To solve this problem, we need to understand what numbers could be rounded to 13,000 and 12,600 respectively, and find the greatest number that satisfies both conditions.
### Step-by-Step Solution:
1. Identify the possible range for Mae's rounding to the thousands place:
- Any number that Mae rounds to 13,000 must be closer to 13,000 than to any other thousand value.
- The smallest number that rounds to 13,000 when rounded to the nearest thousand is 12,500.
- The largest number that rounds to 13,000 when rounded to the nearest thousand is 13,499.
- Therefore, the range of numbers Mae could be thinking of is 12,500 to 13,499.
2. Identify the possible range for Eli's rounding to the hundreds place:
- Any number that Eli rounds to 12,600 must be closer to 12,600 than to any other hundred value.
- The smallest number that rounds to 12,600 when rounded to the nearest hundred is 12,550.
- The largest number that rounds to 12,600 when rounded to the nearest hundred is 12,649.
- Therefore, the range of numbers Eli could be thinking of is 12,550 to 12,649.
3. Find the overlap between the two ranges to determine the numbers that satisfy both conditions:
- Mae's range: 12,500 to 13,499
- Eli's range: 12,550 to 12,649
- The intersection of these ranges provides the numbers that can be rounded as described by both Mae and Eli.
4. Determine the greatest number within this overlapping range:
- The overlap range is from 12,550 to 12,649.
- The greatest number in this range is 12,649.
Thus, the greatest number that can be rounded to the thousands place as 13,000 by Mae and to the hundreds place as 12,600 by Eli is 12,649.
