Determine The Value Of $x$.$\frac{5}{x} = \frac{25}{65}$x = $

by ADMIN 65 views

Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. One of the most common types of equations is the proportionality equation, where two ratios are equal. In this article, we will focus on solving a proportionality equation involving fractions, and we will determine the value of the variable $x$ in the equation 5x=2565\frac{5}{x} = \frac{25}{65}.

Understanding the Equation

The given equation is 5x=2565\frac{5}{x} = \frac{25}{65}. To solve for $x$, we need to isolate the variable on one side of the equation. The equation involves two fractions, and we can start by cross-multiplying to eliminate the fractions.

Cross-Multiplying

Cross-multiplying is a technique used to eliminate fractions in an equation. It involves multiplying both sides of the equation by the denominators of the fractions. In this case, we will multiply both sides of the equation by $x$ and $65$.

5xâ‹…xâ‹…65=2565â‹…xâ‹…65\frac{5}{x} \cdot x \cdot 65 = \frac{25}{65} \cdot x \cdot 65

Simplifying the equation, we get:

5â‹…65=25â‹…x5 \cdot 65 = 25 \cdot x

Simplifying the Equation

Now that we have eliminated the fractions, we can simplify the equation further. We can start by multiplying the numbers on the left-hand side of the equation.

5â‹…65=3255 \cdot 65 = 325

So, the equation becomes:

325=25â‹…x325 = 25 \cdot x

Solving for $x$

To solve for $x$, we need to isolate the variable on one side of the equation. We can do this by dividing both sides of the equation by $25$.

32525=x\frac{325}{25} = x

Simplifying the equation, we get:

13=x13 = x

Conclusion

In this article, we solved a proportionality equation involving fractions to determine the value of the variable $x$. We started by cross-multiplying to eliminate the fractions, and then we simplified the equation to isolate the variable. Finally, we solved for $x$ by dividing both sides of the equation by $25$. The value of $x$ is $13$.

Tips and Tricks

  • When solving proportionality equations, it's essential to cross-multiply to eliminate the fractions.
  • Make sure to simplify the equation after cross-multiplying to isolate the variable.
  • When solving for $x$, always check your work by plugging the value back into the original equation.

Real-World Applications

Solving proportionality equations has many real-world applications, including:

  • Finance: When calculating interest rates or investment returns, proportionality equations are used to determine the value of investments.
  • Science: In physics and chemistry, proportionality equations are used to describe the relationships between variables, such as the relationship between force and acceleration.
  • Engineering: In engineering, proportionality equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Common Mistakes

  • When cross-multiplying, make sure to multiply both sides of the equation by the denominators of the fractions.
  • When simplifying the equation, make sure to combine like terms and eliminate any unnecessary variables.
  • When solving for $x$, make sure to check your work by plugging the value back into the original equation.

Final Thoughts

Solving proportionality equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve proportionality equations involving fractions and determine the value of unknown variables. Remember to always check your work and simplify the equation to isolate the variable. With practice and patience, you can become proficient in solving proportionality equations and apply them to real-world problems.

Introduction

Solving proportionality equations is a fundamental concept in mathematics that has many real-world applications. However, it can be challenging to understand and apply the concepts, especially for beginners. In this article, we will address some of the most frequently asked questions (FAQs) about solving proportionality equations.

Q: What is a proportionality equation?

A: A proportionality equation is a mathematical equation that describes the relationship between two or more variables. It is often represented as a ratio of two quantities, where the ratio is equal to a constant value.

Q: How do I solve a proportionality equation?

A: To solve a proportionality equation, you need to follow these steps:

  1. Cross-multiply to eliminate the fractions.
  2. Simplify the equation to isolate the variable.
  3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to eliminate fractions in an equation. It involves multiplying both sides of the equation by the denominators of the fractions.

Q: How do I know if I have a proportionality equation?

A: A proportionality equation is often represented as a ratio of two quantities, where the ratio is equal to a constant value. For example, the equation 5x=2565\frac{5}{x} = \frac{25}{65} is a proportionality equation because it describes the relationship between two ratios.

Q: Can I use a calculator to solve proportionality equations?

A: Yes, you can use a calculator to solve proportionality equations. However, it's essential to understand the underlying concepts and principles to ensure that you are using the calculator correctly.

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

A: Some common mistakes to avoid when solving proportionality equations include:

  • Not cross-multiplying to eliminate the fractions.
  • Not simplifying the equation to isolate the variable.
  • Not checking your work by plugging the value back into the original equation.

Q: How do I apply proportionality equations in real-world situations?

A: Proportionality equations have many real-world applications, including:

  • Finance: When calculating interest rates or investment returns, proportionality equations are used to determine the value of investments.
  • Science: In physics and chemistry, proportionality equations are used to describe the relationships between variables, such as the relationship between force and acceleration.
  • Engineering: In engineering, proportionality equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: Can I use proportionality equations to solve problems involving percentages?

A: Yes, you can use proportionality equations to solve problems involving percentages. For example, if you want to find the value of a 20% increase in a quantity, you can use a proportionality equation to determine the new value.

Q: How do I determine if a proportionality equation is linear or nonlinear?

A: A proportionality equation is linear if the ratio of the two quantities is constant. For example, the equation 5x=2565\frac{5}{x} = \frac{25}{65} is a linear proportionality equation because the ratio of the two quantities is constant. A proportionality equation is nonlinear if the ratio of the two quantities is not constant.

Q: Can I use proportionality equations to solve problems involving rates and ratios?

A: Yes, you can use proportionality equations to solve problems involving rates and ratios. For example, if you want to find the rate at which a quantity is increasing or decreasing, you can use a proportionality equation to determine the rate.

Conclusion

Solving proportionality equations is a fundamental concept in mathematics that has many real-world applications. By understanding the concepts and principles outlined in this article, you can apply proportionality equations to solve problems involving ratios, rates, and percentages. Remember to always check your work and simplify the equation to isolate the variable. With practice and patience, you can become proficient in solving proportionality equations and apply them to real-world problems.