Question:medium

DIRECTIONS for questions 45 and 46: Read the information below and answer the question that follows.
It is possible to arrange eight of the nine numbers 2, 3, 4, 5, 7, 10, 11, 12, 13 in the vacant squares of the 3 by 4 array shown below so that the arithmetic average of the numbers in each row and column is the same integer.
115
9
14

46. Which one of the nine numbers must be left out when completing the array?

Show Hint

Find which single number, once removed from the sum of all thirteen entries in the grid, leaves a total divisible by 12.
Updated On: Jul 13, 2026
  • 4
  • 5
  • 7
  • 10
Show Solution

The Correct Option is D

Solution and Explanation

Step 1: Reduce the problem using remainders mod 12.
Since the grid total must be a multiple of 12 (from the row/column average condition, grid total = $12x$), look at the sum of all thirteen numbers modulo 12 instead of testing every candidate one by one.
\[ 1+2+3+4+5+7+9+10+11+12+13+14+15=106 \]
$106 = 8\times12 + 10$, so $106 \equiv 10 \pmod{12}$.

Step 2: Match the remainder to the number that must be dropped.
Removing one number from the total must bring the remainder down to 0 mod 12, since a valid grid total has to be an exact multiple of 12. That means the number removed must itself leave a remainder of 10 when divided by 12.

Step 3: Check which of the nine free numbers fits.
Among 2, 3, 4, 5, 7, 10, 11, 12, 13, only the number 10 leaves a remainder of 10 when divided by 12 (since 10 is less than 12, its remainder is just 10 itself). None of the others, 2, 3, 4, 5, 7, 11, 12 or 13, match.

Step 4: Final Answer.
So the number 10 is the one that must be left out of the array.
\[ \boxed{10} \]
Was this answer helpful?
0

Top Questions on Average

Questions Asked in XAT exam

Read the scenario and answer the question:
The upper hinge of a dataset is the median of all the values to the right of the median of the dataset in an ascending arrangement, while the lower hinge of the dataset is the median of all the values to the left of the median of the dataset in the same arrangement.
For example, consider the dataset 4, 3, 2, 6, 4, 2, 7. When arranged in the ascending order, it becomes 2, 2, 3, 4, 4, 6, 7. The median is 4 (the bold value), and hence the upper hinge is the median of 4, 6, 7, i.e., 6. Similarly, the lower hinge is 2. 
A student has surveyed thirteen of her teachers, and recorded their work experience (in integer years). Two of the values recorded by the student got smudged, and she cannot recall those values. All she remembers is that those two values were unequal, so let us write them as A and B, where A < B. The remaining eleven values, as recorded, are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29. Moreover, the student also remembers the following summary measures, calculated based on all the thirteen values: 
Minimum: 2
Lower Hinge: 6.5 
Median: 12 
Upper Hinge: 21 
Maximum: 29
View Solution
Read the following scenario and answer the questions that follow.
A T20 cricket match consists of two teams playing twenty overs each, numbered 1 to 20. The runs scored in any over is a non-negative integer. The run rate at the end of any over is the average runs scored up to and including that over, i.e., the run rate at the end of the kth over is the average number of runs scored in overs numbered \(1, 2, …, k,\) where \(1 \leq k \leq 20, k\) a positive integer.
The following table indicates the run rate of a team at the end of some of the overs during a T20 cricket match (correct up to 2 decimal places), where \(1 \leq N – 2 < N + 6 \leq 20\), N a positive integer. It is also known that the team did not score less than 6 runs and more than 15 runs in any over.
Over NumberRun Rate
N-28.00
N7.43
N+28.11
N+48.45
N+68.08
View Solution