Favorable Outcomes Choosing Odd Multiples Of 3 Between 1 And 25

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#h1 You Randomly Choose an Integer Between 1 and 25. How Many Favorable Outcomes Are There for Choosing an Odd Multiple of 3?

In the realm of mathematics, particularly probability and number theory, we often encounter problems that require us to determine the number of favorable outcomes within a given set. This article delves into one such problem: If you randomly choose an integer between 1 and 25, how many favorable outcomes are there for choosing an odd multiple of 3? This question combines the concepts of multiples, odd numbers, and range, providing a comprehensive exercise in logical thinking and mathematical application. To solve this, we will systematically identify the multiples of 3 within the specified range, then filter out the odd numbers among them. This exploration will not only provide the answer to the specific question but also enhance your understanding of number properties and problem-solving strategies in mathematics.

Understanding the Problem

To accurately address the question of favorable outcomes, we must first dissect the problem into its core components. We're tasked with selecting a number randomly from the integers between 1 and 25, inclusive. Our objective is to determine how many of these numbers are odd multiples of 3. This requires a clear understanding of what constitutes a 'multiple of 3' and an 'odd number,' and how these two properties intersect within the given range.

  • Multiples of 3: These are numbers that can be obtained by multiplying 3 by an integer. For example, 3, 6, 9, 12, and so on, are multiples of 3.
  • Odd Numbers: These are integers that cannot be divided evenly by 2, leaving a remainder of 1. Examples include 1, 3, 5, 7, and so forth.
  • The Range: Our focus is limited to the integers between 1 and 25, meaning we only consider whole numbers within this boundary. This constraint is crucial because it narrows down the possible numbers we need to evaluate.

By combining these concepts, we aim to pinpoint the numbers that satisfy both conditions: being a multiple of 3 and being odd, all while residing within the 1 to 25 range. This methodical approach ensures we don't miss any favorable outcomes and provides a structured way to solve the problem.

Identifying Multiples of 3 Between 1 and 25

The first step in solving our problem is to identify all the multiples of 3 that fall within the range of 1 to 25. A multiple of 3 is any number that can be obtained by multiplying 3 by an integer. To find these multiples, we can start by multiplying 3 by 1 and continue incrementing the integer until the product exceeds 25. This systematic approach ensures we capture all relevant multiples.

Let's list them out:

  • 3 x 1 = 3
  • 3 x 2 = 6
  • 3 x 3 = 9
  • 3 x 4 = 12
  • 3 x 5 = 15
  • 3 x 6 = 18
  • 3 x 7 = 21
  • 3 x 8 = 24

From this, we can see that the multiples of 3 between 1 and 25 are: 3, 6, 9, 12, 15, 18, 21, and 24. These numbers form the pool from which we will further filter out the odd multiples. Identifying these multiples is a foundational step, as it narrows our focus and allows us to apply the second condition of being an odd number more effectively. This process exemplifies a common problem-solving technique in mathematics: breaking down a complex problem into smaller, manageable steps.

Filtering for Odd Multiples

Having identified all the multiples of 3 within our range (1 to 25), the next critical step is to filter these numbers to find those that are also odd. An odd number, by definition, is any integer that cannot be divided evenly by 2. In other words, when an odd number is divided by 2, it leaves a remainder of 1. This property is fundamental to our filtering process.

Let's revisit the list of multiples of 3 we generated earlier: 3, 6, 9, 12, 15, 18, 21, and 24. We will now examine each number to determine if it fits the criteria of being odd. This involves a simple check: if the number can be divided by 2 without any remainder, it is even; otherwise, it is odd.

Applying this criterion, we find the following:

  • 3 is odd.
  • 6 is even.
  • 9 is odd.
  • 12 is even.
  • 15 is odd.
  • 18 is even.
  • 21 is odd.
  • 24 is even.

Therefore, the odd multiples of 3 between 1 and 25 are: 3, 9, 15, and 21. This filtering process demonstrates the importance of understanding number properties in solving mathematical problems. By applying the definition of odd numbers, we successfully narrowed down our set of multiples to only those that meet our specific criteria. This methodical approach not only provides the correct answer but also reinforces the logical reasoning skills essential in mathematics.

Determining the Favorable Outcomes

After meticulously filtering the multiples of 3 between 1 and 25, we have arrived at the crucial step of determining the favorable outcomes. Recall that the original problem asks for the number of ways to randomly choose an integer between 1 and 25 that is an odd multiple of 3. Through our previous steps, we have successfully identified these numbers.

The odd multiples of 3 within the specified range are 3, 9, 15, and 21. Each of these numbers represents a favorable outcome because they satisfy both conditions: being a multiple of 3 and being an odd number. Now, to answer the question, we simply need to count how many numbers are in this list.

By counting the numbers, we find that there are four odd multiples of 3 between 1 and 25. Therefore, there are four favorable outcomes for randomly choosing an integer that meets the given criteria.

This final step underscores the importance of careful enumeration and accurate counting in problem-solving. It demonstrates how a systematic approach, from identifying multiples to filtering for odd numbers, culminates in a clear and concise answer. Furthermore, it highlights the interconnectedness of mathematical concepts and the value of applying definitions and properties to solve problems effectively.

Conclusion

In conclusion, when randomly choosing an integer between 1 and 25, there are four favorable outcomes for selecting an odd multiple of 3. These outcomes are the numbers 3, 9, 15, and 21. This determination was reached through a systematic process of identifying multiples of 3 within the given range and then filtering those multiples to isolate the odd numbers.

This exercise exemplifies how mathematical problems can be effectively solved by breaking them down into smaller, more manageable steps. By understanding the definitions of multiples and odd numbers, and by applying a logical filtering process, we were able to arrive at the correct answer. This approach not only solves the specific problem but also reinforces the critical thinking and problem-solving skills that are fundamental to mathematics.

Moreover, this problem highlights the interconnectedness of different mathematical concepts. It combines number theory principles, such as multiples and odd numbers, with basic counting and probability concepts. This integration of knowledge underscores the importance of a holistic understanding of mathematics. Ultimately, the solution to this problem demonstrates the power of logical reasoning and methodical application of mathematical principles in arriving at a clear and accurate answer.