Given Equivalent Fractions

Oct 13, 2025 by miodeya.com 111 views

Understanding Equivalent Fractions

Equivalent fractions are fractions that have the same value, even though they may look different. In other words, two fractions are equivalent if they can be simplified to the same ratio. For example, the fractions $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent because they both represent the same value.

Given Equivalent Fractions

We are given two fractions: $\frac{3}{4}$ and $\frac{12}{z}$ . These fractions are equivalent, meaning they have the same value. Our goal is to find the value of $z$ that makes these fractions equivalent.

Setting Up the Equation

To solve for $z$ , we can set up an equation based on the fact that these fractions are equivalent. We can start by setting the two fractions equal to each other:

\frac{3}{4} = \frac{12}{z}

Cross-Multiplying

To eliminate the fractions, we can cross-multiply:

3z = 4 \cdot 12

Simplifying the Equation

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

3z = 48

Solving for z

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

z = \frac{48}{3}

Calculating the Value of z

Now, we can calculate the value of $z$ by dividing $48$ by $3$ :

z = 16

Conclusion

Therefore, the value of $z$ that makes the fractions $\frac{3}{4}$ and $\frac{12}{z}$ equivalent is $16$ .

Answer Choice

The correct answer is:

A. 48 (Incorrect)
B. 36 (Incorrect)
C. 24 (Incorrect)
D. 16 (Correct)

Why the Other Options are Incorrect

The other options are incorrect because they do not satisfy the equation. For example, if $z = 48$ , then the equation becomes:

\frac{3}{4} = \frac{12}{48}

This is not true, because $\frac{12}{48}$ can be simplified to $\frac{1}{4}$ , which is not equal to $\frac{3}{4}$ .

Similarly, if $z = 36$ , then the equation becomes:

\frac{3}{4} = \frac{12}{36}

This is not true, because $\frac{12}{36}$ can be simplified to $\frac{1}{3}$ , which is not equal to $\frac{3}{4}$ .

Finally, if $z = 24$ , then the equation becomes:

\frac{3}{4} = \frac{12}{24}

This is not true, because $\frac{12}{24}$ can be simplified to $\frac{1}{2}$ , which is not equal to $\frac{3}{4}$ .

Final Answer

Therefore, the correct answer is D. 16.

Understanding Equivalent Fractions

Q&A: Solving for z

Q: What is the value of z that makes the fractions $\frac{3}{4}$ and $\frac{12}{z}$ equivalent?

A: To solve for $z$ , we can set up an equation based on the fact that these fractions are equivalent. We can start by setting the two fractions equal to each other:

\frac{3}{4} = \frac{12}{z}

We can then cross-multiply and simplify the equation to find the value of $z$ .

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

A: Two fractions are equivalent if they can be simplified to the same ratio. For example, the fractions $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent because they both represent the same value.

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

A: Equivalent fractions are fractions that have the same value, while equivalent ratios are ratios that have the same value. For example, the fractions $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions, while the ratios $1:2$ and $2:4$ are equivalent ratios.

Q: Can you give an example of how to solve for z using equivalent fractions?

A: Let's say we have the fractions $\frac{2}{3}$ and $\frac{6}{z}$ . We can set up an equation based on the fact that these fractions are equivalent:

\frac{2}{3} = \frac{6}{z}

We can then cross-multiply and simplify the equation to find the value of $z$ :

2z = 3 \cdot 6

2z = 18

z = \frac{18}{2}

z = 9

Q: What are some common mistakes to avoid when solving for z using equivalent fractions?

A: Some common mistakes to avoid when solving for $z$ using equivalent fractions include:

Not setting up the equation correctly
Not cross-multiplying correctly
Not simplifying the equation correctly
Not checking the answer to make sure it is correct

Conclusion

Solving for $z$ using equivalent fractions can be a challenging task, but with practice and patience, you can become proficient in solving these types of problems. Remember to always set up the equation correctly, cross-multiply, and simplify the equation to find the value of $z$ .