Question

The last math test that you took had 100 regular points and 10 bonus points. You received a total score of \(87,\) which included 9 bonus points. What would your score have been without any bonus points?

Short answer

Without any bonus points, the score would have been 78.

Step-by-step solution

Identify the total score and bonus points

From the problem, it's given that the overall score is 87 points and we're also informed that this total includes 9 bonus points.

Subtract the bonus points from total score

In order to calculate the score without any bonus points, subtract the bonus points from the total. So, \( 87 - 9 \) must be calculated.

Determine the score without bonus points

The subtraction in the previous step gives the score without the bonus points. This means that \( 87 - 9 = 78 \).

Key concepts

Subtraction in Algebra

Understanding subtraction in algebra is akin to learning the basic rules of the road before driving; it's essential for navigating mathematical problems efficiently. When you're presented with a total and need to determine a part of it, subtraction is the tool you'll use. In the context of algebra, instead of concrete numbers, we often deal with unknowns, but the principles remain the same.

Consider the equation \( X - Y = Z \) as a basic formula, where \( X \) is our starting point, much like the total points in a test, \( Y \) is what we're removing or subtracting from it, analogous to the bonus points, and \( Z \) is the result after the subtraction, akin to the score without bonus points. It's critical to execute this operation accurately to ensure that the algebraic expressions we're working with reflect the true state of the situation at hand. This straightforward operation of 'taking away' provides a foundation for more complex algebraic problem-solving.

Bonus Points Calculation

Bonus points often bring a smile to students' faces, as they offer a way to boost one's score on tests and assignments. However, it's important to understand exactly how these additional points affect your total score.

To calculate the impact of bonus points, one must identify both the total combined score and the amount attributed solely to bonuses. The procedure is simple: you subtract the bonus points from the total score to assess the base score. This action clears the cloudy skies and reveals how you would have fared without the extra help. Knowing this not only helps in evaluating your performance but also in setting realistic academic goals. For instance, in the given exercise, the student deducted 9 bonus points from the total score of 87, unveiling a base score of 78. Bonus points should always be seen as a happy extra, never a crutch, and clear calculation methods help maintain that perspective.

Interpreting Math Problems

Successfully interpreting math problems is a skill akin to deciphering a code; it requires careful reading and a sharp understanding of what is being asked. The first step is to thoroughly read the problem and identify key information, including numbers, operations to perform, and what the question ultimately seeks.

In the example provided, the facts are clear: 100 regular points, 10 bonus points, and a total score of 87 with 9 of those as bonus points. The task? Determine the score without the bonus points. This translates to a need for subtraction. To bolster comprehension, draw parallels to real-world scenarios, organize the information in a structured format, or even re-phrase the problem in simple terms. Using these strategies, students can transform mathematical confusion into a series of manageable steps, eventually leading to the correct answer and, critically, a deeper understanding of the mathematical concepts at play.