A112
B114
C115
D117
Answer:
C. 115
Read Explanation:
Understanding Average and Replacement Problems
The average (or mean) of a set of numbers is calculated by dividing the sum of all numbers by the count of the numbers. That is, Average = Sum / Number of Items.
Consequently, the Sum of Numbers = Average × Number of Items. This relationship is fundamental for solving problems involving averages in competitive exams.
Impact of Replacing a Number on Average
When a number in a group is replaced by another number, the total sum of the group changes, which directly affects the average.
If the average increases after replacement, it indicates that the newly added number is greater than the number that was removed.
Conversely, if the average decreases, the newly added number is smaller than the removed number.
Calculating the Change in Total Sum
In this specific problem, there are 15 numbers in the group.
The average increased by 3 after the replacement.
The total increase in the sum of all numbers is determined by multiplying the increase in the average by the total number of items: Total Increase in Sum = Change in Average × Number of Items.
Therefore, the Total Increase in Sum = 3 × 15 = 45.
Determining the Newly Added Number
The calculated increase in the total sum (45) represents the exact difference by which the new number exceeds the old number that was replaced.
The number that was removed from the group was 70.
Thus, the Newly Added Number = Old Number Replaced + Total Increase in Sum.
Substituting the values: Newly Added Number = 70 + 45 = 115.
Key Formula for Competitive Exams
For quick and efficient problem-solving in competitive examinations, remember this general formula for number replacement problems:
Newly Added Number = Old Number Replaced ± (Change in Average × Number of Items)
Use the '+' sign if the average increases, and the '-' sign if the average decreases.
Applying this formula to the given problem: New Number = 70 + (3 × 15) = 70 + 45 = 115.