Question

If the average test score of four students is \(85,\) which of the following scores could a fifth student receive such that the average of all five scores is greater than 84 and less than \(86 ?\) Indicate all such scores. a 88 b 86 c 85 d 83 e 80

Short answer

The fifth score can be 88, 86, 85, or 83.

Step-by-step solution

- Find the total of the first four scores

The average score of the four students is 85. To find the total score of these four students, multiply the average by the number of students: \[ 85 \times 4 = 340 \]

- Define the total score with the fifth student

Let the score of the fifth student be denoted as \( x \). The total score of all five students will then be \( 340 + x \).

- Set up the inequality for the average of all five scores

The average score of all five students should be greater than 84 and less than 86. Therefore, set up the inequality: \[ 84 < \frac{340 + x}{5} < 86 \]

- Solve the inequality

Solve each part of the inequality separately: First part: \[ 84 < \frac{340 + x}{5} \]Multiply both sides by 5: \[ 420 < 340 + x \]Subtract 340 from both sides: \[ 80 < x \]Second part: \[ \frac{340 + x}{5} < 86 \]Multiply both sides by 5: \[ 340 + x < 430 \]Subtract 340 from both sides: \[ x < 90 \]

- Determine the possible scores

Combine the inequalities to find the range of scores for the fifth student: \[ 80 < x < 90 \]Evaluate all given options to see which fall within this range:\(88\): Yes\(86\): Yes\(85\): Yes\(83\): Yes\(80\): No (because it must be greater than 80)

Key concepts

average calculation

Understanding the concept of average is essential for solving this problem. The average (or mean) of a set of numbers is the sum of the numbers divided by the count of numbers.
In this problem, the average score of four students is given as 85.
To find the total score of the four students, simply multiply the average by the number of students. Thus, \(85 \times 4 = 340\).
The concept of average helps us comprehend how each score contributes to the overall group score.

inequality solving

Inequality solving helps us determine the possible values that satisfy the given conditions.
In this exercise, we want the average of all five students' scores to be greater than 84 and less than 86. This involves setting up and solving an inequality, which is a form of algebraic manipulation.
Here’s how to solve this step-by-step:

First, set up the inequality for the average scores:
\( 84 < \frac{340 + x}{5} < 86 \)
Next, solve the inequality in two parts:

• First part: \( 84 < \frac{340 + x}{5} \). Multiply both sides by 5: \(420 < 340 + x \). Subtract 340 from both sides: \(80 < x\).
• Second part: \( \frac{340 + x}{5} < 86 \). Multiply both sides by 5: \(340 + x < 430 \). Subtract 340 from both sides: \( x < 90\).

Combining both parts, the resulting inequality is: \( 80 < x < 90 \).
This framework helps us determine the valid scores for the fifth student.

algebraic manipulation

Algebraic manipulation involves using basic algebra rules to simplify or solve equations and inequalities. This problem requires manipulating the given inequality to find the range of acceptable scores for the fifth student.

We start by identifying all parts of the given inequality: \(84 < \frac{340 + x}{5} < 86\).

Next, isolate and solve for \(x\):

• For \( 84 < \frac{340 + x}{5} \), multiply by 5 to clear the fraction: \(420 < 340 + x\). Subtract 340 to isolate \(x\): \(80 < x\).
• For \( \frac{340 + x}{5} < 86\), multiply by 5 to clear the fraction: \(340 + x < 430\). Subtract 340 to isolate \(x\): \( x < 90\).

The combined range tells us \(80 < x < 90\).
Evaluate possible scores to identify permissible values: \( 88, 86, 85, \text{and} 83\) fall within this range, while \(80\) does not.