Question

There are 20 questions on an exam, each worth 5 points. Your percent grade \(p\) varies directly with the number \(n\) of correct answers. Find an equation that relates \(p\) and \(n .\) What is your percent grade if you get 17 correct answers?

Short answer

The percent grade for 17 correct answers is 85%.

Step-by-step solution

Setting up the direct variation equation

Since percent grade \(p\) varies directly with \(n\), we can set up the direct variation equation as \(p = kn\), where \(k\) is the constant of variation. Because the maximum grade is 100% when all 20 questions are answered correctly, we substitute \(p = 100\) and \(n = 20\) into the equation. Solving for \(k\) gives \(k = p/n = 100/20 = 5\).

Substitute the constant \(k\) into the equation

Substitute \(k = 5\) into \(p = kn\) to get \(p = 5n\). This is the equation that relates percent grade \(p\) and the number of correct answers \(n\).

Finding the percent grade for 17 correct answers

For finding the percent grade with 17 correct answers, you need to substitute \(n = 17\) into the equation \(p = 5n\), which renders \(p = 5*17 = 85\).

Key concepts

Algebraic Equations

Algebraic equations are the bread and butter of algebra, a fundamental area of mathematics. These equations involve variables, constants, and arithmetic operations. Understanding how to set up and solve an equation is crucial for finding values that satisfy certain conditions.

For instance, in our problem, we used the direct variation to establish that the percent grade on an exam \(p\) is dependent on the number of questions correctly answered \(n\). The relationship was expressed as \(p = kn\), which is a simple algebraic equation. Here, \(k\) is our constant of variation, and it represents the rate at which the percent grade changes with each additional correct answer.

By understanding such algebraic equations, students can solve a wide range of problems that involve direct or indirect relationships, including those in real-life scenarios such as calculating grades, distances, or financial interest.

Percent Grade Calculation

Calculating percent grades is a practical and common application of direct variation in algebra. The percent grade represents what portion of the total points available has been earned. To find the percent grade, you often set a ratio that compares the number of points earned to the total possible points, and then multiply by 100 to get a percentage.

In our problem, the equation \(p = 5n\) arises from the fact that each question on the exam is worth 5 points. Hence, the percent grade \(p\) directly varies with the number \(n\) of correct answers. The total number of points is \(20 \times 5 = 100\), matching the concept that a perfect score corresponds to 100%. Once you know the total possible points and the value for each correct answer, you can easily calculate the percent grade for any number of correct answers.

Constant of Variation

The constant of variation, symbolized as \(k\) in direct variation problems, plays a pivotal role. It is the factor that relates how one variable changes in relation to another. In direct variation, when one variable increases, the other increases at a rate determined by the constant of variation.

For our grade calculation exercise \(p = 5n\), the constant of variation \(k = 5\) signifies that for each additional correct answer, the percent grade increases by 5%. It's derived by dividing the total percent (100%, for full marks) by the total number of questions (20). This constant remains the same throughout the problem and is essential for finding specific values, such as the percent grade for 17 correct answers, which equates to \(85%\) when applying the constant in the equation.