Thrice the square of a natural number decreased by four times the number equals 50 more than the number. What is that natural number?

Introduction / Context:
This question is an example of forming and solving a quadratic equation from a verbal statement about a natural number. Such problems are very common in aptitude tests as they assess your ability to translate words into algebra and then solve the resulting equation. Here the relationships involve the square of a number, multiples of the number, and a constant difference of 50.

Given Data / Assumptions:
- Let the natural number be n.- Thrice the square of the number means 3 * n^2.- Four times the number means 4 * n.- The expression thrice the square decreased by four times the number equals 50 more than the number.

Concept / Approach:
The phrase thrice the square of a natural number decreased by 4 times the number describes an algebraic expression that we can write as 3 * n^2 - 4 * n. The phrase 50 more than the number describes n + 50. The problem states that these two quantities are equal. Therefore, we set up the equation 3 * n^2 - 4 * n = n + 50 and then bring all terms to one side to obtain a standard quadratic equation of the form a n^2 + b n + c = 0. We then solve this quadratic and interpret which solution is acceptable as a natural number.

Step-by-Step Solution:
- Start from the verbal condition: thrice the square decreased by four times the number equals 50 more than the number.- Write it algebraically: 3 * n^2 - 4 * n = n + 50.- Move all terms to the left side: 3 * n^2 - 4 * n - n - 50 = 0.- Simplify the like terms: 3 * n^2 - 5 * n - 50 = 0.- This is a quadratic equation with a = 3, b = -5, c = -50.- Compute the discriminant D = b^2 - 4 * a * c = (-5)^2 - 4 * 3 * (-50).- D = 25 + 600 = 625.- The square root of 625 is 25.- The solutions are n = (5 ± 25) / (2 * 3) = (5 ± 25) / 6.- So n = 30 / 6 = 5 or n = -20 / 6 which simplifies to -10 / 3.- Since n is a natural number, discard the negative non integer value.- Therefore n = 5.

Verification / Alternative check:
Substitute n = 5 back into the original statement. Thrice the square is 3 * 5^2 = 3 * 25 = 75. Four times the number is 4 * 5 = 20. So thrice the square decreased by four times the number is 75 - 20 = 55. The right side is 50 more than the number, which is 5 + 50 = 55. Both sides match, so n = 5 satisfies the condition perfectly.

Why Other Options Are Wrong:
- Option 4: If n = 4, 3 * 4^2 - 4 * 4 = 48 - 16 = 32, but 50 more than 4 is 54, so it fails.- Option 7: If n = 7, 3 * 7^2 - 4 * 7 = 147 - 28 = 119, while 7 + 50 = 57, not 119.- Option 6: If n = 6, 3 * 6^2 - 4 * 6 = 108 - 24 = 84, but 6 + 50 = 56.- Option 8: If n = 8, 3 * 8^2 - 4 * 8 = 192 - 32 = 160, while 8 + 50 = 58.

Common Pitfalls:
Candidates sometimes misinterpret the phrase decreased by and write 4 * n - 3 * n^2 instead of 3 * n^2 - 4 * n. Another frequent mistake is to forget to move all terms to one side of the equation, leading to incorrect quadratic coefficients. Always rewrite verbal expressions carefully, verify the signs, and then solve the quadratic using the discriminant or factorization when possible. Finally, remember to check which solutions satisfy the requirement of being a natural number.

Final Answer:
The required natural number is 5.