Two-Digit Numbers: Four Times The Sum Of Digits

by TextBrain Team 48 views

Hey guys! Let's dive into a fun mathematical puzzle. We're going to explore the fascinating world of two-digit numbers and figure out which ones are equal to four times the sum of their digits. You know, like how 12 is four times its digit sum because 1 + 2 = 3, and 3 multiplied by 4 is indeed 12. Let's roll up our sleeves and find out what other numbers fit this cool pattern!

Understanding the Problem

Okay, first things first, let's break down what we're actually trying to solve. We need to find two-digit numbers. What exactly does that mean? Well, it simply refers to numbers from 10 up to 99. Each of these numbers has two digits: a tens digit and a ones digit. For example, in the number 25, '2' is the tens digit, and '5' is the ones digit.

Now, what about the "sum of the digits" part? That’s pretty straightforward too. It just means we add the tens digit and the ones digit together. In our 25 example, the sum of the digits would be 2 + 5 = 7.

The core of the puzzle lies in this: we're looking for numbers where the entire number is exactly four times the sum of its digits. So, if we take the sum of the digits and multiply it by 4, we should get the original two-digit number back. This is the key concept that will guide our exploration and help us identify the numbers we're looking for. We need to keep this relationship firmly in mind as we start our search. Think of it as a kind of mathematical detective work, where we are piecing together clues to solve a mystery!

Setting Up the Equation

To tackle this problem systematically, let’s translate our word puzzle into an algebraic equation. This will give us a powerful tool to find our solutions.

Let's use some variables: Let t represent the tens digit and u represent the ones digit. Any two-digit number can be expressed as 10 times the tens digit plus the ones digit. So, our two-digit number is 10t + u.

The sum of the digits is simply t + u. According to the problem, the two-digit number is equal to four times the sum of its digits. This can be written as the equation:

10t + u = 4(t + u)

This equation is the mathematical representation of our problem. It encapsulates the relationship we're trying to find. Now, our task is to solve this equation. But don't worry, it's not as daunting as it might seem! We're going to use some basic algebraic techniques to simplify it and find the possible values of t and u. This equation gives us a clear and concise way to explore the problem, and it sets the stage for our next step: simplifying the equation to find potential solutions.

Solving the Equation

Alright, guys, it's time to put on our algebraic hats and simplify the equation we set up! We have:

10t + u = 4(t + u)

First, we need to distribute the 4 on the right side of the equation. This means multiplying both t and u inside the parentheses by 4. This gives us:

10t + u = 4t + 4u

Now, let’s gather the t terms on one side and the u terms on the other. We can subtract 4t from both sides:

10t - 4t + u = 4t - 4t + 4u

Which simplifies to:

6t + u = 4u

Next, we subtract u from both sides to isolate the u terms:

6t + u - u = 4u - u

This simplifies to:

6t = 3u

Now, we can divide both sides by 3 to make the equation even simpler:

6t / 3 = 3u / 3

Which gives us:

2t = u

This is a crucial equation! What does it tell us? It tells us that the ones digit (u) is twice the tens digit (t). This is a significant finding because it narrows down the possibilities considerably. We've transformed a complex-looking equation into a simple relationship between the tens and ones digits. With this relationship in hand, we can start plugging in different values for t and see what u values we get. This simplified equation is the key to unlocking the solutions to our puzzle.

Finding the Numbers

Now comes the fun part – using our simplified equation, 2t = u, to find the actual numbers that fit the criteria. Remember, t represents the tens digit, and u represents the ones digit. Since these are digits, they can only be whole numbers from 0 to 9.

Let's start by trying different values for t and calculating the corresponding value for u:

  • If t = 1, then u = 2 * 1 = 2. This gives us the number 12.
  • If t = 2, then u = 2 * 2 = 4. This gives us the number 24.
  • If t = 3, then u = 2 * 3 = 6. This gives us the number 36.
  • If t = 4, then u = 2 * 4 = 8. This gives us the number 48.

What happens if we try t = 5? Then u = 2 * 5 = 10. But wait a minute! The ones digit u can't be 10, because it has to be a single digit (0-9). So, we've hit the limit. Any value of t greater than 4 will result in a u that is greater than 9, which is not a valid digit.

Therefore, we've found all the possible numbers! By systematically substituting values for t and using our equation, we've identified the numbers that fit the conditions of our problem. This is the power of using a mathematical approach to solve puzzles – it allows us to explore possibilities in a logical and organized way.

Verifying the Solutions

Before we declare victory, it's always a good idea to double-check our answers. Let's make sure that each of the numbers we found is indeed equal to four times the sum of its digits.

We found the numbers 12, 24, 36, and 48. Let's test them:

  • For 12: The sum of the digits is 1 + 2 = 3. Four times the sum is 4 * 3 = 12. Check!
  • For 24: The sum of the digits is 2 + 4 = 6. Four times the sum is 4 * 6 = 24. Check!
  • For 36: The sum of the digits is 3 + 6 = 9. Four times the sum is 4 * 9 = 36. Check!
  • For 48: The sum of the digits is 4 + 8 = 12. Four times the sum is 4 * 12 = 48. Check!

All the numbers we found satisfy the condition! This verification step is crucial in any problem-solving process. It gives us confidence that our solutions are correct and that we haven't made any mistakes along the way. By checking our work, we ensure the accuracy of our results and solidify our understanding of the problem.

Conclusion

So, guys, we've successfully solved our mathematical puzzle! We set out to find two-digit numbers that are equal to four times the sum of their digits, and we did it. Through a combination of understanding the problem, setting up an equation, simplifying it, finding potential solutions, and verifying our answers, we discovered that the numbers are 12, 24, 36, and 48. Isn't that awesome?

This exercise wasn't just about finding numbers, though. It was also about the process of problem-solving. We used mathematical thinking, algebraic techniques, and logical reasoning to arrive at our solutions. These are valuable skills that can be applied to many different situations in life, not just math problems. So, the next time you encounter a puzzle or a challenge, remember the steps we took here: understand the problem, create a plan, execute the plan, and check your results. You might be surprised at what you can accomplish!