Problem 31 - Elementary Algebra

Question

A gymnast scores 8.8 on the vault, 9.3 on the uneven bars, 9.1 on the balance beam, and 9.8 on the floor exercise. What is her overall average?

Step-by-Step Solution

Answer

The gymnast's overall average score is 9.25.

Step 1: List the Scores

First, we need to list down all the scores the gymnast received during her performances. These scores are: - Vault: 8.8 - Uneven Bars: 9.3 - Balance Beam: 9.1 - Floor Exercise: 9.8.

Step 2: Find the Sum of the Scores

To find the overall average, we start by calculating the sum of all the scores. We do this by adding the scores together:\[8.8 + 9.3 + 9.1 + 9.8.\]Performing the addition:\[8.8 + 9.3 = 18.1\]\[18.1 + 9.1 = 27.2\]\[27.2 + 9.8 = 37.0\]Thus, the total sum of the scores is 37.0.

Step 3: Count the Number of Events

Next, we determine the number of events for which scores were received. In this case, the gymnast has scores for four events: vault, uneven bars, balance beam, and floor exercise, so there are 4 events.

Step 4: Calculate the Average Score

Now we calculate the average score by dividing the total sum of the scores by the number of events. The formula for the average is:\[\text{Average} = \frac{\text{Total Sum of Scores}}{\text{Number of Events}}\]Substituting the values we have:\[\text{Average} = \frac{37.0}{4} = 9.25\]So, the gymnast's overall average score is 9.25.

Key Concepts

Addition

When calculating an average, one of the initial steps is to add up all the values. This is known as addition. In our example, the gymnast has scores from several events that need to be added together. Addition involves combining the scores into a single total. Here, it means:

• Starting with the first score, 8.8, and adding each subsequent score one by one.
• The careful addition of each number ensures the final sum is correct.

Breaking down the addition, we proceed like this:

• Add 8.8 and 9.3 to get 18.1.
• Next, add 9.1, bringing your total to 27.2.
• Finally, add 9.8 to reach a total of 37.0.

This stepping process helps check and confirm the correct sum of 37.0.

Division

Once we know the total sum, we use division to find the average. Division is the process of splitting a total into equal parts. To find the average score:

• Divide the sum of all scores by the number of scores.
• This divides the total score into equal parts across the events.

In this particular case, we have a total of four scores, which means the calculation for the gymnast's average is:

• Take the sum, 37.0, and divide it by 4.
• This gives an average score of 9.25.

This process shows how each event contributes equally to the overall score.

Algebraic Expressions

Algebra often involves expressions that use letters to represent numbers, but here we're using the concept of an expression in a numerical sense. An algebraic expression for the average can be seen in the formula.An average formula looks like this:\[\text{Average} = \frac{\text{Sum of all scores}}{\text{Number of scores}}\]Inserting our values, we reformulate it specifically as:\[\text{Average} = \frac{37.0}{4}\]This expression shows clearly what numbers and operations are involved in finding the average. By using clear expressions, it helps us recognize and solve for parts of the problem effectively.

Problem Solving

Problem solving is a process that involves finding the solution to a question or issue. In our example, we're tackling how to determine the gymnast's average score. It requires steps such as:

• Understanding the problem by listing all scores and knowing what you need: the average.
• Planning by deciding on the steps to take: addition first, division next.
• Solving by carrying out your plan: add all scores and then divide by the number of events.
• Checking by verifying if each step adds up correctly and if your solution makes sense in context.

By following these organized steps, we ensure that the solution we find is correct and reliable. This methodical approach is crucial for success in math problems and beyond!