Question

$$ \begin{aligned} &\text { Elena's math scores }\\\ &\begin{array}{|l|l|} \hline \text { Test 1 } & 78 \\ \hline \text { Test 2 } & 83 \\ \hline \text { Test 3 } & 81 \\ \hline \text { Midterm exam } & 85 \\ \hline \text { Test 4 } & 89 \\ \hline \text { Test 5 } & 94 \\ \hline \text { Final Exam } & 96 \\ \hline \end{array} \end{aligned} $$ If the final exam carries double weight, what would Elena's new average be? A. 87 B. 88 C. 89 D. 90

Short answer

B. 88

Step-by-step solution

Understand the weights

The final exam carries double weight compared to other tests. This means in the calculation, count the final exam score twice.

List all scores with weights

The scores are: \[78, 83, 81, 85, 89, 94, 96, 96\] (Note that 96 is counted twice)

Calculate the total sum of scores

First, add up all the scores: \[78 + 83 + 81 + 85 + 89 + 94 + 96 + 96 = 702\]

Count the number of scores

There are 7 tests and 1 exam, but since the final exam counts twice, the total number of scores to average is 8.

Calculate the average

Divide the total sum by the number of scores: \[\frac{702}{8} = 87.75\]

Determine the closest answer

The closest option to 87.75 is 88.

Key concepts

Mean Calculation

Calculating the mean is a fundamental statistical concept. The mean, also known as the average, is determined by summing all the values in a data set and then dividing by the count of those values. For instance, in Elena's case, she has several math scores which need to be summed up first. These scores are adjusted to account for the final exam, which carries double weight. Summing them up gives a total, which is then divided by the number of scores, taking the weighted counts into consideration. This division by the total number of counts gives the average score, which offers a central measure of her performance. Regularly breaking down the steps in mean calculation can make it much easier to grasp.

Weights in Data Sets

When calculating the mean for a set of data, certain values might be more significant than others. These 'weights' determine how much influence each value has on the final average. In Elena's scenario, the final exam score is twice as important as each test score. To reflect this in the average calculation, we count the final exam score twice. This shifts the balance because the higher weight makes this score more impactful on the average. Understanding weights in data sets is crucial for interpreting data accurately, as it provides a clearer representation of different values' relevances.

Averaging Scores

Averaging scores involves totaling all the individual scores and then dividing by the number of scores. This gives a single number that represents an overall measure of performance. In weighted situations, we adjust the scores accordingly before averaging. For example, if one score carries double weight, we include it twice in our calculations. By averaging in this manner, we make sure that more important scores have the proper influence over the final average. This method ensures a fair calculation, especially in academic settings where different assessments may vary in importance.

Mathematical Reasoning

Mathematical reasoning is the ability to use logical thinking to solve problems. This involves understanding concepts, breaking down problems, and executing calculations step-by-step. Elena's score average problem requires understanding weights, summing values, and then dividing them. Each step must be reasoned out: identifying the correct weights, properly summing the adjusted scores, and correctly applying division to find the mean. Each error-free step ensures that the final solution is accurate. Developing strong mathematical reasoning skills helps in all areas of life, as it forms the basis for solving complex problems systematically and correctly.