Assess which explanation on how to obtain the value of x in the equation 4/x = 36/81 is reasonable.

Answer 1

To solve the equation \( \frac{4}{x} = \frac{36}{81} \), we can follow these steps:

1. Cross-multiply to get rid of the fractions:

   \[ 4 \cdot 81 = 36 \cdot x \]

   This simplifies to:

   \[ 324 = 36x \]

2. Solve for \( x \) by dividing both sides by 36:

   \[ x = \frac{324}{36} \]

3. Simplify \( \frac{324}{36} \):

   \[ x = 9 \]

Therefore, the value of \( x \) is 9.

Reasonable Explanation

A reasonable explanation for these steps involves clarifying the logic behind each operation. You can explain that by cross-multiplying, we eliminate the fractions to make the equation easier to work with. Then, dividing by 36 isolates \( x \) on one side, allowing us to find its value. Finally, simplifying the fraction gives the solution clearly and shows that \( x \) equals 9.

If any alternative explanation suggests solving the equation in a way that does not adhere to these mathematical principles, it would not be considered reasonable.

Answer 2

So, would this work: divide 81 by 9 to obtain the value of x?

Answer 3

Dividing 81 by 9 does not provide a correct method to obtain the value of \( x \) in the equation \( \frac{4}{x} = \frac{36}{81} \).

To clarify why this does not work:

1. Understanding the Equation: The equation \( \frac{4}{x} = \frac{36}{81} \) requires you to find \( x \) such that the two fractions are equal. You can't simply manipulate one side (like dividing 81 by 9) to find \( x \).

2. Calculating \( \frac{36}{81} \): We know \( \frac{36}{81} \) simplifies to \( \frac{4}{9} \) because both the numerator and denominator can be divided by 9. Thus, our original equation rewrites to \( \frac{4}{x} = \frac{4}{9} \).

3. Cross-Multiplying: With \( \frac{4}{x} = \frac{4}{9} \), we can cross-multiply:

   • \( 4 \cdot 9 = 4 \cdot x \)
   • This simplifies to \( 36 = 4x \).
   • Dividing both sides by 4 gives \( x = 9 \).

Therefore, dividing 81 by 9 does not relate correctly to solving for \( x \) in this context. The appropriate method is as described earlier: cross-multiplying and then isolating \( x \). The value of \( x \) remains 9.

Answer 4

What about: Multiply the numerators and denominators. Equate the product to calculate the value of x?

Answer 5

The approach of multiplying the numerators and denominators directly in the equation \( \frac{4}{x} = \frac{36}{81} \) is not a valid method to solve for \( x \).

Here's Why:

To solve the equation accurately, it needs to maintain the equality and relationship between the two sides. The proper method typically involves cross-multiplication or manipulating the equation to isolate \( x \).

Steps to Solve Correctly:

1. Set up the equation: \[ \frac{4}{x} = \frac{36}{81} \]

2. Cross-multiply: \[ 4 \cdot 81 = 36 \cdot x \] Which gives: \[ 324 = 36x \]

3. Solve for \( x \): \[ x = \frac{324}{36} = 9 \]

Explanation of the Incorrect Method:

• Multiplying the numerators (4 and 36) and the denominators (x and 81) separately does not maintain the correct relationships required to solve for \( x \). The resulting products would be nonsensical in this context, as they do not lend themselves to isolating \( x \) properly.

To summarize, using multiplication of numerators and denominators is not a mathematically sound method in this case, and the logical steps provided above (specifically cross-multiplication) are needed to find \( x \) accurately.

Answer 6

What about: Divide 36 by 81, and then multiply by 4 to determine the value of x?

Answer 7

Using the method of dividing \( 36 \) by \( 81 \) and then multiplying by \( 4 \) does not directly give the value of \( x \) in the equation \( \frac{4}{x} = \frac{36}{81} \). However, let’s evaluate your suggestion step-by-step to clarify and see how it ties in with the original equation.

