Problem 43 worked answer
The Fullest Gallery Case
Designed for Grade 3 · Uses Middle Grade 3 math · About 4–11 minutes
At a glance
- Part 1: The six totals are 615, 627, 679, 706, 758, and 770.
- Part 2: Choose trays A and D. Their total is 679 glass tiles.
Key idea
The display case has a limit, so a pair must pass two tests:
- Its total must be 700 or less.
- Among the totals that pass the first test, it must be the greatest.
A total above 700 cannot win, even when it is very close to 700.
1. Complete the pair table
There are four trays. The supplied table lists each pair exactly once.
| Pair | Addition | Total | Fits 700 or less? |
|---|---|---|---|
| A + B | 615 | Yes | |
| A + C | 627 | Yes | |
| A + D | 679 | Yes | |
| B + C | 706 | No | |
| B + D | 758 | No | |
| C + D | 770 | No |
2. Choose the fullest pair that fits
The allowed totals are 615, 627, and 679. The greatest of these is 679, so the Gallery should choose trays A and D.
The total 706 is closer to 700 than 679 is, but . That pair does not fit, so it cannot be the best allowed choice.
Check and completeness
Check the chosen sum by place value:
The result fits because .
The table is complete: A is paired with B, C, and D; B is then paired with C and D; and C is paired with D. That covers every way to choose two different trays, with no pair repeated. Every other allowed total is smaller than 679, and every larger total is above 700. Therefore trays A and D are the unique best choice.
Teaching note
A complete explanation does not need the phrase unique best choice. It is enough for the learner to identify the allowed totals, compare them, and state that all larger totals go over 700.
Technical fit and rating
Operations and computation · Number and quantity
How the rating works →