Multiples Of 3 Between 5 And 41 A Step-by-step Solution
Hey guys! Have you ever stumbled upon a math problem that seems tricky at first glance? Well, today we're going to break down one of those problems together. We'll be tackling a question that involves finding multiples of a number within a specific range. Specifically, we're going to figure out how many multiples of 3 there are between 5 and 41. Sounds like fun, right? Let's dive in!
Understanding Multiples
Before we jump into solving the problem, let's quickly recap what multiples are. In simple terms, a multiple of a number is the result you get when you multiply that number by an integer (a whole number). For example, the multiples of 3 are 3, 6, 9, 12, and so on. Each of these numbers is obtained by multiplying 3 by an integer (3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, etc.). Understanding this concept is crucial for solving our problem.
Identifying the Range
Our problem asks us to find the multiples of 3 between 5 and 41. This means we need to identify the multiples of 3 that are greater than 5 and less than 41. We won't include 5 or 41 in our count because the question specifies "between" these numbers. This is a common trick in math problems, so it's important to pay close attention to the wording!
Methods to Find Multiples of 3
There are a couple of ways we can approach this. One way is to simply list out the multiples of 3 and see which ones fall within our range. The other way is to use division to find the first and last multiples of 3 within the range and then calculate how many multiples there are in total. Let's explore both methods.
Method 1: Listing Multiples
This method is straightforward and easy to understand. We'll start listing the multiples of 3 until we reach a number greater than 41. Here's how it goes:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42...
Now, we need to identify the multiples that fall between 5 and 41. Looking at the list, we can see that the multiples of 3 within our range are:
6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39
Counting these numbers, we find that there are 12 multiples of 3 between 5 and 41.
Advantages and Disadvantages
This method is easy to grasp and doesn't require any complex calculations. However, it can be time-consuming if the range is very large or if we're dealing with a number that has many multiples. In such cases, the division method might be more efficient.
Method 2: Using Division
This method involves using division to find the first and last multiples of 3 within our range. Here's how it works:
Finding the First Multiple
To find the first multiple of 3 greater than 5, we can divide 5 by 3.
5 ÷ 3 = 1 with a remainder of 2.
This tells us that 3 goes into 5 once, with a remainder of 2. To find the next multiple of 3, we need to multiply 3 by the next whole number after 1, which is 2.
3 x 2 = 6
So, the first multiple of 3 greater than 5 is 6.
Finding the Last Multiple
To find the last multiple of 3 less than 41, we divide 41 by 3.
41 ÷ 3 = 13 with a remainder of 2.
This tells us that 3 goes into 41 thirteen times, with a remainder of 2. So, the last multiple of 3 less than 41 is:
3 x 13 = 39
Calculating the Number of Multiples
Now that we know the first and last multiples of 3 within our range (6 and 39), we can calculate the total number of multiples. We can do this by dividing the last multiple by 3 and subtracting the result of dividing the first multiple by 3, and then adding 1. This might sound a bit confusing, so let's break it down.
First multiple = 6
Last multiple = 39
Now, divide both multiples by 3:
6 ÷ 3 = 2
39 ÷ 3 = 13
Subtract the results and add 1:
13 - 2 + 1 = 12
So, there are 12 multiples of 3 between 5 and 41.
Why Does This Method Work?
This method works because we're essentially finding the position of the first and last multiples in the sequence of multiples of 3. By subtracting the positions and adding 1, we get the total number of multiples within the range. For example, 6 is the second multiple of 3 (3 x 2), and 39 is the thirteenth multiple of 3 (3 x 13). The difference between their positions (13 - 2) gives us the number of multiples between them, but we need to add 1 to include the first multiple itself.
The Answer
Using both methods, we've found that there are 12 multiples of 3 between 5 and 41. So, the correct answer is A) 12.
Checking the Options
The problem provides us with the following options:
A) 12
B) 13
C) 14
D) 15
Our calculations confirm that option A) 12 is the correct answer. It's always a good idea to check your answer against the given options to make sure you haven't made any mistakes.
Justifying the Answer
To justify our answer, we can explain the steps we took to find the number of multiples of 3 between 5 and 41. Here's a concise justification:
"To find the number of multiples of 3 between 5 and 41, we can either list out the multiples or use division. Listing the multiples of 3 (6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39) shows that there are 12 multiples. Alternatively, dividing 5 and 41 by 3 and using the quotients to calculate the number of multiples also gives us 12. Therefore, the answer is 12."
This justification clearly explains the methods we used and the reasoning behind our answer. It's important to be able to explain your solutions in a clear and logical way, especially in math problems.
Why This Problem Matters
You might be wondering, why is it important to know how to find multiples of a number? Well, this skill is fundamental in many areas of mathematics, including:
- Number theory: Multiples are essential in understanding prime numbers, divisibility rules, and factorization.
- Algebra: Multiples are used in simplifying expressions, solving equations, and working with polynomials.
- Real-life applications: Understanding multiples can help you with tasks like scheduling, budgeting, and measuring.
By mastering this concept, you're building a solid foundation for more advanced mathematical topics. Plus, it's a great mental exercise that sharpens your problem-solving skills!
Tips for Solving Similar Problems
If you encounter similar problems in the future, here are some tips to keep in mind:
- Read the question carefully: Pay close attention to the wording, especially the range specified (e.g., between, inclusive). A slight change in wording can significantly affect the answer.
- Choose the right method: Decide whether listing multiples or using division is more efficient for the given problem. For smaller ranges, listing might be quicker, but for larger ranges, division is usually the better option.
- Double-check your work: Always double-check your calculations and make sure your answer makes sense in the context of the problem.
- Practice makes perfect: The more you practice solving these types of problems, the better you'll become at them. Try solving similar problems with different numbers and ranges to build your confidence.
Conclusion
So, guys, we've successfully tackled the problem of finding the multiples of 3 between 5 and 41. We explored two different methods, understood why they work, and justified our answer. Remember, the key to solving math problems is to break them down into smaller steps and understand the underlying concepts. Keep practicing, and you'll become a math whiz in no time!
If you have any questions or want to explore more math problems, feel free to ask. Keep learning and keep shining!