Original Equation:

\[ \frac{4}{x} = \frac{36}{81} \]

Step 1: Analyzing the Right Side

1. Calculate \( \frac{36}{81} \):
   • Simplifying \( \frac{36}{81} \) gives: \[ \frac{36 \div 9}{81 \div 9} = \frac{4}{9} \]

Step 2: Set the Equation

Now our equation becomes: \[ \frac{4}{x} = \frac{4}{9} \]

Step 3: Cross-Multiply

From here, we can simply cross-multiply or equate directly: \[ 4 \cdot 9 = 4 \cdot x \] This simplifies to: \[ 36 = 4x \]

Step 4: Solve for \( x \)

Dividing both sides by \( 4 \): \[ x = \frac{36}{4} = 9 \]

Evaluation of Your Method

If you divide \( 36 \) by \( 81 \), you get \( \frac{4}{9} \), and then if you multiply by \( 4 \): \[ \frac{4}{9} \times 4 = \frac{16}{9} \] This result, \( \frac{16}{9} \), is not used to find \( x \) in this case. Instead, you would need to equate the fractions correctly from the outset or cross-multiply.

Conclusion

So while your steps do not directly yield \( x \), they are related because they simplify the right side of the equation. The correct way to find \( x \) is still by using the equality derived from \( \frac{4}{x} = \frac{4}{9} \) and solving for \( x \) directly. The answer remains \( x = 9 \).

To summarize: dividing \( 36 \) by \( 81 \) and multiplying by \( 4 \) does not yield the desired value of \( x \) directly, and the best method to solve it is by cross-multiplying or equating the fractions at the start.

Answer 8

Then what about: Reduce the second ratio by dividing both numbers by 9, and then multiplying the result by 4?

Answer 9

Let's analyze the proposed method of reducing the second ratio \( \frac{36}{81} \) by dividing both numbers by 9, and then multiplying the result by 4 to find the value of \( x \) in the equation \( \frac{4}{x} = \frac{36}{81} \).

Step 1: Simplify the Second Ratio

First, we simplify \( \frac{36}{81} \) by dividing both the numerator and the denominator by 9:

\[ \frac{36 \div 9}{81 \div 9} = \frac{4}{9} \]

Step 2: Set Up the Equation

Now, using the simplified ratio, we rewrite the equation we are trying to solve:

\[ \frac{4}{x} = \frac{4}{9} \]

Step 3: Compare or Cross-Multiply

At this point, you can either compare the two fractions directly or cross-multiply:

1. Direct Comparison: Since the numerators are equal (both are 4), we can set the denominators equal:

\[ x = 9 \]

2. Cross-Multiplication: Alternatively, if we cross-multiply, we get:

\[ 4 \cdot 9 = 4 \cdot x \Rightarrow 36 = 4x \] Solving for \( x \) gives:

\[ x = \frac{36}{4} = 9 \]

Evaluation of Your Method

If we follow your method explicitly:

1. First, reduce \( \frac{36}{81} \) to \( \frac{4}{9} \).
2. Then multiply the simplified second ratio (the result of the first step) by 4, although what we multiply by 4 is not the same context as \( x \):

If you take \( \frac{4}{9} \) and multiply it by 4, you get:

\[ \frac{4}{9} \times 4 = \frac{16}{9} \]

This \( \frac{16}{9} \) does not correspond to \( x \), but rather just changes the fraction.

Conclusion

The correct and direct approach is to recognize that once you simplify \( \frac{36}{81} \) to \( \frac{4}{9} \), you can conclude that \( \frac{4}{x} = \frac{4}{9} \), leading directly to the conclusion that \( x = 9 \).

Therefore, while your method of simplifying \( \frac{36}{81} \) is valid and leads you to the correct form of the equation, doing additional multiplication afterward does not help directly in finding \( x \). The result \( x = 9 \) stands true based on comparing the simplified fractions or cross-multiplying.