CAT 2023 Slot 2 — DILR Question 18
Answer the following questions based on the information given below:
There are nine boxes arranged in a 3×3 array as shown in Tables 1 and 2. Each box contains three sacks. Each sack has a certain number of coins, between 1 and 9, both inclusive.
The average number of coins per sack in the boxes are all distinct integers. The total number of coins in each row is the same. The total number of coins in each column is also the same.
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Table 1 gives information regarding the median of the numbers of coins in the three sacks in a box for some of the boxes. In Table 2 each box has a number which represents the number of sacks in that box having more than 5 coins. That number is followed by a * if the sacks in that box satisfy exactly one among the following three conditions, and it is followed by ** if two or more of these conditions are satisfied.
- The minimum among the numbers of coins in the three sacks in the box is 1.
- The median of the numbers of coins in the three sacks is 1.
- The maximum among the numbers of coins in the three sacks in the box is 9.
For how many boxes are the average and median of the numbers of coins contained in the three sacks in that box the same?
Answer & solution
Answer: 4
Easy
Same solved grid as the rest of the set. For each box compute its average () and its median (the middle of the three ascending values), then count where they're equal.
Average vs median for all nine boxes:
| Box | Sacks | Average | Median | Equal? |
|---|---|---|---|---|
| C1R1 | 1, 1, 7 | 3 | 1 | — |
| C1R2 | 1, 2, 9 | 4 | 2 | — |
| C1R3 | 7, 8, 9 | 8 | 8 | ✓ |
| C2R1 | 3, 9, 9 | 7 | 9 | — |
| C2R2 | 1, 2, 3 | 2 | 2 | ✓ |
| C2R3 | 1, 8, 9 | 6 | 8 | — |
| C3R1 | 1, 6, 8 | 5 | 6 | — |
| C3R2 | 9, 9, 9 | 9 | 9 | ✓ |
| C3R3 | 1, 1, 1 | 1 | 1 | ✓ |
Boxes where average median: C1R3, C2R2, C3R2, C3R3.
(Notice these are exactly the boxes whose three sacks are symmetric or all-equal — e.g. and — where mean and median always coincide.)