Average of 13 number is 52. Average of first 7 number is 51. Average of last 7 number is 54. Find `7^(th)` number.

Answer

To find the 7th number given the average of 13 numbers, the average of the first 7 numbers, and the average of the last 7 numbers, we can follow these steps:

### Step 1: Calculate the sum of the 13 numbers
The average of 13 numbers is given as 52.
To find the sum of these numbers, we use the formula:
\[ \text{Sum} = \text{Average} \times \text{Number of Observations} \]
So,
\[ \text{Sum of 13 numbers} = 52 \times 13 = 676 \]

### Step 2: Calculate the sum of the first 7 numbers
The average of the first 7 numbers is given as 51.
Using the same formula:
\[ \text{Sum of first 7 numbers} = 51 \times 7 = 357 \]

### Step 3: Calculate the sum of the last 7 numbers
The average of the last 7 numbers is given as 54.
Again using the formula:
\[ \text{Sum of last 7 numbers} = 54 \times 7 = 378 \]

### Step 4: Set up the equation for the 7th number
In the sums calculated, the 7th number is included in both the sum of the first 7 numbers and the sum of the last 7 numbers. Therefore, we can express the relationship as:
\[ \text{Sum of 13 numbers} = \text{Sum of first 7 numbers} + \text{Sum of last 7 numbers} - \text{7th number} \]
Substituting the values we found:
\[ 676 = 357 + 378 - \text{7th number} \]

### Step 5: Solve for the 7th number
Now, we can rearrange the equation to find the 7th number:
\[ \text{7th number} = 357 + 378 - 676 \]
Calculating the right side:
\[ \text{7th number} = 735 - 676 = 59 \]

Thus, the 7th number is **59**.

---