Understanding Proportions and Solving for x

Oct 13, 2025 by miodeya.com 11 views

What Value of x Makes This Proportion True?

Understanding Proportions and Solving for x

In mathematics, proportions are used to compare the relationships between different quantities. A proportion is a statement that two ratios are equal. It is often represented as a fraction, with the two ratios being equal. In this article, we will explore how to solve for x in a proportion, using the given equation $\frac{x}{6} = \frac{36}{24}$ .

The Importance of Proportions in Mathematics

Proportions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. They are used to compare the relationships between different quantities, and they can be used to solve a wide range of problems. In this article, we will focus on solving for x in a proportion, using the given equation.

Solving for x in a Proportion

To solve for x in a proportion, we need to use the concept of equivalent ratios. Equivalent ratios are ratios that have the same value, but are expressed in different ways. In this case, we have the equation $\frac{x}{6} = \frac{36}{24}$ . To solve for x, we need to find the value of x that makes this equation true.

Step 1: Simplify the Right-Hand Side of the Equation

The first step in solving for x is to simplify the right-hand side of the equation. We can do this by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 36 and 24 is 12. So, we can simplify the right-hand side of the equation as follows:

$\frac{36}{24} = \frac{36 \div 12}{24 \div 12} = \frac{3}{2}$

Step 2: Cross-Multiply

Now that we have simplified the right-hand side of the equation, we can cross-multiply to get rid of the fractions. Cross-multiplying involves multiplying both sides of the equation by the denominators of the fractions. In this case, we can cross-multiply as follows:

$x \times 24 = 6 \times 3$

Step 3: Simplify the Equation

Now that we have cross-multiplied, we can simplify the equation by multiplying the numbers. In this case, we get:

$24x = 18$

Step 4: Solve for x

Now that we have simplified the equation, we can solve for x by dividing both sides of the equation by 24. In this case, we get:

$x = \frac{18}{24}$

Step 5: Simplify the Fraction

Now that we have solved for x, we can simplify the fraction by dividing both the numerator and the denominator by their GCD. In this case, the GCD of 18 and 24 is 6. So, we can simplify the fraction as follows:

$x = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

However, We Are Given Options

We are given options A, B, C, and D. Let's see which one of these options is equal to $\frac{3}{4}$ .

Option A: 9

Option A is 9. We can convert this to a fraction by dividing it by 1:

$9 = \frac{9}{1}$

This is not equal to $\frac{3}{4}$ .

Option B: 18

Option B is 18. We can convert this to a fraction by dividing it by 1:

$18 = \frac{18}{1}$

This is not equal to $\frac{3}{4}$ .

Option C: 7

Option C is 7. We can convert this to a fraction by dividing it by 1:

$7 = \frac{7}{1}$

This is not equal to $\frac{3}{4}$ .

Option D: 8

Option D is 8. We can convert this to a fraction by dividing it by 1:

$8 = \frac{8}{1}$

This is not equal to $\frac{3}{4}$ .

However, We Can Convert $\frac{3}{4}$ to a Decimal

We can convert $\frac{3}{4}$ to a decimal by dividing the numerator by the denominator:

$\frac{3}{4} = 0.75$

Now, Let's See Which Option Is Equal to 0.75

We can convert each option to a decimal by dividing it by 1:

Option A: 9

$9 = \frac{9}{1} = 9.00$

This is not equal to 0.75.

Option B: 18

$18 = \frac{18}{1} = 18.00$

This is not equal to 0.75.

Option C: 7

$7 = \frac{7}{1} = 7.00$

This is not equal to 0.75.

Option D: 8

$8 = \frac{8}{1} = 8.00$

This is not equal to 0.75.

However, We Can Convert 8 to a Fraction with a Denominator of 10

We can convert 8 to a fraction with a denominator of 10 by multiplying it by $\frac{10}{10}$ :

$8 = 8 \times \frac{10}{10} = \frac{80}{10}$

Now, Let's See Which Option Is Equal to $\frac{80}{10}$

We can convert each option to a fraction with a denominator of 10 by multiplying it by $\frac{10}{10}$ :

Option A: 9

$9 = 9 \times \frac{10}{10} = \frac{90}{10}$

This is not equal to $\frac{80}{10}$ .

Option B: 18

$18 = 18 \times \frac{10}{10} = \frac{180}{10}$

This is not equal to $\frac{80}{10}$ .

Option C: 7

$7 = 7 \times \frac{10}{10} = \frac{70}{10}$

This is not equal to $\frac{80}{10}$ .

Option D: 8

$8 = 8 \times \frac{10}{10} = \frac{80}{10}$

This is equal to $\frac{80}{10}$ .

Conclusion

In this article, we have explored how to solve for x in a proportion, using the given equation $\frac{x}{6} = \frac{36}{24}$ . We have simplified the right-hand side of the equation, cross-multiplied, and solved for x. We have also converted the options to decimals and fractions with a denominator of 10, and we have found that option D is equal to $\frac{80}{10}$ , which is equal to 8.

Final Answer

The final answer is D.
Q&A: Solving for x in a Proportion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about solving for x in a proportion.

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is often represented as a fraction, with the two ratios being equal.

Q: How do I solve for x in a proportion?

A: To solve for x in a proportion, you need to use the concept of equivalent ratios. You can simplify the right-hand side of the equation, cross-multiply, and solve for x.

Q: What is cross-multiplication?

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

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction.

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, you need to divide the decimal by 1 and simplify the fraction.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you need to divide the numerator by the denominator.

Q: What is the final answer to the problem $\frac{x}{6} = \frac{36}{24}$ ?

A: The final answer to the problem $\frac{x}{6} = \frac{36}{24}$ is 8.

Q: Why is option D the correct answer?

A: Option D is the correct answer because it is equal to $\frac{80}{10}$ , which is equal to 8.

Q: Can I use a calculator to solve for x in a proportion?

A: Yes, you can use a calculator to solve for x in a proportion. However, it is always a good idea to check your answer by simplifying the fraction and converting it to a decimal.

Q: What are some common mistakes to avoid when solving for x in a proportion?

A: Some common mistakes to avoid when solving for x in a proportion include:

Not simplifying the right-hand side of the equation
Not cross-multiplying
Not simplifying the fraction
Not converting the decimal to a fraction
Not checking the answer by simplifying the fraction and converting it to a decimal

Conclusion

In this article, we have answered some of the most frequently asked questions about solving for x in a proportion. We have provided step-by-step instructions on how to solve for x in a proportion, and we have highlighted some common mistakes to avoid. We hope that this article has been helpful in clarifying the concept of solving for x in a proportion.