Impact of Changes on Averages

1. When a New Item is Added

Let's start with the most common scenario: a new person joins a group, a new score is added, and so on. This changes both the total sum and the number of items.

The Fundamental Approach

The most basic way to solve this is to go back to the core formula: Sum = Average × Count.

1. Calculate the original sum: Original Sum = Original Average × Original Count.
2. Calculate the new sum: New Sum = Original Sum + Value of New Item.
3. Calculate the new average: New Average = New Sum / (Original Count + 1).

This method always works, but it can involve large numbers.

The Formulaic Shortcut

Mathematicians have rearranged the fundamental steps into a convenient formula. It's useful to see how it's derived, as it reinforces the underlying logic.

The Deviation Method (Recommended)

This is the fastest and most intuitive method, building directly on our last lesson. The core idea is to see how the new item's value compares to the old average.

Let's use an example from the BYJU's article:
The average of a batsman in 16 innings is 36. In the next innings, he scores 70 runs. What is his new average?

1. Compare the new value to the old average. The old average is 36. The new score is 70. The batsman scored 70 - 36 = 34 runs more than the average. This is the surplus.
2. Distribute the surplus. This surplus of 34 runs will be distributed among all innings, including the new one. There are now 16 + 1 = 17 innings.
3. Calculate the increase in average. The average will increase by Surplus / New Count = 34 / 17 = 2 runs.
4. Find the new average. New Average = Old Average + Increase = 36 + 2 = 38.

This method avoids large calculations like 16 × 36 and is much quicker.

To solidify this, let's look at a few more examples using this exact shortcut technique.

2. When an Item is Removed

Now, let's consider the case where an item is taken out of the group. The logic is the reverse of adding an item.

The Fundamental Approach

1. Calculate the original sum: Original Sum = Original Average × Original Count.
2. Calculate the new sum: New Sum = Original Sum - Value of Removed Item.
3. Calculate the new average: New Average = New Sum / (Original Count - 1).

The Formulaic Shortcut

Again, there's a pre-packaged formula for this, which is useful to know.

The Deviation Method (Recommended)

Here's how to adapt the deviation logic for removal:

Think about the "hole" left by the removed item.

• If the removed item's value was higher than the original average, it took a surplus with it. The remaining group's average will decrease.
• If the removed item's value was lower than the original average, it took a deficit with it. The remaining group's average will increase.

Let's take an example:
The average weight of 10 people is 60 kg. A person weighing 80 kg leaves the group. What is the new average weight?

1. Compare the removed value to the old average. The old average is 60 kg. The person who left weighed 80 kg. They were 80 - 60 = 20 kg heavier than the average. This is the surplus that was removed.
2. Distribute the impact of the loss. The group has lost a surplus of 20 kg. This loss must be absorbed by the remaining 10 - 1 = 9 people.
3. Calculate the decrease in average. The average will decrease by Removed Surplus / New Count = 20 / 9 kg.
4. Find the new average. New Average = Old Average - Decrease = 60 - 20/9 ≈ 60 - 2.22 = 57.78 kg.

3. When an Item is Replaced

This is often the easiest case to handle with the deviation method because the number of items remains the same.

The Fundamental Approach

1. Calculate the original sum.
2. Calculate the new sum: New Sum = Original Sum - Value of Old Item + Value of New Item.
3. Calculate the new average: New Average = New Sum / Original Count.

The Deviation Method (Highly Recommended)

The logic here is beautifully simple:

1. Calculate the net change in the total sum. Net Change = Value of New Item - Value of Old Item.
2. Distribute this net change. This change is distributed equally among all members of the group (the count doesn't change).
3. Find the change in the average. Change in Average = Net Change / Count.
4. Calculate the new average. New Average = Old Average + Change in Average.

Let's see this in action with an example from the Wordpandit article:
The average age of 6 students is increased by 2 years when one student whose age was 13 years is replaced by a new boy. Find the age of the new boy.

This is a reverse problem, but the logic holds.

1. Find the total change in sum. The average increased by 2 years across 6 students. Total Change = Change in Average × Count = 2 × 6 = 12 years.
2. Relate to the replaced item. This total increase of 12 years happened because the new student was heavier than the old one. So, Value of New Boy - Value of Old Boy = 12.
3. Calculate the new boy's age. Value of New Boy = Value of Old Boy + 12 = 13 + 12 = 25 years.

Now let's quickly review the formula for this case, which essentially formalizes the steps we just took.

Conclusion

Today we've systematically broken down how averages change when items are added, removed, or replaced. While the fundamental Sum = Average × Count method is a reliable fallback, your goal for speed and accuracy in the CAT should be to master the deviation method.

Key Takeaways:

• Core Principle: All these problems can be solved by thinking about the net change to the total sum and distributing that change across the members of the final group.
• Adding an Item: Find the surplus/deficit of the new item compared to the old average. Distribute this over the new, larger group.
• Removing an Item: Find the surplus/deficit of the removed item. This amount is now gone, so its effect is distributed over the new, smaller group. Removing a surplus lowers the average; removing a deficit raises it.
• Replacing an Item: Find the net difference between the new and old items. Distribute this difference over the unchanged number of items.

In our next lesson, we will venture into the topic of mixtures. You will see that the concepts of weighted average and the deviation method are the bedrock for a powerful shortcut technique used to solve mixture problems, known as the rule of alligation.