Average Replacement Problems with Correction Formula
Average Replacement Problems involve finding the value of a person or object removed from a group and the value that replaces it. The method uses the change in the group’s total, calculated from the old and new averages. This page explains the main formulas, multiple-replacement cases, shortcut methods and solved aptitude examples.
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What Are Average Replacement Problems?
If a group has n members, its total is n multiplied by its average. When one value is replaced, the change in the total equals n × (new average − old average). Therefore, the difference between the replacing and removed values is found directly from the change in average.
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Average Replacement Problems
Understand how averages change when one or more values are replaced, with classroom-based examples illustrating the calculation of revised averages.
Average Replacement Problems Formula & Tricks
Important Formulas
Use a positive result when the average increases and a negative result when the average decreases.
This applies when one value replaces another in a group of n members.
Use this when the replacing value and both averages are known.
Use the total difference when several members are replaced and the group size remains unchanged.
Quick Tricks
Find the increase or decrease in average before working with individual values. Then multiply that change by the fixed group size.
If the new average is lower, use a negative change. This automatically shows that the replacing value is smaller than the removed value.
When several values are replaced, do not calculate each value separately unless required. Compare the sum of the new values with the sum of the old values.
Average Replacement Problems Concepts
Finding the Removed Value
For a group of n members, first calculate n × (new average − old average). Subtract this result from the replacing value. If the average decreases, the change is negative, so the removed value becomes greater than the replacing value.
Finding the Replacing Value
Calculate the change in average and multiply it by the unchanged number of members. Add the result to the removed value. A positive change gives a larger replacing value, while a negative change gives a smaller one.
Multiple Members Replaced
If k values are replaced but the group still contains n members, use: sum of replacing values − sum of removed values = n × change in average. The number of replaced members does not replace n in this formula; n is the total group size.
Replacement with Addition or Removal
When a member is added, new total = old total + added value. When a member is removed, new total = old total − removed value. The old and new totals are calculated using their respective group sizes and averages.
Average Replacement Problems Data Table
Use the structured data below while following the explanation and examples.
Average Change and Total Change
The sign of the average change determines the direction of the total change.
| Situation | Average change | Change in total |
|---|---|---|
| Average increases by d | +d | +n × d |
| Average decreases by d | −d | −n × d |
| Average remains unchanged | 0 | 0 |
Note: Here, n is the total number of members before and after the replacement.
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Average Replacement Problems Revision Points
Use these rules to revise the calculation method.
- Total = Number of items × Average.
- For a fixed-size group, change in total = n × (new average − old average).
- Replacing value − removed value = n × change in average.
- For multiple replacements, compare the sums of all replacing and removed values.
- Use the total group size n, not merely the number of replaced members.
- If members are added or removed, calculate old and new totals using their different group sizes.
- A decrease in average must be represented by a negative change.
Average Replacement Problems FAQs
What is the average replacement problems formula?
For a fixed-size group, replacing value − removed value = n × (new average − old average), where n is the total number of members.
A group of 20 students has its average increase by 3 marks after one student is replaced. What is the change in total marks?
The change in total marks is 20 × 3 = 60 marks.
The average of 15 numbers falls by 2 after replacement. How does the total change?
The total decreases by 15 × 2 = 30. Thus, the change in total is −30.
The average age of 12 people rises by 1.5 years when a 20-year-old person is replaced. What is the new person's age?
Increase in total age = 12 × 1.5 = 18 years. New person's age = 20 + 18 = 38 years.
Does n represent the number of replaced members?
No. In a fixed-size replacement problem, n represents the total number of members in the group. The number of replaced members is used only when identifying the sums of the old and new values.
How is a multiple-replacement problem solved?
Use: sum of new values − sum of old values = n × (new average − old average). For example, if n = 25 and the average rises by 2, the sum difference is 25 × 2 = 50.