The average of 13 numbers is 75. If the average of the first six numbers is 69 and the average of the last six numbers is 80, what is the 7th number?

The correct answer is Option 81 (or the first option, which is 81).

To find the 7th number, we can use the concept of total sum of numbers based on their averages.

Step 1: Calculate the total sum of all 13 numbers.

We are given that the average of 13 numbers is 75.

$\text{Total sum of 13 numbers}=13\times 75$

$\text{Total sum of 13 numbers}=975$

Step 2: Calculate the sum of the first six numbers.

The average of the first 6 numbers is given as 69.

$\text{Sum of first 6 numbers}=6\times 69$

$\text{Sum of first 6 numbers}=414$

Step 3: Calculate the sum of the last six numbers.

The average of the last 6 numbers is given as 80.

$\text{Sum of last 6 numbers}=6\times 80$

$\text{Sum of last 6 numbers}=480$

Step 4: Find the 7th number.

The 13 numbers consist of the first 6 numbers, the 7th number, and the last 6 numbers (since 6 + 1 + 6 = 13).

$\text{Total sum}=\left(\text{Sum of first 6}\right)+\text{7th number}+\left(\text{Sum of last 6}\right)$

$975=414+\text{7th number}+480$

$975=894+\text{7th number}$

$\text{7th number}=975-894$

$\text{7th number}=81$

Thus, the 7th number is 81.