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Let's solve the problem step by step:

We are given:

- \( A = 300,000 \) to the nearest 100,000.
- \( B = 300,000 \) to the nearest 10,000.

### Step 1: Range of values for \( A \)
Rounding \( A = 300,000 \) to the nearest 100,000 means that:
- The smallest possible value of \( A \) is \( 250,000 \), since any number less than \( 250,000 \) would round to \( 200,000 \).
- The largest possible value of \( A \) is \( 349,999 \), since any number above that would round to \( 400,000 \).

So, \( 250,000 \leq A \leq 349,999 \).

### Step 2: Range of values for \( B \)
Rounding \( B = 300,000 \) to the nearest 10,000 means that:
- The smallest possible value of \( B \) is \( 295,000 \), since any number less than \( 295,000 \) would round to \( 290,000 \).
- The largest possible value of \( B \) is \( 304,999 \), since any number above that would round to \( 310,000 \).

So, \( 295,000 \leq B \leq 304,999 \).

### Solving the questions:

**a) What is the greatest possible value of \( A + B \)?**

To maximize \( A + B \), we use the largest possible values for \( A \) and \( B \):
- \( A = 349,999 \)
- \( B = 304,999 \)

Thus, the greatest possible value of \( A + B \) is:
\[
349,999 + 304,999 = 654,998
\]

**b) What is the smallest possible value of \( A + B \)?**

To minimize \( A + B \), we use the smallest possible values for \( A \) and \( B \):
- \( A = 250,000 \)
- \( B = 295,000 \)

Thus, the smallest possible value of \( A + B \) is:
\[
250,000 + 295,000 = 545,000
\]

**c) What is the greatest possible value of \( A - B \)?**

To maximize \( A - B \), we use the largest possible value for \( A \) and the smallest possible value for \( B \):
- \( A = 349,999 \)
- \( B = 295,000 \)

Thus, the greatest possible value of \( A - B \) is:
\[
349,999 - 295,000 = 54,999
\]

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Would you like any further details on this, or explanations?

Here are 5 related questions to explore further:

1. What would be the smallest possible value of \( A - B \)?
2. How would the ranges change if \( A \) were rounded to the nearest 50,000?
3. What if both \( A \) and \( B \) were rounded to the nearest 1,000 instead?
4. How does rounding impact the results for different types of numbers (e.g., decimals)?
5. Can you find examples where rounding leads to counterintuitive results in real-world contexts?

**Tip:** When rounding, always think about both the upper and lower bounds of the range to get a full understanding of the possible values!

A and B are integers. A = 300,000 to the nearest 100,000. B = 300,000 to the nearest 10,000. a) What is the greatest possible value of A + B? b) What is the smallest possible value of A + B? c) What is the greatest possible value of A - B?

Solve the problem where A and B are rounded integers. A is rounded to the nearest 100,000, and B to the nearest 10,000. Discover the greatest and smallest possible values of A + B and A - B. Suitable for students in Grades 6-8.

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