quantities remains constant, but the quantities themselves are not necessarily equal.

Q: How do I determine if a relationship is proportional or not?

A: To determine if a relationship is proportional or not, look for a constant ratio or fraction between the two quantities. If the ratio or fraction remains constant, then the relationship is proportional.

Q: What are some real-world applications of proportional relationships?

A: Proportional relationships have numerous real-world applications, such as:

Finance: When calculating interest rates or investment returns, proportional relationships are used to determine the amount of money that will be earned or lost.
Science: When studying the behavior of physical systems, proportional relationships are used to model the relationships between different variables.
Engineering: When designing systems or structures, proportional relationships are used to ensure that the system or structure is stable and efficient.

Q: How do I use proportional relationships to solve problems in real-world situations?

A: To use proportional relationships to solve problems in real-world situations, identify the proportional relationship between the two quantities, and then use the equation to solve for the unknown quantity.

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

A: Some common mistakes to avoid when working with proportional relationships include:

Not identifying the correct equation: Make sure that you have identified the correct equation that models the proportional relationship between the two quantities.
Not multiplying both sides by the same value: When multiplying both sides of the equation by a value, make sure that you multiply both sides by the same value.
Not simplifying the equation: Simplify the equation by combining like terms and canceling out any common factors.

Conclusion

In conclusion, proportional relationships are an essential concept in mathematics, and understanding them is crucial for solving problems in various fields, such as science, engineering, and economics. By identifying the correct equation, multiplying both sides by the same value, and simplifying the equation, we can ensure that we solve problems correctly and accurately.