Decoding Numbers Summing To 13.875 An Algebra Puzzle

Hey guys! Today, we're diving into a cool algebra problem that involves figuring out three mysterious numbers. These numbers have a special relationship – they're connected by the position of their decimal points. Sounds intriguing, right? So, let's put on our detective hats and get started!

Decoding the Number Puzzle

In this algebraic puzzle, the sum of three numbers is 13.875. Now, here's the twist: If we take one of these numbers and shift its decimal point one place to the right, we magically get the second number. And if we shift the decimal point two places to the right, voilà, we have the third number! Our mission, should we choose to accept it, is to find out what these three elusive numbers are. To really break down this problem, we'll use a bit of algebraic thinking and turn this word puzzle into a set of equations we can solve. Think of it like translating a secret code – we're just changing words into math symbols.

We will start by identifying the core relationships between the numbers as described in the problem. The shifting decimal points are the key here, as they tell us how the numbers are related in magnitude. This is where our algebra skills come in handy, allowing us to represent these relationships mathematically and ultimately solve for our unknown numbers. So, let's grab our algebraic magnifying glasses and get ready to zoom in on the details of this number mystery!

Setting Up the Algebraic Equation

Let's turn this number puzzle into algebra! The most important first step in tackling any word problem is to translate the words into mathematical expressions. We'll start by assigning a variable to one of the unknown numbers. Let's call the first number 'x'. This sets the stage for expressing the other numbers in terms of 'x', using the information about the decimal point shifts. Remember, shifting a decimal point to the right is the same as multiplying by powers of 10. So, if we shift the decimal one place to the right, we multiply by 10; two places, we multiply by 100, and so on.

Now, according to our problem, the second number is obtained by shifting the decimal point of the first number ('x') one place to the right. Mathematically, this means the second number is 10 times the first number, or 10x. Similarly, the third number is found by shifting the decimal point of the first number two places to the right, which is the same as multiplying by 100. So, the third number is represented as 100 times the first number, or 100x. We have now successfully expressed all three numbers in terms of a single variable, 'x'. This is a crucial step, as it allows us to combine these expressions into a single equation.

With our numbers represented algebraically, we can now use the information about their sum to form an equation. The problem states that the sum of these three numbers is 13.875. So, we can write this as an equation: x + 10x + 100x = 13.875. This equation beautifully captures the essence of our problem in a concise mathematical form. It is the key to unlocking the values of our unknown numbers. Our next step is to solve this equation for 'x', which will reveal the value of our first number and set off a chain reaction to find the other two. So, let's sharpen our algebra pencils and get ready to solve!

Solving for the Unknown

Now comes the fun part – solving for 'x'! We've got our equation: x + 10x + 100x = 13.875. The first thing we want to do is simplify the left side of the equation by combining like terms. Think of 'x' as having an invisible coefficient of 1, so we have 1x + 10x + 100x. Adding these together, we get 111x. So, our equation simplifies to 111x = 13.875. We're getting closer to uncovering the value of 'x'.

To isolate 'x' and find its value, we need to undo the multiplication by 111. We do this by dividing both sides of the equation by 111. Remember, in algebra, what we do to one side of the equation, we must also do to the other side to keep the equation balanced. So, we divide both 111x and 13.875 by 111. This gives us x = 13.875 / 111. Now, it's time for a little bit of arithmetic. You can use a calculator or good old-fashioned long division to find the result. When we perform this division, we find that x = 0.125.

Eureka! We've found the value of 'x'. This is our first number. But we're not done yet. We still need to find the other two numbers. But now that we know 'x', this becomes much easier. We'll use the relationships we established earlier to find the second and third numbers. Remember, the second number is 10x and the third number is 100x. So, let's plug in our value of 'x' and reveal the rest of the mystery!

Unveiling the Numbers

With the value of 'x' in our grasp, we're ready to reveal the other two numbers. We found that x = 0.125. Now, let's recall how the other numbers are related to 'x'. The second number is 10x, meaning it's 10 times the value of 'x'. So, to find the second number, we simply multiply 0.125 by 10. This is a quick calculation – just shift the decimal point one place to the right! So, 10 * 0.125 = 1.25. There's our second number!

Moving on to the third number, we know it's 100x, which means it's 100 times the value of 'x'. Again, we multiply 0.125, but this time by 100. Multiplying by 100 is like shifting the decimal point two places to the right. So, 100 * 0.125 = 12.5. And there you have it – our third number! We've successfully found all three numbers that solve our puzzle. Let's recap: the first number is 0.125, the second number is 1.25, and the third number is 12.5.

But before we celebrate our victory, it's always a good idea to check our work. We can do this by adding the three numbers together to see if they indeed sum up to 13.875. So, let's add 0.125 + 1.25 + 12.5. If you do the math, you'll find that they do indeed add up to 13.875. This confirms that our solution is correct. We've cracked the code and unveiled the three mysterious numbers!

Summing Up the Solution

Awesome job, guys! We've successfully navigated this algebraic puzzle and discovered the three hidden numbers. To recap, we started with the problem statement: the sum of three numbers is 13.875, and these numbers are related by shifts in their decimal points. We then translated this word problem into an algebraic equation by representing the numbers as x, 10x, and 100x. This crucial step allowed us to use the power of algebra to solve for the unknown.

We formed the equation x + 10x + 100x = 13.875, which we simplified to 111x = 13.875. Solving for 'x' by dividing both sides by 111, we found that x = 0.125. This gave us our first number. Then, using the relationships between the numbers, we found the second number (10x) to be 1.25 and the third number (100x) to be 12.5. Finally, we checked our work by adding the three numbers together, confirming that their sum is indeed 13.875. Our final answer is: the three numbers are 0.125, 1.25, and 12.5.

This problem beautifully illustrates the power of algebra in solving real-world puzzles. By translating words into mathematical expressions and equations, we can unlock solutions that might seem hidden at first glance. So, keep practicing your algebra skills, and you'll become a master puzzle solver in no time! Keep an eye out for more exciting math adventures coming your way!