Problem 42 - Algebra 2

Question

ACT/SAT Amanda wanted to determine the average of her 6 test scores. She added the scores correctly to get \(T,\) but divided by 7 instead of 6 The result was 12 less than her actual average. Which equation could be used to determine the value of \(T ?\) $$ \begin{array}{l}{\text { A } 6 T+12=7 T} \\ {\text { B } \frac{T}{7}=\frac{T-12}{6}} \\ {\text { C } \frac{T}{7}+12=\frac{T}{6}} \\\ {\text { D } \frac{T}{6}=\frac{T-12}{7}}\end{array} $$

Step-by-Step Solution

Answer

The correct equation is C: \( \frac{T}{7} + 12 = \frac{T}{6} \).

Step 1: Understanding the Problem

Amanda calculated her average test score by mistakenly dividing the total of her scores by 7 instead of 6. This incorrect average is 12 less than the correct one. We need to find the equation that represents this situation.

Step 2: Expressing the Correct Average

The correct average of Amanda's test scores should be calculated by dividing the total sum of her scores, denoted as \( T \), by 6. Thus, the correct average can be expressed as \( \frac{T}{6} \).

Step 3: Expressing the Incorrect Average

Amanda mistakenly divided the total sum of her scores by 7, resulting in an incorrect average. Therefore, the incorrect average is \( \frac{T}{7} \).

Step 4: Relating the Incorrect and Correct Average

We know from the problem statement that the incorrect average is 12 less than the correct average. Therefore, we can write the equation \( \frac{T}{7} = \frac{T}{6} - 12 \).

Step 5: Rearranging for Equation Format

We rearrange the equation \( \frac{T}{7} = \frac{T}{6} - 12 \) to align with the options provided. Thus, \( \frac{T}{7} + 12 = \frac{T}{6} \), which matches option C.

Key Concepts

Average Calculation

The concept of average is a fundamental topic in algebra and statistics. It's often used to summarize a set of numbers into a single value that represents the center of the data set. This single value is what's known as the "mean." To calculate the average, you follow these basic steps:

• Add up all the numbers in the set to find their sum.
• Divide the total by the number of values in the set.

For instance, if Amanda was calculating the average of her 6 test scores correctly, she would first add up her scores to get a total, which we call \( T \). She would then divide \( T \) by the total number of test scores, which is 6. This gives us \( \frac{T}{6} \), the correct average of her test scores.
However, in scenarios like Amanda's, it's crucial to divide by the correct number to get an accurate mean. Misunderstanding either step can lead to errors like dividing by an incorrect number, resulting in a faulty average.

Equation Solving

Solving algebraic equations is an essential skill in mathematics. It involves finding the value of unknowns that make the equation true. In Amanda's problem, we're tasked with finding \( T \), the total sum of her test scores.

The step-by-step approach to solving equations involves:

• Identifying what you need to find; in this case, the correct equation to find \( T \).
• Understanding the relationship between different parts of the problem.

For Amanda, we know:

• The correct average calculated as \( \frac{T}{6} \).
• The incorrect average, \( \frac{T}{7} \), is 12 less than the correct one.

This leads us to form the equation \( \frac{T}{7} = \frac{T}{6} - 12 \). Rearranging different parts of this equation manipulates it into the form discussed in options, which is \( \frac{T}{7} + 12 = \frac{T}{6} \).
When solving, it's important to keep track of all your operations to ensure no mistake is made in aligning terms on either side.

Test Scores Analysis

Analyzing test scores often requires understanding various metrics, including averages and their impacts when miscalculated. In this scenario, Amanda's situation highlights how an error in computation can influence the perceived outcome.

• When test scores are summed up to get a total \( T \), each number in the set contributes to the final average.
• By calculating the average incorrectly — dividing by 7 instead of 6 — Amanda undervalues her actual performance by 12 points.

Understanding this difference helps in ensuring that scores are interpreted accurately during test results analysis.
By forming equations to solve for errors, one can verify that each step along the analysis is correct and precisely reflects the real performance. Utilizing equations as shown is a powerful way to check and balance academic analysis.