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The problem presents two integers \( A \) and \( B \) rounded as follows:

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

The questions are:

**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 \( A - B \)?

### Let's analyze:

1. **A = 300,000 to the nearest 100,000:**
   - The possible values of \( A \) are between \( 250,000 \) and \( 350,000 \), inclusive.

2. **B = 300,000 to the nearest 10,000:**
   - The possible values of \( B \) are between \( 295,000 \) and \( 305,000 \), inclusive.

### a) Greatest possible value of \( A + B \):

- Maximum \( A = 350,000 \)
- Maximum \( B = 305,000 \)

So, the greatest possible value of \( A + B = 350,000 + 305,000 = 655,000 \).

### b) Smallest possible value of \( A + B \):

- Minimum \( A = 250,000 \)
- Minimum \( B = 295,000 \)

So, the smallest possible value of \( A + B = 250,000 + 295,000 = 545,000 \).

### c) Greatest possible value of \( A - B \):

- Maximum \( A = 350,000 \)
- Minimum \( B = 295,000 \)

So, the greatest possible value of \( A - B = 350,000 - 295,000 = 55,000 \).

Would you like further details on any of these calculations?

Here are 5 related questions for further practice:
1. What is the smallest possible value of \( A - B \)?
2. If \( A = 200,000 \) to the nearest 100,000, what would be the possible values for \( A \)?
3. How would the results change if \( B \) were rounded to the nearest 5,000?
4. What is the largest possible difference between \( A + B \) and \( A - B \)?
5. What is the difference between rounding to the nearest thousand and nearest ten thousand?

**Tip:** When rounding numbers, remember that the possible values will fall within a range centered around the rounded value, based on the place value being rounded to.

A = 300,000 to the nearest 100,000 and B = 300,000 to the nearest 10,000. What are the greatest and smallest possible values of A + B, and the greatest possible value of A - B?

Solve for the greatest and smallest possible values of A + B, and the greatest possible value of A - B when A is rounded to the nearest 100,000 and B to the nearest 10,000. A detailed arithmetic breakdown suitable for middle school students.

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