Finding Three Numbers With A Given Mean And Ratio

by TextBrain Team 50 views

Hey guys! Ever stumbled upon a math problem that looks like a puzzle? Well, let's dive into one together! This is a classic problem involving means, ratios, and a little bit of algebraic thinking. We're going to break it down step by step, so it's super easy to follow. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what's going on. The core of the problem revolves around finding three unknown numbers. We know a couple of things about them: First, we know the mean (or average) of a set of nine numbers. Remember, the mean is just the sum of all the numbers divided by how many numbers there are. In our case, the mean of nine numbers is 12. Second, we're given six of these numbers explicitly: 3, 5, 6, 10, 12, and 18. Third, and this is crucial, the remaining three numbers are in a specific ratio: 1:2:6. This ratio tells us how these numbers relate to each other. It means if the first number is x, the second is 2x, and the third is 6x. Our goal is to find the actual values of these three numbers. To do this effectively, we need to combine our knowledge of means and ratios. We'll start by using the information about the mean to figure out the total sum of all nine numbers. Then, we'll use the given six numbers to find the sum of the remaining three. Finally, we'll apply the ratio to break down that sum and find the individual numbers. So, with a clear understanding of the problem, we can strategize and solve it effectively.

Setting Up the Equation

Alright, let's translate this word problem into a mathematical equation. This is a crucial step in solving any math problem. We know the mean of nine numbers is 12. Remember, the mean is calculated by summing all the numbers and dividing by the total count. So, we can write this as:

(Sum of all nine numbers) / 9 = 12

To find the sum of all nine numbers, we simply multiply both sides of the equation by 9:

Sum of all nine numbers = 12 * 9 = 108

Okay, great! Now we know the total sum of all nine numbers is 108. We also know six of these numbers: 3, 5, 6, 10, 12, and 18. Let's find the sum of these six numbers:

3 + 5 + 6 + 10 + 12 + 18 = 54

So, the sum of the six known numbers is 54. This means the sum of the remaining three numbers must be the total sum minus the sum of the known numbers:

Sum of the three unknown numbers = 108 - 54 = 54

Excellent! We've narrowed it down. We know the sum of the three unknown numbers is 54, and we also know they are in the ratio 1:2:6. Now, we can use this information to set up another equation. Let's represent the three numbers as x, 2x, and 6x (based on the ratio). Their sum is:

x + 2x + 6x = 54

This equation is our key to unlocking the values of the three unknown numbers. We've successfully translated the problem into a manageable algebraic equation. Next, we'll solve this equation to find the value of x and then determine the three numbers.

Solving for the Unknown

Now comes the fun part – actually solving for our unknown! We've got the equation:

x + 2x + 6x = 54

The first step is to combine the like terms on the left side of the equation. We're essentially adding up all the x terms:

(1 + 2 + 6)x = 54

9x = 54

Now, we need to isolate x. To do this, we divide both sides of the equation by 9:

x = 54 / 9

x = 6

Woohoo! We found the value of x. But hold on, we're not quite done yet. Remember, x is just the first number in our ratio. We need to find the other two numbers as well. This is a classic mistake – stopping too early in the problem! We know the three numbers are in the ratio 1:2:6, and we've found that x (the 1 part of the ratio) is 6. So:

  • The first number is x = 6
  • The second number is 2x = 2 * 6 = 12
  • The third number is 6x = 6 * 6 = 36

So, there you have it! We've found the three unknown numbers: 6, 12, and 36. We used our knowledge of means, ratios, and basic algebra to crack this problem. Now, let's double-check our work to make sure we got it right.

Verifying the Solution

It's always a good idea to check your work, guys. Verification is key to ensuring we've got the correct solution. We found the three numbers to be 6, 12, and 36. Let's make sure these numbers fit the conditions of the problem.

First, let's check the ratio. Are 6, 12, and 36 in the ratio 1:2:6? Well:

  • 6 / 6 = 1
  • 12 / 6 = 2
  • 36 / 6 = 6

Yep, the ratio checks out! They are indeed in the ratio 1:2:6.

Now, let's check the mean. We know the mean of all nine numbers (3, 5, 6, 10, 12, 18, 6, 12, and 36) should be 12. Let's add them up:

3 + 5 + 6 + 10 + 12 + 18 + 6 + 12 + 36 = 108

And now divide by 9 (since there are nine numbers):

108 / 9 = 12

Awesome! The mean is 12, just like the problem stated. So, our solution satisfies both the ratio condition and the mean condition. We can be confident that we've found the correct three numbers. This step is super important because it confirms our understanding and application of the concepts. We've not only solved the problem but also verified our answer, making sure we haven't made any silly mistakes along the way.

Key Takeaways

So, what did we learn from this math adventure? There are a few key takeaways that we can apply to similar problems in the future:

  1. Understanding the Problem: Before you start crunching numbers, make sure you fully understand what the problem is asking. Break it down into smaller parts and identify the key information. In this case, understanding the concepts of mean and ratio was crucial.
  2. Translating to Equations: Turn word problems into mathematical equations. This makes the problem much easier to solve. We translated the information about the mean and the ratio into algebraic equations.
  3. Solving Step-by-Step: Don't try to do everything at once. Solve the equation step-by-step, showing your work clearly. This helps prevent errors and makes it easier to follow your logic. We first found the value of x and then used it to find the other two numbers.
  4. Verifying the Solution: Always, always, always check your answer! Make sure your solution makes sense and fits the conditions of the problem. We verified that our numbers satisfied both the ratio and the mean conditions.
  5. Combining Concepts: Many math problems require you to combine different concepts. This problem combined the concepts of mean, ratio, and algebra. Practice identifying and applying the relevant concepts.

By mastering these skills, you'll be well-equipped to tackle a wide range of math problems. So, keep practicing, keep exploring, and remember to have fun with it!

Practice Problems

Want to put your new skills to the test? Here are a couple of practice problems similar to the one we just solved:

  1. The mean of seven numbers is 15. Four of the numbers are 2, 8, 12, and 17. The remaining three numbers are in the ratio 2:3:5. Find the three numbers.
  2. The mean of ten numbers is 20. Seven of the numbers are 4, 6, 10, 15, 22, 25, and 30. The remaining three numbers are in the ratio 1:4:5. Find the three numbers.

Try solving these problems on your own. Use the steps we outlined above: understand the problem, set up the equation, solve for the unknown, and verify your solution. You've got this!

Solving math problems like this is like cracking a code, guys! You've got the tools, you've got the knowledge, now go out there and conquer those numbers! Keep practicing, and you'll become a math whiz in no time. And remember, math can actually be fun when you approach it with a curious and problem-solving mindset. So, keep exploring, keep learning, and most importantly, keep believing in yourself! You've got this!