The arithmetic mean (average) of seventy-five numbers is 35. If each of these numbers is increased by 5, then what will be the new mean of the resulting numbers?

Introduction / Context:
This problem tests a very basic but powerful property of averages. When every number in a data set is increased or decreased by the same amount, the average also increases or decreases by that same amount. Recognizing this saves time and avoids unnecessary calculations in exams.

Given Data / Assumptions:

• There are 75 numbers in total.
• The current arithmetic mean of these 75 numbers is 35.
• Each number is increased by 5.
• We must find the new arithmetic mean after the increase.

Concept / Approach:
The average of a set of numbers is defined as the total sum divided by the count of numbers. If each number in the set is increased by a constant value k, the total sum increases by k times the number of elements, but the count itself does not change. Therefore, the average also increases by exactly k. In this question, k is +5, so the new average is simply the old average plus 5.

Step-by-Step Solution:
Number of values = 75.Old mean = 35.Total sum of the original 75 numbers = 35 * 75.Each number is increased by 5, so the total increase in sum = 5 * 75.New total sum = old total sum + 5 * 75.New mean = (old total sum + 5 * 75) / 75.This simplifies to (35 * 75) / 75 + (5 * 75) / 75 = 35 + 5 = 40.Hence the new mean is 40.

Verification / Alternative check:
Instead of using sums, we can apply the direct property: adding the same constant to every term shifts the mean by that constant. Since each number increases by 5, the average must also increase by 5. Starting from 35, we get the new mean as 35 + 5 = 40. Both the formal and intuitive reasoning give the same result.

Why Other Options Are Wrong:
Options 30 and 35 would mean that the average has decreased or remained unchanged, which is impossible when all numbers increase by 5. Options 50 and 60 represent much larger increases in the mean and do not correspond to the constant increase of only 5 per number. Only 40 is consistent with the basic property of averages in this situation.

Common Pitfalls:
Some students try to recalculate the total sum explicitly, which is unnecessary and time consuming, but not wrong. Others mistakenly add 5 only once to the average of 35/75 or confuse the number of observations. Remember, when every value increases by k, the mean also increases by k, regardless of how many values there are.

Final Answer:
The new arithmetic mean of the numbers is 40.