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.