Question
Assess which explanation on how to obtain the value of x in the equation 4/x = 36/81 is reasonable.
Answers
GPT-4o mini
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.
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.
Denki Kaminari
So, would this work: divide 81 by 9 to obtain the value of x?
GPT-4o mini
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.
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.
Denki Kaminari
What about: Multiply the numerators and denominators. Equate the product to calculate the value of x?
GPT-4o mini
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.
### 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.
Denki Kaminari
What about: Divide 36 by 81, and then multiply by 4 to determine the value of x?
GPT-4o mini
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.
### 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.
Denki Kaminari
Then what about: Reduce the second ratio by dividing both numbers by 9, and then multiplying the result by 4?
GPT-4o mini
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.
### 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.