Unraveling The Numbers: A Math Puzzle

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Unraveling the Numbers: A Math Puzzle

Hey math enthusiasts! Today, we're diving into a fun number puzzle. This problem gives us some clues about five numbers and challenges us to find them. The cool part? We know the average of these numbers and some relationships between them. Let's break it down and solve it together! Get ready to put on your thinking caps, guys!

The Problem Unpacked: Identifying the Clues

Alright, let's get down to the nitty-gritty. The problem states that we have five numbers. Here's what we know:

  • The fifth number is 21.5. That's our first piece of the puzzle! Knowing one number right off the bat is always a good start.
  • The first number is 1.6 times smaller than the second. This means if we know the first number, we can easily find the second one. Or, if we know the second, we can get the first. It's all about that multiplicative relationship!
  • The third number is 2.1 times greater than the first. This gives us another direct link. We can find the third number just by knowing the first one.
  • The fourth number is 1.8 times greater than the first. Another direct relationship to the first number. Similar to the third, knowing the first unlocks the fourth.
  • The average (arithmetic mean) of all five numbers is 14.7. This is super important because it connects everything! The average gives us a total sum we can use to tie it all together.

So, with these clues, we need to uncover the values of all five numbers. No sweat, right? Let's figure this out!

Setting Up the Equations: Translating Words to Math

Okay, guys, let's translate the problem's clues into mathematical expressions. This is where we build the foundation for solving the puzzle.

Let's denote the first number as x. This will be our base variable, allowing us to express all other numbers in terms of x.

Here’s how we can represent each number:

  • First number: x (our base)
  • Second number: 1.6x (since the first is 1.6 times smaller, the second is bigger, and is thus 1.6 times the first, so 1.6x)
  • Third number: 2.1x (2.1 times the first number, since its bigger)
  • Fourth number: 1.8x (1.8 times the first number, because its bigger)
  • Fifth number: 21.5 (given)

Now, let's tackle the average. The average is the sum of all the numbers divided by the total count (which is 5 in this case). We know the average is 14.7. So, we can write the equation like this:

(x + 1.6x + 2.1x + 1.8x + 21.5) / 5 = 14.7

See? We've just turned all those words into one neat equation. The hardest part is over, I promise! Now we just need to solve for x.

Solving for x: The First Number Revealed!

Time to crack the code and find the value of x! We'll start by simplifying our equation.

Remember the equation is: (x + 1.6x + 2.1x + 1.8x + 21.5) / 5 = 14.7.

First, multiply both sides of the equation by 5 to get rid of the fraction:

x + 1.6x + 2.1x + 1.8x + 21.5 = 14.7 * 5

Which simplifies to:

x + 1.6x + 2.1x + 1.8x + 21.5 = 73.5

Next, combine all the x terms on the left side:

6.5x + 21.5 = 73.5

Now, subtract 21.5 from both sides to isolate the x term:

6.5x = 73.5 - 21.5

So we get:

6.5x = 52

Finally, divide both sides by 6.5 to solve for x:

x = 52 / 6.5

This gives us:

x = 8

Ta-da! We've found that the first number (x) is 8. Isn't that neat? Now we can use this to find all the other numbers, guys!

Unveiling the Numbers: Finishing the Puzzle

Awesome, we've found the first number! Now, let's use that information and the relationships we defined earlier to find the rest of the numbers. It's like finding treasure, but with numbers, haha!

Remember how we defined each number in terms of x?

  • First number: x = 8
  • Second number: 1.6 * x = 1.6 * 8 = 12.8
  • Third number: 2.1 * x = 2.1 * 8 = 16.8
  • Fourth number: 1.8 * x = 1.8 * 8 = 14.4
  • Fifth number: 21.5 (given)

And there we have it! We've successfully found all five numbers. We've gone from a bunch of clues to a complete set of values. Pretty amazing, right?

Checking Our Work: Ensuring Accuracy

Okay, before we declare victory, let's do a quick check to make sure our answers are correct. After all, it's always smart to verify your work! We'll use the average as our key to check.

Remember, the average of all five numbers should equal 14.7. So let's calculate the average using the numbers we've found:

(8 + 12.8 + 16.8 + 14.4 + 21.5) / 5 = ?

Let's add those numbers up: 8 + 12.8 + 16.8 + 14.4 + 21.5 = 73.5

Now divide by 5: 73.5 / 5 = 14.7

And there it is! Our calculated average is 14.7, which matches the average given in the problem. This means our calculations are spot-on. Hooray!

Wrapping Up: Celebrating the Solution

And that, my friends, concludes our number puzzle adventure! We successfully found all five numbers by carefully interpreting the clues, setting up equations, and solving for our unknown variables. Remember, guys, math can be super fun when you break down problems step by step.

Here’s a quick recap of the numbers we found:

  • First number: 8
  • Second number: 12.8
  • Third number: 16.8
  • Fourth number: 14.4
  • Fifth number: 21.5

We started with a bunch of relationships and an average, and through careful calculation, we cracked the code! Keep practicing, and you'll become a number puzzle master in no time! Until next time, happy calculating!