Estimate the folloowing by rounding off the numbers to their greatest place:
a. `8576-:289` b. `6514-:78`

Answer

To solve the problem of estimating the following by rounding off the numbers to their greatest place, we will break it down into two parts as given in the question.

### Part a: Estimate `8576 ÷ 289`

1. **Identify the greatest place of 8576**:
- The greatest place in 8576 is the thousands place, which is 8000 (since 8576 is closer to 9000 than to 8000).

**Hint**: Look for the highest place value in the number.

2. **Round 289 to the nearest hundred**:
- The greatest place in 289 is the hundreds place. Since 289 is closer to 300 than to 200, we round it to 300.

**Hint**: Determine which hundred the number is closest to.

3. **Set up the estimation**:
- Now we can write the estimation as:
\[ 8576 \approx 9000 \quad \text{and} \quad 289 \approx 300 \]
- Therefore, we have:
\[ 9000 ÷ 300 \]

4. **Perform the division**:
- Simplifying \(9000 ÷ 300\):
\[ 9000 ÷ 300 = 30 \]

**Hint**: Divide the numbers after rounding to simplify the calculation.

### Part b: Estimate `6514 ÷ 78`

1. **Identify the greatest place of 6514**:
- The greatest place in 6514 is the thousands place, which is 6000 (since 6514 is closer to 7000 than to 6000).

**Hint**: Again, look for the highest place value in the number.

2. **Round 78 to the nearest ten**:
- The greatest place in 78 is the tens place. Since 78 is closer to 80 than to 70, we round it to 80.

**Hint**: Determine which ten the number is closest to.

3. **Set up the estimation**:
- Now we can write the estimation as:
\[ 6514 \approx 7000 \quad \text{and} \quad 78 \approx 80 \]
- Therefore, we have:
\[ 7000 ÷ 80 \]

4. **Perform the division**:
- Simplifying \(7000 ÷ 80\):
\[ 7000 ÷ 80 = 87.5 \]
- This can also be approximated to 88 for a rounded estimate.

**Hint**: Divide the rounded numbers to get an approximate result.

### Final Answers:
- Part a: \( 8576 ÷ 289 \approx 30 \)
- Part b: \( 6514 ÷ 78 \approx 88 \)