Problem 10 - Algebra and Trigonometry

Question

\(7-18 \cdot\) Express the given quantity in terms of the indicated variable. The average of four quiz scores if each of the first three scores is \(8 ; \quad q=\) fourth quiz score

Step-by-Step Solution

Answer

The average is \( \frac{24 + q}{4} \).

Step 1: Identify Known and Unknown Variables

We know there are four quiz scores. The first three are each 8, and the fourth score is represented by the variable \( q \). We need to express the average of these scores in terms of \( q \).

Step 2: Setup the Expression for Average

The average of four numbers is calculated by adding the numbers together and dividing by 4. Therefore, the average of the four quiz scores can be expressed as \( \frac{8 + 8 + 8 + q}{4} \).

Step 3: Simplify the Expression

First, add the known scores: \( 8 + 8 + 8 = 24 \). Then substitute back into the expression, resulting in \( \frac{24 + q}{4} \).

Step 4: Final Expression for the Average

The expression for the average of the four quiz scores in terms of \( q \) is \( \frac{24 + q}{4} \).

Key Concepts

Expression Simplification

Simplifying an expression means breaking it down into a simpler or more manageable form without changing its value. In algebra, this often involves combining like terms or performing arithmetic operations. For example, when calculating the average of the four quiz scores given in the problem, we first add up the three constant scores. They are all 8. Therefore, we compute the total as:

• First quiz score: 8
• Second quiz score: 8
• Third quiz score: 8

When these are summed, the result is 24: \[ 8 + 8 + 8 = 24 \].
This step already simplifies the expression considerably by reducing the three individual scores into a single constant. Inserting the variable for the fourth score, the expression becomes: \[ 24 + q \].
This expresses three out of the four scores into one term, simplifying our task of finding the average. The final expression offers a neat and concise representation which is easy to work with, illustrating the power of simplification in algebra.

Variables in Algebra

Variables are symbols used to represent unknown or general values. In algebra, they allow us to formulate equations and expressions where certain values might change or be unknown. In the given problem, the variable in use is \( q \), representing the unknown fourth quiz score.
Using \( q \) allows us to keep the expression flexible. Suppose the value of the fourth quiz score changes or isn't known initially, \( q \) gives us a way to continue solving or rearranging until that information is available.
This dynamic quality makes variables essential in algebra. They provide a way to work with unknown values and prepare us to solve more complex or real-world problems where not all information is immediately available.
Variables help structure the problem, making it easier to understand and manage later on.

Average Calculation

Calculating an average is a fundamental concept in mathematics that provides a measure of central tendency or a typical value. The average can offer insights about a set of numbers, helping to understand trends or typical outcomes. To find the average of a set of numbers, you add them up and then divide by the count of numbers.
In our scenario, we have four quiz scores: three of them are fixed at 8, and one is represented by \( q \), the variable. Calculating the average involves:

• Summing all quiz scores: \( 8 + 8 + 8 + q \).
• The sum so far is 24, thus the total becomes \( 24 + q \).

To find the average, this sum is divided by the number of scores, which is four: \[ \frac{24 + q}{4} \].
This expression demonstrates the average score in terms of \( q \), illustrating how each score, including unknown values, contributes equally to the overall average.