Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the number of ways in which a computer can randomly generate one or more such integers, or pairs of integers, from 1 through 15. A pair of integers whose sum is 20

Step-by-Step Solution

Answer

The number of ways in which the computer can generate the pairs is 3.

Step 1: Define the integers range

The range of integers should be between 1 and 15.

Step 2: Determine the pairs

Find pairs of two different integers in the range that result in a sum of 20. In this case, there are three possible pairs that consist of two distinct integers: (5, 15), (6, 14), (7, 13)

Step 3: Calculate the possible ways

To calculate the number of ways such pairs could be generated, simply count the number of pairs obtained in Step 2. This results in a total of 3 pairs, or ways, in which such numbers could be generated.

Key Concepts

Understanding Integer Pairs

Integer pairs are combinations of two whole numbers that can be plotted within a defined range, and they work together to satisfy a particular condition or equation. In this exercise, the focus is on pairs of integers where each pair results in a specific summed value.
For example, when we talk about pairs of integers that add up to a given number, we are looking for two numbers within a specified range that, when combined through addition, equal that target number.
To find integer pairs:

• Determine the target sum you want the pairs to achieve.
• Set a range for your integers (e.g., 1 to 15).
• Identify two distinct numbers within this range whose sum is the target number.

This exercise is a great example of using integer pairs, where the sum needs to be 20 and the integers fall between 1 and 15.

Calculating the Sum of Integers

The sum of integers is a fundamental concept in mathematics, often used in problem solving and data analysis. It is the result of adding two or more integers together. Understanding how to calculate and manipulate sums is crucial.
Given a range of integers, our task is to identify those instances where the sum equals a specified number.
For the problem at hand:

• We need to find pairs of integers from the numbers 1 through 15.
• These pairs should add up to exactly 20.

In this case, the solution demonstrates that there are three pairs: (5, 15), (6, 14), and (7, 13), each satisfying the condition. This shows the simplicity of addition but emphasizes the method of finding specific solutions within constraints.

The Range of Integers

When working with integers, clearly defining the range is crucial, as it sets the boundaries within which you can work. The range specifies the lowest and highest values that your integers can take.
For example, the exercise posits a range from 1 to 15. This implies any integer you consider for forming pairs must lie within these bounds.
Working with a defined integer range ensures:

• Every considered integer fits within your specified criteria.
• Your results remain accurate and relevant to the given problem.

The defined range plays a fundamental role in limiting possibilities, making large number problems more tractable by focusing only on the most promising candidate solutions.