The average of 13 numbers is 42. If a `14^(th)` number is included, then the average becomes 44. What is the `14^(th)` number ?

Options

A. 70
B. 62
C. 66
D. 68

Answer

To find the 14th number when the average of 13 numbers is given, we can follow these steps:

### Step 1: Calculate the total of the first 13 numbers
Given that the average of 13 numbers is 42, we can find the total of these numbers by multiplying the average by the number of numbers:

\[ \text{Total of 13 numbers} = \text{Average} \times \text{Number of numbers} = 42 \times 13 \]

Calculating this gives:

\[ 42 \times 13 = 546 \]

### Step 2: Calculate the total of 14 numbers
When the 14th number is included, the average becomes 44. We can find the total of these 14 numbers by multiplying the new average by the total number of numbers:

\[ \text{Total of 14 numbers} = \text{New Average} \times \text{Total Numbers} = 44 \times 14 \]

Calculating this gives:

\[ 44 \times 14 = 616 \]

### Step 3: Find the 14th number
Now, we can find the 14th number by subtracting the total of the first 13 numbers from the total of the 14 numbers:

\[ \text{14th number} = \text{Total of 14 numbers} - \text{Total of 13 numbers} = 616 - 546 \]

Calculating this gives:

\[ 616 - 546 = 70 \]

Thus, the 14th number is **70**.

### Summary of Steps:
1. Calculate the total of the first 13 numbers: \( 42 \times 13 = 546 \).
2. Calculate the total of the 14 numbers: \( 44 \times 14 = 616 \).
3. Find the 14th number: \( 616 - 546 = 70 \).