Select The Correct Answer.If 3 4 \frac{3}{4} 4 3 ​ And 12 X \frac{12}{x} X 12 ​ Are Equivalent, What Is The Value Of X X X ?A. 48 B. 36 C. 24 D. 16

Understanding Equivalent Fractions

Equivalent fractions are fractions that have the same value, but may appear different. In other words, two fractions are equivalent if they represent the same proportion or ratio. For example, the fractions $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent because they both represent the same proportion of the whole.

The Problem at Hand

We are given two fractions: $\frac{3}{4}$ and $\frac{12}{x}$. We are told that these two fractions are equivalent, and we need to find the value of $x$. To do this, we can use the concept of equivalent fractions and algebraic manipulation.

Setting Up the Equation

Since the two fractions are equivalent, we can set up an equation to represent this relationship. We can start by setting the two fractions equal to each other:

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

Cross-Multiplying

To solve for $x$, we can cross-multiply the two fractions. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa:

$$3x = 12 \times 4$$

Simplifying the Equation

Now, we can simplify the equation by multiplying $12$ and $4$:

$$3x = 48$$

Solving for x

To solve for $x$, we can divide both sides of the equation by $3$:

$$x = \frac{48}{3}$$

Evaluating the Expression

Now, we can evaluate the expression on the right-hand side of the equation:

$$x = 16$$

Conclusion

Therefore, the value of $x$ is $16$. This makes sense, because if we substitute $x = 16$ into the original equation, we get:

$$\frac{3}{4} = \frac{12}{16}$$

Why is this the Correct Answer?

This is the correct answer because it satisfies the condition that the two fractions are equivalent. In other words, the two fractions have the same value, and $x = 16$ is the value that makes this true.

Alternative Solutions

There are alternative solutions to this problem, but they are not as straightforward as the one we presented. For example, we could have multiplied both sides of the equation by $x$ to get:

$$3x^2 = 12 \times 4$$

Simplifying the Equation

Now, we can simplify the equation by multiplying $12$ and $4$:

$$3x^2 = 48$$

Solving for x

To solve for $x$, we can divide both sides of the equation by $3$:

$$x^2 = \frac{48}{3}$$

Evaluating the Expression

Now, we can evaluate the expression on the right-hand side of the equation:

$$x^2 = 16$$

Solving for x

To solve for $x$, we can take the square root of both sides of the equation:

$$x = \pm \sqrt{16}$$

Evaluating the Expression

Now, we can evaluate the expression on the right-hand side of the equation:

$$x = \pm 4$$

Conclusion

Therefore, the value of $x$ is either $4$ or $-4$. However, this is not the correct answer, because it does not satisfy the condition that the two fractions are equivalent.

Why is this not the Correct Answer?

This is not the correct answer because it does not satisfy the condition that the two fractions are equivalent. In other words, the two fractions do not have the same value when $x = 4$ or $x = -4$.

Conclusion

In conclusion, the value of $x$ is $16$. This is the correct answer because it satisfies the condition that the two fractions are equivalent.

Final Answer

The final answer is: $\boxed{16}$

Q: What is the definition of equivalent fractions?

A: Equivalent fractions are fractions that have the same value, but may appear different. In other words, two fractions are equivalent if they represent the same proportion or ratio.

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

A: To determine if two fractions are equivalent, you can cross-multiply the two fractions and see if the resulting equation is true. If the equation is true, then the two fractions are equivalent.

Q: What is the process of cross-multiplying?

A: Cross-multiplying involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This is represented by the equation:

$$\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$$

Q: How do I solve for x in an equation involving equivalent fractions?

A: To solve for x in an equation involving equivalent fractions, you can cross-multiply the two fractions and then solve for x using algebraic manipulation.

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

A: Equivalent fractions are fractions that have the same value, but may appear different. Proportional fractions, on the other hand, are fractions that have the same ratio, but may have different values.

Q: Can you give an example of equivalent fractions?

A: Yes, here is an example of equivalent fractions:

$$\frac{1}{2} = \frac{2}{4}$$

Q: Can you give an example of proportional fractions?

A: Yes, here is an example of proportional fractions:

$$\frac{1}{2} = \frac{3}{6}$$

Q: How do I know if two fractions are proportional?

A: To determine if two fractions are proportional, you can divide the numerator of the first fraction by the denominator of the first fraction, and then divide the numerator of the second fraction by the denominator of the second fraction. If the two resulting ratios are equal, then the two fractions are proportional.

Q: What is the relationship between equivalent fractions and algebraic manipulation?

A: Equivalent fractions and algebraic manipulation are closely related. Algebraic manipulation involves using mathematical operations to simplify or solve equations, and equivalent fractions are often used in algebraic manipulation to solve equations involving fractions.

Q: Can you give an example of using equivalent fractions in algebraic manipulation?

A: Yes, here is an example of using equivalent fractions in algebraic manipulation:

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

To solve for x, we can cross-multiply the two fractions and then solve for x using algebraic manipulation.

Q: What is the importance of equivalent fractions in real-world applications?

A: Equivalent fractions are important in real-world applications because they allow us to simplify complex fractions and solve equations involving fractions. This is useful in a variety of fields, including mathematics, science, and engineering.

Q: Can you give an example of using equivalent fractions in a real-world application?

A: Yes, here is an example of using equivalent fractions in a real-world application:

Suppose we want to find the area of a rectangle with a length of 6 inches and a width of 4 inches. We can use equivalent fractions to simplify the calculation:

$$\frac{6}{4} = \frac{3}{2}$$

This allows us to find the area of the rectangle more easily.

Q: What are some common mistakes to avoid when working with equivalent fractions?

A: Some common mistakes to avoid when working with equivalent fractions include:

• Not cross-multiplying the fractions correctly
• Not simplifying the fractions correctly
• Not solving for x correctly

Q: How can I practice working with equivalent fractions?

A: You can practice working with equivalent fractions by:

• Solving problems involving equivalent fractions
• Creating your own problems involving equivalent fractions
• Using online resources or worksheets to practice working with equivalent fractions

Q: What are some resources for learning more about equivalent fractions?

A: Some resources for learning more about equivalent fractions include:

• Online tutorials or videos
• Math textbooks or workbooks
• Online resources or worksheets
• Math classes or tutoring sessions