If the numbers A and B when rounded to three decimal places are 3.724 and 2.251 respectively. Evaluate the approximation B - A and discuss the error in the approximation.

Let's begin by approximating the subtraction of \( B - A \), using the rounded values provided.

### Given:
- \( A = 3.724 \) (rounded to three decimal places)
- \( B = 2.251 \) (rounded to three decimal places)

### Step 1: Perform the subtraction

\[
B - A = 2.251 - 3.724 = -1.473
\]

### Step 2: Discussing the error in the approximation

Since both \( A \) and \( B \) are rounded to three decimal places, there is an inherent uncertainty in their exact values. Let’s explore the range of possible errors:

- \( A \) could be any number in the range \( 3.7235 \leq A < 3.7245 \).
- \( B \) could be any number in the range \( 2.2505 \leq B < 2.2515 \).

Now, we evaluate the worst-case scenario for both over- and under-estimations:

#### Maximum possible value for \( B - A \):
- Use the maximum possible value of \( B \) and the minimum possible value of \( A \):
  \[
  B - A = 2.2515 - 3.7235 = -1.4720
  \]

#### Minimum possible value for \( B - A \):
- Use the minimum possible value of \( B \) and the maximum possible value of \( A \):
  \[
  B - A = 2.2505 - 3.7245 = -1.4740
  \]

Thus, the actual value of \( B - A \) lies in the range:
\[
-1.4740 \leq B - A \leq -1.4720
\]

The approximation using the rounded values is \( -1.473 \), and the possible error due to rounding is approximately \( \pm 0.001 \).

### Conclusion:
The subtraction \( B - A = -1.473 \) is accurate to within \( \pm 0.001 \), meaning the error in this approximation is no larger than 0.001.

Would you like more details or have any further questions?

Here are 5 related questions to consider:
1. What happens if \( A \) and \( B \) are rounded to fewer decimal places?
2. How can you calculate absolute and relative errors in such approximations?
3. Why is it important to consider rounding errors in scientific measurements?
4. How can rounding errors affect the results of complex calculations involving multiple operations?
5. What strategies can be used to minimize rounding errors in practical applications?

**Tip**: When working with approximations, always account for potential rounding errors, especially in cases where many operations are involved, as these errors can accumulate.

Learn how to approximate the value of B - A, given that A and B are rounded to three decimal places (3.724 and 2.251). Explore error analysis due to rounding, and find the range of possible results. Suitable for students in Grades 10-12, this problem involves key concepts like approximation, rounding, and error estimation.

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