The average age of 14 girls and their teacher's age is 15 years. If the teacher's age is excluded, the average reduces by 1. What is the teacher's age?

Options

A. 35 years
B. 32 years
C. 30 years
D. 29 years

Answer

To find the teacher's age based on the information provided, we can follow these steps:

### Step 1: Calculate the total age of the 14 girls and the teacher
The average age of 14 girls and their teacher is 15 years. The formula for average is:

\[ \text{Average} = \frac{\text{Sum of ages}}{\text{Number of persons}} \]

Let \( T \) be the teacher's age. The total number of persons is 15 (14 girls + 1 teacher).

Using the average:

\[ 15 = \frac{\text{Sum of ages of 14 girls} + T}{15} \]

Multiplying both sides by 15 gives:

\[ \text{Sum of ages of 14 girls} + T = 15 \times 15 = 225 \]

### Step 2: Express the sum of ages of the 14 girls
From the equation above, we can express the sum of the ages of the 14 girls as:

\[ \text{Sum of ages of 14 girls} = 225 - T \]

### Step 3: Calculate the new average without the teacher
When the teacher's age is excluded, the average age of the 14 girls is reduced by 1 year, making the new average 14 years.

Using the average formula again for the 14 girls:

\[ 14 = \frac{\text{Sum of ages of 14 girls}}{14} \]

Multiplying both sides by 14 gives:

\[ \text{Sum of ages of 14 girls} = 14 \times 14 = 196 \]

### Step 4: Set up the equation
Now, we have two expressions for the sum of the ages of the 14 girls:

1. From Step 2: \( 225 - T \)
2. From Step 3: \( 196 \)

Setting these equal to each other gives:

\[ 225 - T = 196 \]

### Step 5: Solve for the teacher's age
Rearranging the equation to solve for \( T \):

\[ T = 225 - 196 \]
\[ T = 29 \]

### Conclusion
The teacher's age is **29 years**.

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