Complete The Equation:${ \frac{3}{5} = \frac{}{10} }$

Mar 11, 2025 by ADMIN 55 views

Solving the Equation: A Step-by-Step Guide

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Introduction

In mathematics, equations are a fundamental concept that help us understand and solve problems. 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 specific proportionality equation: $\frac{3}{5} = \frac{x}{10}$ . We will break down the solution step by step, using simple language and providing examples to illustrate each concept.

Understanding the Equation

The given equation is a proportionality equation, where two ratios are equal. The left-hand side of the equation is $\frac{3}{5}$ , and the right-hand side is $\frac{x}{10}$ . Our goal is to find the value of $x$ that makes the two ratios equal.

What is a Proportionality Equation?

A proportionality equation is a statement that two ratios are equal. In other words, if we have two quantities, $a$ and $b$ , and we know that $\frac{a}{c} = \frac{b}{d}$ , then we can say that $a$ is proportional to $b$ with a constant of proportionality $k = \frac{c}{d}$ .

Why is this Equation Important?

Solving proportionality equations is an essential skill in mathematics, as it helps us understand and solve problems in various fields, such as physics, engineering, economics, and more. In this article, we will focus on solving a specific proportionality equation, but the concepts and techniques we will learn can be applied to a wide range of problems.

Solving the Equation

To solve the equation $\frac{3}{5} = \frac{x}{10}$ , we can use the following steps:

Step 1: Cross-Multiply

The first step in solving a proportionality equation is to cross-multiply. This means that we multiply the numerator of the left-hand side by the denominator of the right-hand side, and vice versa.

$\frac{3}{5} = \frac{x}{10}$

Cross-multiplying gives us:

$3 \times 10 = 5 \times x$

Step 2: Simplify the Equation

Now that we have cross-multiplied, we can simplify the equation by multiplying the numbers.

$30 = 5x$

Step 3: Divide Both Sides by 5

To isolate $x$ , we need to divide both sides of the equation by 5.

$\frac{30}{5} = x$

Step 4: Simplify the Right-Hand Side

Now that we have divided both sides by 5, we can simplify the right-hand side of the equation.

$6 = x$

Conclusion

In this article, we have solved the equation $\frac{3}{5} = \frac{x}{10}$ using the steps of cross-multiplication, simplification, and division. We have learned that the value of $x$ that makes the two ratios equal is 6. This is an essential skill in mathematics, as it helps us understand and solve problems in various fields.

Frequently Asked Questions

Q: What is a proportionality equation?

A: A proportionality equation is a statement that two ratios are equal.

Q: Why is solving proportionality equations important?

A: Solving proportionality equations is an essential skill in mathematics, as it helps us understand and solve problems in various fields, such as physics, engineering, economics, and more.

Q: How do I solve a proportionality equation?

A: To solve a proportionality equation, you can use the steps of cross-multiplication, simplification, and division.

Additional Resources

For more information on solving proportionality equations, you can check out the following resources:

Khan Academy: Proportionality Equations
Mathway: Proportionality Equations
Wolfram Alpha: Proportionality Equations

Final Thoughts

Solving proportionality equations is an essential skill in mathematics, as it helps us understand and solve problems in various fields. In this article, we have learned how to solve a specific proportionality equation using the steps of cross-multiplication, simplification, and division. We have also learned that the value of $x$ that makes the two ratios equal is 6. With practice and patience, you can become proficient in solving proportionality equations and apply this skill to a wide range of problems.

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Introduction

In our previous article, we discussed how to solve proportionality equations using the steps of cross-multiplication, simplification, and division. However, we know that there are many more questions and concerns that readers may have. In this article, we will address some of the most frequently asked questions about solving proportionality equations.

Q&A

Q: What is a proportionality equation?

A: A proportionality equation is a statement that two ratios are equal. In other words, if we have two quantities, $a$ and $b$ , and we know that $\frac{a}{c} = \frac{b}{d}$ , then we can say that $a$ is proportional to $b$ with a constant of proportionality $k = \frac{c}{d}$ .

Q: Why is solving proportionality equations important?

A: Solving proportionality equations is an essential skill in mathematics, as it helps us understand and solve problems in various fields, such as physics, engineering, economics, and more.

Q: How do I solve a proportionality equation?

A: To solve a proportionality equation, you can use the steps of cross-multiplication, simplification, and division.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to solve proportionality equations. It involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa.

Q: How do I simplify an equation?

A: To simplify an equation, you can combine like terms and eliminate any unnecessary variables.

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

A: A proportionality equation is a statement that two ratios are equal, while an algebraic equation is a statement that two expressions are equal.

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

A: Yes, you can use a calculator to solve proportionality equations. However, it's always a good idea to check your work by hand to ensure that you have the correct solution.

Q: How do I know if I have the correct solution?

A: To check if you have the correct solution, you can plug your answer back into the original equation and see if it's true.

Q: What if I get stuck on a proportionality equation?

A: If you get stuck on a proportionality equation, don't worry! You can try breaking down the problem into smaller steps, or you can ask for help from a teacher or tutor.

Tips and Tricks

Tip 1: Read the problem carefully

Before you start solving a proportionality equation, make sure you read the problem carefully and understand what's being asked.

Tip 2: Use a diagram to visualize the problem

Using a diagram can help you visualize the problem and make it easier to understand.

Tip 3: Check your work

It's always a good idea to check your work by hand to ensure that you have the correct solution.

Tip 4: Practice, practice, practice!

The more you practice solving proportionality equations, the more comfortable you'll become with the steps and techniques.

Conclusion

Solving proportionality equations is an essential skill in mathematics, and with practice and patience, you can become proficient in solving these types of equations. In this article, we have addressed some of the most frequently asked questions about solving proportionality equations, and we have provided some tips and tricks to help you succeed.

Additional Resources

For more information on solving proportionality equations, you can check out the following resources:

Khan Academy: Proportionality Equations
Mathway: Proportionality Equations
Wolfram Alpha: Proportionality Equations

Final Thoughts

Solving proportionality equations is a skill that takes practice and patience to develop. However, with the right techniques and strategies, you can become proficient in solving these types of equations and apply this skill to a wide range of problems. Remember to read the problem carefully, use a diagram to visualize the problem, check your work, and practice, practice, practice!