Problem 146 - Beginning and Intermediate Algebra

Question

Find the average of each group of numbers. All integers between -15 and \(-10,\) including both -15 and -10

Step-by-Step Solution

Answer

The average is -12.5.

Step 1 - Identify the range of integers

Write down all the integers between -15 and -10, including both -15 and -10: -15, -14, -13, -12, -11, -10

Step 2 - Count the total number of integers

Count the numbers in the list: there are 6 integers in total.

Step 3 - Find the sum of the integers

Add all the integers from the list: -15 + (-14) + (-13) + (-12) + (-11) + (-10) = -75

Step 4 - Calculate the average

To find the average, divide the sum by the number of integers: \[ \text{Average} = \frac{\text{Sum}}{\text{Total number of integers}} = \frac{-75}{6} = -12.5 \]

Key Concepts

integer operations

Integers are whole numbers that include positive numbers, negative numbers, and zero. Integer operations involve basic math tasks like addition, subtraction, multiplication, and division.
When we work with operations involving multiple integers, it is essential to follow specific rules:

• Adding two positive integers results in a positive integer.
• Adding two negative integers gives a negative integer.
• When adding positive and negative integers, subtract the smaller absolute value from the larger absolute value and keep the sign of the larger absolute value.
• For subtraction, convert the problem to an addition problem by adding the opposite. For example, subtracting -10 is the same as adding +10.

This foundational understanding helps in tasks like finding the average of a group of integers, which involves multiple integer operations.

summing integers

Summing integers is straightforward: it's all about adding the numbers together. Let's go over the steps to sum a series of integers:
Start with the numbers listed: -15, -14, -13, -12, -11, -10.
Use addition rules mentioned earlier:

• Start by adding -15 and -14 to get -29.
• Add -13 to -29 to get -42.
• Continue this process: -42 + (-12) = -54, then -54 + (-11) = -65, and finally -65 + (-10) to get -75.

The sum of the numbers -15, -14, -13, -12, -11, and -10 is -75.
Understanding how to sum integers is crucial, as it is a foundational step in calculating the arithmetic mean or average.

arithmetic mean

The arithmetic mean, commonly referred to as the average, of a set of numbers is found by dividing the sum of the numbers by the count of the numbers. Here's the formula:
\[\text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of elements}}\]
Let's go through the steps to find the average of a set of integers:

• First, list all the integers in the range given (e.g., -15, -14, -13, -12, -11, -10).
• Next, sum these integers, as shown in the previous section, which gives -75.
• Count the number of integers (6 in this case).
• Finally, divide the sum by the number of integers: \(\frac{-75}{6} = -12.5\).

The average of the integers from -15 to -10 is -12.5.
Calculating the average is a common mathematical operation that helps to understand the central tendency of a set of numbers.