Hey guys! Today, we're diving into a fun math problem that involves finding the value of 'm' in an ordered set of observations. This type of question often pops up in statistics and helps us understand how the median works. So, let's get started and break it down step by step.
Understanding the Problem
The core of the problem lies in understanding the median of an ordered set. The median is the middle value in a set of numbers that are arranged in ascending order. When we have an odd number of observations, the median is simply the number in the very middle. However, when we have an even number of observations, like in our case, the median is the average of the two middle numbers.
In this specific problem, we're given the ordered set: 2, 3, (4m - 3), (3m + 1), 11, and 13. We also know that the median of this set is 9. Our mission, should we choose to accept it, is to find the value of 'm' that makes this true. Sounds like a math adventure, right? To really nail this, we need to dissect each part of the problem. First, let's talk about the median. It's super important because it represents the central tendency of our data. Think of it as the balancing point of the dataset. Unlike the mean (or average), the median isn't affected much by extreme values or outliers. This makes it a robust measure in situations where you have some really high or low numbers that could skew the average. For example, if we had a dataset of incomes, the median income would give us a better sense of the 'typical' income than the average income, especially if there are a few billionaires in the mix. Next, we have an ordered set of observations. The fact that the set is ordered is crucial because the median calculation depends on it. We can't just find the median of any old jumbled list of numbers. They have to be arranged in ascending order, from smallest to largest. This ordering helps us easily identify the middle value(s). In our case, the set includes algebraic expressions (4m - 3) and (3m + 1), which adds a layer of complexity. We don't know their exact values until we figure out 'm'. However, the problem tells us that the set is already in ascending order, which gives us a valuable clue about the relative values of these expressions. The expressions (4m - 3) and (3m + 1) are the mystery ingredients in our mathematical recipe. These expressions contain the variable 'm', which we're trying to find. They represent two of the numbers in our ordered set, but their exact values depend on the value of 'm'. This means we'll need to use some algebra to solve for 'm'. The fact that they are part of the ordered set also gives us some hints. For example, we know that (4m - 3) must be greater than 3 (since 3 is the number before it in the set) and (3m + 1) must be less than 11 (since 11 is the number after it). These inequalities could be helpful in checking our final answer.
Setting up the Equation
Now, let's put our detective hats on and figure out how to use the information we have to set up an equation. Since we have six numbers in our set (an even number), the median is the average of the two middle numbers. Looking at our ordered set – 2, 3, (4m - 3), (3m + 1), 11, and 13 – the two middle numbers are (4m - 3) and (3m + 1). We know the median is 9, so we can set up the following equation:
[(4m - 3) + (3m + 1)] / 2 = 9
This equation is the key to unlocking the value of 'm'. It represents the mathematical relationship between the two middle numbers, their average, and the given median. To really understand how we got here, let's break down the equation piece by piece. The left side of the equation, [(4m - 3) + (3m + 1)] / 2, represents the average of the two middle numbers in our ordered set. We know that the median of an even-numbered set is calculated by adding the two middle numbers together and dividing by 2. In our case, the two middle numbers are represented by the algebraic expressions (4m - 3) and (3m + 1). So, we add these expressions together: (4m - 3) + (3m + 1). Then, we divide the sum by 2 to find the average. This gives us the expression [(4m - 3) + (3m + 1)] / 2. The right side of the equation, 9, represents the median of the set, which is given in the problem. We know that the median is the middle value, or in the case of an even-numbered set, the average of the two middle values. The problem explicitly states that the median is 9. Now, let's talk about why this equation works. It works because it directly translates the definition of the median into a mathematical statement. We know that the median is the average of the two middle numbers, and we know that this average is equal to 9. So, we can set the expression representing the average equal to 9. This creates an equation that we can solve for 'm'. The equation is a powerful tool because it allows us to use algebra to solve the problem. We've taken a word problem and turned it into a symbolic representation that we can manipulate to find the answer. This is a fundamental skill in mathematics and problem-solving. By setting up the equation correctly, we've laid the groundwork for solving for 'm'. The next step is to simplify the equation and isolate 'm' on one side.
Solving for m
Alright, let's roll up our sleeves and solve for 'm'. First, we need to simplify the equation. Let's start by getting rid of the fraction. We can do this by multiplying both sides of the equation by 2:
[(4m - 3) + (3m + 1)] / 2 * 2 = 9 * 2
This simplifies to:
(4m - 3) + (3m + 1) = 18
Now, let's combine like terms on the left side of the equation. We have 4m and 3m, which add up to 7m. We also have -3 and +1, which add up to -2. So, our equation becomes:
7m - 2 = 18
Next, we want to isolate the term with 'm' in it. To do this, we can add 2 to both sides of the equation:
7m - 2 + 2 = 18 + 2
This simplifies to:
7m = 20
Finally, to solve for 'm', we need to divide both sides of the equation by 7:
7m / 7 = 20 / 7
This gives us:
m = 20 / 7
So, the value of 'm' is 20/7. Now that we've found a potential value for 'm', it's super important to check our answer. This is a crucial step in problem-solving because it helps us catch any mistakes we might have made along the way. In this case, we need to make sure that our value of 'm' makes sense in the context of the original problem. Specifically, we need to check two things. First, we need to make sure that the expressions (4m - 3) and (3m + 1) are in the correct order within the ordered set. In other words, we need to make sure that (4m - 3) is greater than 3 and (3m + 1) is less than 11. If these inequalities aren't true, then our value of 'm' doesn't work. Second, we need to make sure that the median is actually 9 when we plug in our value of 'm'. This is a direct check of the original problem statement. If the median isn't 9, then we know we've made a mistake somewhere. Let's start by checking the order of the expressions. We found that m = 20/7. So, let's plug this value into the expressions (4m - 3) and (3m + 1). For (4m - 3), we have 4 * (20/7) - 3 = 80/7 - 3 = 80/7 - 21/7 = 59/7, which is approximately 8.43. This is greater than 3, so that's good. For (3m + 1), we have 3 * (20/7) + 1 = 60/7 + 1 = 60/7 + 7/7 = 67/7, which is approximately 9.57. This is less than 11, so that's also good. Now, let's check the median. When m = 20/7, our ordered set becomes 2, 3, 59/7, 67/7, 11, and 13. The two middle numbers are 59/7 and 67/7. The average of these numbers is (59/7 + 67/7) / 2 = (126/7) / 2 = 18 / 2 = 9. So, the median is indeed 9, which confirms our solution.
Final Answer
So, after all that math magic, we've found that the value of m is 20/7. We set up the equation based on the definition of the median, solved for 'm', and then double-checked our answer to make sure it fits the original problem. Great job, guys! This is a classic example of how understanding the core concepts and using algebra can help us solve some pretty interesting problems. Keep up the awesome work, and I'll catch you in the next math adventure!
What is the value of m, given that the median of the ordered set 2, 3, (4m-3), (3m+1), 11, and 13 is 9?
Finding m The Median of an Ordered Set Problem