Perform the indicated operations assuming all numbers are approximate. The answer should be rounded using the following procedure. Keep all digits in the intermediate​ steps, but keep track of​ (perhaps by​ underlining) the significant digits that would be retained according to the appropriate rounding rule for each step. Then round off the final answer according to the last operation that is performed.
StartFraction 23.546 left parenthesis 0.02372 right parenthesis Over 9.431 minus 7.324 EndFraction

To solve the expression:

\[
\frac{23.546 \times 0.02372}{9.431 - 7.324}
\]

we will follow the steps below, rounding appropriately at each step.

### Step 1: Subtraction in the Denominator
Calculate the subtraction in the denominator:

\[
9.431 - 7.324 = 2.107
\]

**Significant Digits:** 
- \(9.431\) has 4 significant digits.
- \(7.324\) has 4 significant digits.
- The result of subtraction should have 4 significant digits, so 2.107 is retained as 4 significant digits.

### Step 2: Multiplication in the Numerator
Calculate the multiplication in the numerator:

\[
23.546 \times 0.02372 = 0.55861672
\]

**Significant Digits:** 
- \(23.546\) has 5 significant digits.
- \(0.02372\) has 4 significant digits.
- The result of multiplication should have 4 significant digits, so 0.55861672 is rounded to **0.5586**.

### Step 3: Division
Now, divide the result from the multiplication by the result from the subtraction:

\[
\frac{0.5586}{2.107} \approx 0.265160655
\]

**Significant Digits:** 
- The numerator \(0.5586\) has 4 significant digits.
- The denominator \(2.107\) has 4 significant digits.
- The final result should have 4 significant digits, so 0.265160655 is rounded to **0.2652**.

### Final Answer
The final answer, rounded to the appropriate number of significant digits, is:

\[
\boxed{0.2652}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the result change if the numbers had different significant digits?
2. What are the general rules for rounding in multiplication and division?
3. Why is it important to track significant digits during calculations?
4. Can you apply these steps to a different arithmetic expression?
5. How would errors propagate if intermediate rounding were not performed carefully?

**Tip:** Always keep track of significant digits during calculations to ensure the final answer's precision is accurate.

Learn how to perform arithmetic operations with significant figures using a step-by-step guide. Understand the rules for rounding at each step to ensure accuracy. Suitable for high school students studying mathematics.

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