Find the average of all prime numbers between 20 and 50.

Introduction / Context:
This question asks you to work with prime numbers in a given interval and compute their average. You need to identify all primes between 20 and 50, sum them and divide by how many there are. It tests your ability to recognize prime numbers, perform accurate addition and correctly apply the formula for the arithmetic mean.

Given Data / Assumptions:

- We are interested in all prime numbers strictly between 20 and 50. - A prime number has exactly two distinct positive divisors: 1 and the number itself. - We must compute their arithmetic mean, that is, sum divided by count.

Concept / Approach:
The first step is to list all primes between 20 and 50 by checking each integer for divisibility by smaller primes such as 2, 3, 5 and 7. After listing the primes, we add them together and then divide the sum by the number of primes. Using careful arithmetic avoids errors in both the list and the calculations. Since the options are given with decimal values, we may express the average as a fraction first and then convert to decimal form.

Step-by-Step Solution:
Step 1: List integers between 20 and 50 and identify which are prime: 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49. Step 2: Check each candidate. - 23 is prime (no divisors other than 1 and 23). - 29 is prime. - 31 is prime. - 37 is prime. - 41 is prime. - 43 is prime. - 47 is prime. Step 3: Composite numbers like 21, 22, 24, 25, 26, 27, 28, 30, 32 and so on have obvious divisors and are not primes. Step 4: The full list of primes between 20 and 50 is therefore: 23, 29, 31, 37, 41, 43 and 47. Step 5: Count them: there are 7 prime numbers. Step 6: Compute their sum: 23 + 29 = 52; 52 + 31 = 83; 83 + 37 = 120; 120 + 41 = 161; 161 + 43 = 204; 204 + 47 = 251. Step 7: The average is sum divided by count: 251 / 7. Step 8: Divide 251 by 7. Since 7 * 35 = 245, the remainder is 6, so 251 / 7 = 35 + 6/7. Step 9: 6/7 is approximately 0.857 (because 6 ÷ 7 ≈ 0.857). So the average is about 35.857.

Verification / Alternative check:
You can verify the decimal more precisely. 6/7 = 0.857142..., which rounds to 0.857 to three decimal places. Adding this to 35 gives 35.857. Comparing with the options shows that 35.857 exactly matches option A. This cross check ensures both the prime list and arithmetic are correct.

Why Other Options Are Wrong:
Values 35.657, 34.857 and 36.657 correspond to averages computed from incorrect sums or wrong counts of primes. For example, missing one prime or including a composite number would change both the sum and the count, producing a different average. None of these alternatives simplify to 251 / 7.

Common Pitfalls:
Students sometimes forget to include 23 or 47 or mistakenly classify a composite like 33 or 39 as prime. Others may misadd the primes, especially if done quickly. Carefully checking divisibility and summing step by step helps ensure accuracy. Using the fraction 251 / 7 before converting to decimal also reduces rounding errors.

Final Answer:
The average of all prime numbers between 20 and 50 is 35.857, which corresponds to option A.