Question

Choose the statement below that is true about the given numbers. A) The number in column A is greater. (B) The number in column B is greater. (C) The two numbers are equal. (D) The relationship cannot be determined from the given information. $$\begin{array}{|c|c|}\hline \text { Column A } & \text { Column B } \\\\\hline \text { The mean of } -2,-2,-2,4, & \text { The mean of } -2,-2,-3,-3,-3 \\\\\hline\end{array}$$

Short answer

(A) The number in column A is greater.

Step-by-step solution

Calculate Mean of Column A

In order to find the mean of the numbers in Column A (-2,-2,-2,4), the numbers are added together: \[ -2 - 2 - 2 + 4 = -2 \] and then divided by the count of numbers, which is 4: \[ -2 ÷ 4 = -0.5 \] Hence, the mean of the set of numbers in Column A is -0.5.

Calculate Mean of Column B

In order to find the mean of the numbers in Column B (-2,-2,-3,-3,-3), the numbers are added together: \[ -2 - 2 -3 -3 -3 = -13 \] and then divided by the count of numbers, which is 5: \[ -13 ÷ 5 = -2.6 \] Hence, the mean of the set of numbers in Column B is -2.6.

Compare Mean of Column A and Column B

After calculation, it has been found that the mean of the numbers in Column A is -0.5 and the mean of the numbers in Column B is -2.6. Clearly, -0.5 is greater than -2.6. Therefore, the number in Column A is greater than the number in Column B.

Key concepts

Mean Calculation

When solving problems involving averages, one vital operation is the mean calculation. The mean, often referred to as the average, is found by summing up all the numbers in a dataset and then dividing by the quantity of elements present in that dataset.

The formula for calculating the mean is expressed as: \
\( \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Number of elements}} \).

For example, if we’re given a set of numbers like -2, -2, -2, and 4 in Column A, we add them to get -2. To find the mean, we divide by the number of elements, which is 4, resulting in a mean of -0.5. This simple yet powerful tool allows us to compare sets of data at a glance, revealing which of two sets is larger or smaller on average.

Comparing Numbers

Comparing numbers is a foundational skill in mathematics, used to evaluate the magnitude of different values. When comparing numbers like means, it’s crucial to understand the numerical ordering, where on the number line each value lies, and how negative numbers work.

Negative numbers are less than zero, and in a comparison, the more negative a number is, the smaller it is considered. For example, when comparing -0.5 and -2.6, we observe that -0.5 is closer to zero and thus larger than -2.6. By understanding this concept and applying it to means, one can determine that the set with the higher (or less negative) mean is the set with the higher average value.

Algebraic Operations

Algebraic operations, such as addition, subtraction, multiplication, and division, are essential tools for manipulating and solving equations. In the case of mean calculation, we primarily rely on addition and division.

By adding all the numbers within a data set, we compile their total value. Subsequently, to balance the operation and extract the average, we use division, distributing the total evenly among the elements. This process, while it may seem straightforward, is a cornerstone of algebraic thinking and is applied in various contexts, from simple averages to complex equations.