Solving A Game Show Puzzle Determining Correct Answers

In the captivating realm of television game shows, contestants embark on intellectual duels, their minds sharpened to a keen edge, vying for the coveted title of champion. Amidst the dazzling lights and pulsating music, a complex scoring system often lies beneath the surface, a numerical labyrinth that determines victory or defeat. Consider the enthralling scenario where contestants accumulate points for correct answers while facing deductions for incorrect responses. This seemingly simple dynamic conceals a wealth of mathematical intrigue, a challenge that beckons us to unravel its secrets.

Let's delve into a specific case, a television game show where contestants earn 5 points for each correct answer, a reward for their mental prowess. However, the path to glory is not without its pitfalls, as a penalty of 2 points is levied for each wrong answer, a humbling reminder of the game's demanding nature. On a recent episode, the victorious contestant emerged triumphant with a total score of 80 points, a testament to their intellectual fortitude. This champion had faced a barrage of 30 questions, a gauntlet of inquiries designed to test their knowledge and acumen. Now, a tantalizing question arises: How many questions did this astute contestant answer correctly? This is a mathematical puzzle that beckons us to solve, a challenge that requires us to decipher the interplay between correct answers, wrong answers, and the ultimate score.

To embark on our mathematical quest, let's introduce a variable, a symbolic representation of the unknown. Let x be the number of questions the contestant answered correctly, a value that holds the key to our solution. Since the contestant answered a total of 30 questions, the number of questions answered incorrectly can be expressed as 30 - x, a simple subtraction that unveils a crucial piece of the puzzle. Now, let's translate the game's scoring system into mathematical language. The contestant earns 5 points for each correct answer, so the total points earned from correct answers can be represented as 5x, a multiplication that reflects the accumulation of knowledge. Conversely, the contestant loses 2 points for each wrong answer, resulting in a deduction of 2(30 - x) points, a penalty for missteps along the way. The contestant's final score is the difference between the points earned from correct answers and the points lost from wrong answers, an equation that encapsulates the essence of the game's dynamics. This final score is given as 80 points, a target that guides our mathematical pursuit. Thus, we can formulate the following equation:

5x - 2(30 - x) = 80

This equation is the cornerstone of our solution, a mathematical expression that encapsulates the relationship between correct answers, wrong answers, and the ultimate score.

Now, let's embark on the process of solving the equation, a journey that requires us to navigate the intricacies of algebra. Our goal is to isolate x, the variable that represents the number of correct answers, and unveil its true value.

First, we need to expand the equation, a process that involves distributing the -2 across the terms within the parentheses:

5x - 60 + 2x = 80

Next, let's combine like terms, a simplification that streamlines our equation:

7x - 60 = 80

To isolate the term containing x, we need to add 60 to both sides of the equation, maintaining the balance of the mathematical scales:

7x = 140

Finally, to unveil the value of x, we divide both sides of the equation by 7, completing our algebraic journey:

x = 20

Thus, the contestant answered 20 questions correctly, a testament to their knowledge and acumen. This solution is the culmination of our mathematical efforts, the answer that we sought to uncover. To verify our solution, we can substitute x = 20 back into the original equation:

5(20) - 2(30 - 20) = 100 - 2(10) = 100 - 20 = 80

The equation holds true, confirming the validity of our solution. The contestant's performance was indeed impressive, a display of intellectual prowess that led to victory.

In the realm of television game shows, mathematical equations often lie hidden beneath the surface, waiting to be deciphered. By carefully translating the game's rules into mathematical language, we can unravel the mysteries of scoring systems and determine the outcome of intellectual battles. In this particular scenario, we successfully determined that the winning contestant answered 20 questions correctly, a testament to their knowledge and skill. This exercise not only demonstrates the practical application of algebra but also highlights the importance of logical reasoning and problem-solving in everyday life. So, the next time you find yourself engrossed in a television game show, remember that mathematics may be playing a crucial role behind the scenes, shaping the destiny of contestants and determining the ultimate champion.

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