The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a

Q: Wait, why is the denominator still 25? Shouldn't it be 24 when the teacher leaves?

That is a classic trap! While there is a fleeting moment where only 24 teachers are present, the problem states a new teacher is appointed *in his place*. By the time we calculate the *new mean*, the group is back to its original size of 25. Always look at the 'before' and 'after' states of the system.

Q: Is there a faster way to solve this without calculating the total sum of 1000?

Absolutely! You can use the 'deviation' method. The mean dropped by $1$ year ($40 - 39 = 1$). Since there are $25$ teachers, the total age dropped by $25 \times 1 = 25$ years. The retiring teacher was $60$, and the new one is $25$ years younger than him. So, $60 - 25 = 35$. Both methods are elegant, but understanding the 'Total Sum' method builds a stronger foundation for more complex problems.

Q: Why does the mean drop if the new teacher is younger? I get confused by the direction of change.

Think of the mean as the 'center of gravity' of the ages. If you remove a heavy weight (a $60$-year-old) and replace it with a lighter weight (a $35$-year-old), the center of gravity must shift downwards. The mean is simply the mathematical representation of that center of gravity.

Question

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is ______.

Sigma Percentile · Accepted Answer

### The Philosophy of the Average: A Journey into Data

Welcome, future engineer. Today, we are not just solving a statistics problem; we are learning to visualize the flow of numbers.

When you look at a problem involving the 'mean,' I want you to stop seeing it as a dry formula. Instead, imagine a balance scale where the mean is the point where the scale balances perfectly. When we change the components of that scale, the balance shifts.

### Phase 1: The Initial State

Imagine a school staff room with $25$ teachers. We are told their mean age is $40$.

In the world of JEE, never leave the mean as an abstract concept. Immediately convert it into the 'Total Age Pool.'

Mathematically, we define the mean as:

$$
\bar{x} = \frac{\sum X_i}{n}
$$

Therefore, the total sum is $\sum X_i = n 	imes \bar{x}$. Plugging in our values, we get $25 	imes 40 = 1000$.

This $1000$ is the 'Total Age Pool' of the school. It is the bedrock upon which we will build our solution.

### Phase 2: The Departure

A change occurs. A senior teacher, aged $60$, retires. He walks out of the staff room, and with him, he takes his $60$ years of experience.

Our 'Total Age Pool' is no longer $1000$. It has been depleted.

We perform a simple subtraction: $1000 - 60 = 940$. This $940$ represents the sum of the ages of the remaining $24$ teachers.

### Phase 3: The Arrival and the New Balance

A new teacher joins. Let us call their age $N$. The staff room is once again full, with $25$ teachers.

The new total sum of ages is now $940 + N$. We are told that the new mean age of the school is $39$.

This is the crucial moment where the physics of the problem meets the algebra. We set up our equation:

$$
\frac{940 + N}{25} = 39
$$

This equation is the bridge between the old state and the new state. It tells us that the new total, distributed across $25$ people, results in an average of $39$.

### Phase 4: The Algebraic Resolution

Now, let us solve for $N$. We multiply both sides by $25$ to clear the denominator:

$$
940 + N = 39 	imes 25
$$

Calculating $39 	imes 25$ can be done quickly if you think of it as $(40 - 1) 	imes 25$. This gives us $1000 - 25 = 975$.

Now, the equation is simple: $940 + N = 975$. Subtracting $940$ from both sides, we find **$N = 35$**. The new teacher is $35$ years old.

### The Takeaway

Notice how the mean dropped from $40$ to $39$. This makes perfect physical sense.

We removed a $60$-year-old (who was well above the average) and replaced them with a $35$-year-old (who is below the average). The 'center of gravity' of the ages had to shift downwards.

You have just mastered the logic of replacement in statistics. Keep this visualization in your toolkit, and no variation of this problem will ever stump you again.

Enter Numerical Value:

Visualized Solution

The Initial Setup

The Concept of Mean

Calculating Initial Total Age

Initial Total Age

A Teacher Retires

Sum After Retirement

A New Teacher Joins

The New Mean

Setting up the Equation

Solving for the New Sum

Calculating the Right Side

Finding the Final Answer

The Sigma Insight: Measures of Central Tendency (Mean, Median, Mode)

The Philosophy of the Average

Phase 1

Phase 2

Phase 3

Phase 4

The Takeaway

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