Solve For X X X :${ \frac{14}{35} = \frac{x}{24} }$

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Introduction to Solving Equations with Fractions

When it comes to solving equations with fractions, it's essential to understand the concept of equivalent ratios. In this article, we will focus on solving the equation $\frac{14}{35} = \frac{x}{24}$, which involves simplifying fractions and finding the value of $x$. We will break down the solution into manageable steps, making it easier to understand and apply the concept to similar problems.

Understanding Equivalent Ratios

Equivalent ratios are fractions that have the same value, but may appear different. For example, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent ratios because they both represent the same value. To solve the equation $\frac{14}{35} = \frac{x}{24}$, we need to find the equivalent ratio that has the same value as $\frac{14}{35}$.

Simplifying Fractions

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder. In this case, the GCD of 14 and 35 is 7.

Simplifying the Left-Hand Side of the Equation

To simplify the left-hand side of the equation, we need to divide both the numerator and denominator by the GCD.

$\frac{14}{35} = \frac{14 \div 7}{35 \div 7} = \frac{2}{5}$

Simplifying the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we need to divide both the numerator and denominator by the GCD.

$\frac{x}{24} = \frac{x \div 1}{24 \div 1} = \frac{x}{24}$

Cross-Multiplying

Now that we have simplified both sides of the equation, we can cross-multiply to solve for $x$. Cross-multiplying involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa.

$2 \times 24 = 5 \times x$

Solving for $x$

To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by dividing both sides of the equation by 5.

$\frac{2 \times 24}{5} = x$

Calculating the Value of $x$

Now that we have isolated $x$, we can calculate its value.

$x = \frac{2 \times 24}{5} = \frac{48}{5} = 9.6$

Conclusion

In this article, we have solved the equation $\frac{14}{35} = \frac{x}{24}$ by simplifying fractions and cross-multiplying. We have found that the value of $x$ is $\frac{48}{5}$, which is equal to 9.6. This problem demonstrates the importance of understanding equivalent ratios and simplifying fractions when solving equations with fractions.

Frequently Asked Questions

• What is the greatest common divisor (GCD) of 14 and 35?
• How do you simplify a fraction?
• What is cross-multiplying?
• How do you solve for $x$ in an equation with fractions?

Additional Resources

• Khan Academy: Solving Equations with Fractions
• Mathway: Solving Equations with Fractions
• Wolfram Alpha: Solving Equations with Fractions

Final Thoughts

Solving equations with fractions requires a solid understanding of equivalent ratios and simplifying fractions. By following the steps outlined in this article, you can solve equations with fractions and find the value of $x$. Remember to always simplify fractions and cross-multiply to solve for $x$. With practice and patience, you will become proficient in solving equations with fractions.

Introduction

Solving equations with fractions can be a challenging task, but with the right guidance, it can be made easier. In this article, we will address some of the most frequently asked questions related to solving equations with fractions. Whether you are a student or a teacher, this article will provide you with the answers you need to tackle these types of problems.

Q: What is the greatest common divisor (GCD) of 14 and 35?

A: The greatest common divisor (GCD) of 14 and 35 is 7. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

Q: How do you simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator. Then, divide both the numerator and denominator by the GCD. For example, to simplify the fraction $\frac{14}{35}$, you would divide both the numerator and denominator by 7, resulting in $\frac{2}{5}$.

Q: What is cross-multiplying?

A: Cross-multiplying is a technique used to solve equations with fractions. It involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. For example, to solve the equation $\frac{14}{35} = \frac{x}{24}$, you would cross-multiply by multiplying 14 by 24 and 35 by x.

Q: How do you solve for $x$ in an equation with fractions?

A: To solve for $x$ in an equation with fractions, you need to follow these steps:

1. Simplify both sides of the equation.
2. Cross-multiply.
3. Isolate $x$ on one side of the equation.
4. Calculate the value of $x$.

Q: What is the difference between equivalent ratios and equivalent fractions?

A: Equivalent ratios and equivalent fractions are two related concepts. Equivalent ratios are fractions that have the same value, but may appear different. Equivalent fractions are fractions that have the same value and are in the same ratio.

Q: How do you determine if two fractions are equivalent?

A: To determine if two fractions are equivalent, you need to check if they have the same numerator and denominator, or if they can be simplified to the same fraction.

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

A: Some common mistakes to avoid when solving equations with fractions include:

• Not simplifying both sides of the equation.
• Not cross-multiplying correctly.
• Not isolating $x$ on one side of the equation.
• Not calculating the value of $x$ correctly.

Q: How can I practice solving equations with fractions?

A: You can practice solving equations with fractions by working through examples and exercises. You can also use online resources, such as Khan Academy or Mathway, to practice solving equations with fractions.

Q: What are some real-world applications of solving equations with fractions?

A: Solving equations with fractions has many real-world applications, including:

• Calculating proportions and ratios in cooking and recipes.
• Determining the cost of goods and services.
• Calculating interest rates and investments.
• Solving problems in physics and engineering.

Conclusion

Solving equations with fractions can be a challenging task, but with the right guidance, it can be made easier. By understanding the concepts of equivalent ratios, simplifying fractions, and cross-multiplying, you can solve equations with fractions and find the value of $x$. Remember to practice regularly and avoid common mistakes to become proficient in solving equations with fractions.

Additional Resources

• Khan Academy: Solving Equations with Fractions
• Mathway: Solving Equations with Fractions
• Wolfram Alpha: Solving Equations with Fractions

Final Thoughts

Solving equations with fractions requires a solid understanding of equivalent ratios and simplifying fractions. By following the steps outlined in this article, you can solve equations with fractions and find the value of $x$. Remember to always simplify fractions and cross-multiply to solve for $x$. With practice and patience, you will become proficient in solving equations with fractions.