Problem:
A number is multiplied by 3 times itself and then 61 is subtracted from the product obtained. If the final result is 9200, then the number can be represented as 7k. Find the value of k.

Solution:

Let's break down the problem into steps to find the value of k.

Step 1: Translate the given problem statement into a mathematical equation.

The problem states that a number is multiplied by 3 times itself and then 61 is subtracted from the product obtained. Mathematically, we can represent this as:

3k^2 - 61 = 9200

Step 2: Simplify the equation.

To find the value of k, we need to solve the equation. Let's rearrange the equation by bringing 9200 to the left side:

3k^2 - 61 - 9200 = 0

Simplifying further:

3k^2 - 9261 = 0

Step 3: Factorize the quadratic equation.

To factorize the quadratic equation, we need to find two numbers whose product is -9261 and whose sum is 0. The factors of -9261 are -1, 1, -3, 3, -7, 7, -9, 9, -11, 11, -21, 21, -27, and 27.

By trial and error, we find that the factors -3 and 3099 satisfy the conditions:

(3k + 3)(k - 3099) = 0

Step 4: Solve for k.

From the factorized equation, we have two possible solutions:

3k + 3 = 0 or k - 3099 = 0

Solving the first equation:

3k = -3
k = -3/3
k = -1

Solving the second equation:

k = 3099

Step 5: Determine the value of k.

Since k represents the value of the number multiplied by 7, we can conclude that k = -1 or k = 3099.

Therefore, the value of k is 3099.

Summary:

By following the steps outlined above, we found that the value of k is 3099.

K = 441