A student was asked to find the arithmetic mean of the numbers 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14 and x. He found that the mean of these twelve numbers is 12. What should be the value of x?

Introduction / Context:
This question illustrates how to recover a missing number when the average of the entire set is known. Such problems appear frequently in aptitude tests, and the key is to work with total sums rather than trying to adjust averages directly. Once we know the total sum needed for a given mean, we can subtract the sum of known numbers to find the unknown value.

Given Data / Assumptions:

• The list of numbers is 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14 and x.
• The arithmetic mean of these 12 numbers is 12.
• We need to find the value of x that makes this mean correct.

Concept / Approach:
If the mean of 12 numbers is 12, then the total sum of those 12 numbers must be 12 * 12. We first compute the sum of the 11 known numbers. The difference between the required total sum and this known sum must equal x. This is a simple but very useful application of the definition of arithmetic mean.

Step-by-Step Solution:
We have 12 numbers in total, so required total sum = mean * count = 12 * 12 = 144.Now, find the sum of the known 11 numbers: 3 + 11 + 7 + 9 + 15 + 13 + 8 + 19 + 17 + 21 + 14.Compute this sum stepwise: 3 + 11 = 14, 14 + 7 = 21, 21 + 9 = 30, 30 + 15 = 45, 45 + 13 = 58, 58 + 8 = 66, 66 + 19 = 85, 85 + 17 = 102, 102 + 21 = 123, 123 + 14 = 137.So the sum of the 11 known numbers is 137.Since total required sum is 144, x must satisfy 137 + x = 144.Therefore, x = 144 - 137 = 7.

Verification / Alternative check:
Create the full set including x = 7: 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14, 7. Their sum is 137 + 7 = 144. Divide by 12 to get the mean: 144 / 12 = 12. This matches the given mean, confirming that the calculation is correct.

Why Other Options Are Wrong:
If x were 6, 8, 9 or 10, the total sum would become 143, 145, 146 or 147 respectively, giving means of 143 / 12, 145 / 12, 146 / 12 or 147 / 12, none of which equals exactly 12. Only x = 7 produces the exact mean of 12.

Common Pitfalls:
Students sometimes average the known numbers first and then adjust that average, which can cause confusion. Others miscalculate the sum of the 11 numbers or forget that there are 12 numbers when multiplying the mean. Carefully computing the total required sum and subtracting the known sum is the fastest and most reliable method.

Final Answer:
The value that should replace x is 7.