The arithmetic mean of the numbers 7, 5, 13, x and 9 is 10. What is the value of x?

Introduction / Context:
This question is a straightforward application of the concept of arithmetic mean (average). You are given four explicit numbers and one unknown number x, and told that their overall average is 10. The task is to determine the value of x that makes this true. Such problems are common in basic quantitative aptitude and help reinforce the formula for the mean and simple equation solving.

Given Data / Assumptions:

- The five numbers are 7, 5, 13, x and 9. - Their arithmetic mean is 10. - We must solve for the unknown number x.

Concept / Approach:
The arithmetic mean of a set of numbers is the sum of the numbers divided by the count of numbers. Here, there are five numbers, so the mean is (7 + 5 + 13 + x + 9) / 5. We are told that this expression equals 10. By equating the expression to 10 and simplifying, we get a linear equation in x. Solving this equation gives the required value of x. This is a simple one step algebra problem after computing the sum of the known numbers.

Step-by-Step Solution:
Step 1: Write the formula for the mean of the five numbers: (7 + 5 + 13 + x + 9) / 5 = 10. Step 2: Compute the sum of the known numbers: 7 + 5 + 13 + 9. Step 3: 7 + 5 = 12, 12 + 13 = 25, 25 + 9 = 34. So the total sum is 34 + x. Step 4: Substitute into the mean formula: (34 + x) / 5 = 10. Step 5: Multiply both sides by 5 to clear the denominator: 34 + x = 50. Step 6: Subtract 34 from both sides to isolate x: x = 50 - 34. Step 7: Compute 50 - 34 = 16. Step 8: Therefore, x = 16.

Verification / Alternative check:
Substitute x = 16 back into the list and check the average. The five numbers become 7, 5, 13, 16 and 9. Their sum is 7 + 5 + 13 + 16 + 9 = 50. The mean is 50 / 5 = 10, exactly matching the given condition. This confirms that x = 16 is correct and there is no mistake in the computation.

Why Other Options Are Wrong:
If x were 10, 12 or 14, the total sum of the five numbers would be different from 50, and the resulting average would not be 10. For example, with x = 10, the sum would be 44 and the average would be 44 / 5 = 8.8, which is too small. Only x = 16 produces the required sum of 50 and therefore the correct mean of 10.

Common Pitfalls:
Some learners may forget to multiply the mean by the number of terms to find the total sum or may make a mistake when adding the known numbers. Others might divide by 4 instead of 5, forgetting that there are five numbers in total. Always count the terms carefully and double check the arithmetic when summing the numbers to avoid such simple but costly mistakes.

Final Answer:
The value of x for which the mean is 10 is 16, which corresponds to option D.