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 27.665 left parenthesis 0.03239 right parenthesis Over 9.538 minus 7.876 EndFraction

We are tasked with performing the following mathematical operation while keeping track of significant digits:

\[
\frac{27.665 \times 0.03239}{9.538 - 7.876}
\]

### Step 1: Subtract \(9.538 - 7.876\)

- **Operation**: \(9.538 - 7.876 = 1.662\)
- Both numbers are accurate to **3 decimal places**, so the result should be rounded to **3 decimal places**.
- Thus, the result of this subtraction is **1.662**.

### Step 2: Multiply \(27.665 \times 0.03239\)

- **Operation**: \(27.665 \times 0.03239 = 0.89600235\)
- The number \(27.665\) has **5 significant digits**, and \(0.03239\) has **4 significant digits**.
- The result should be rounded to **4 significant digits** because the number with the fewest significant digits is \(0.03239\).

Thus, we underline the first **4 significant digits** in the product:

\[
0.8960\underline{0235}
\]

So, the result of this multiplication is **0.8960** (rounded to 4 significant digits).

### Step 3: Divide the result from Step 2 by the result from Step 1

- **Operation**: \(\frac{0.8960}{1.662} = 0.539 04\)
- The result of \(0.8960\) has **4 significant digits**, and \(1.662\) has **4 significant digits**.
- So, the result should be rounded to **4 significant digits**.

Thus, we underline the first **4 significant digits** in the quotient:

\[
0.5390\underline{4}
\]

So, the final answer is **0.5390** (rounded to 4 significant digits).

### Final Answer:
\[
0.5390
\]

Would you like further details on any step or have any questions?

Here are 5 related questions:
1. How do you determine the number of significant digits in a calculation?
2. What is the difference between rounding and truncating numbers?
3. Why do we round to the least number of significant digits when multiplying or dividing?
4. How are significant digits affected by addition and subtraction?
5. How does using significant digits help in real-life calculations?

**Tip**: Always keep track of significant digits at each step to maintain accuracy, especially in scientific and engineering calculations.

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 27.665 left parenthesis 0.03239 right parenthesis Over 9.538 minus 7.876 EndFraction.

Learn how to perform operations involving multiplication, division, and subtraction while accurately tracking and rounding significant digits. This detailed step-by-step solution includes examples and explanations on applying significant digit rules in calculations. Suitable for students in Grades 9-12.

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