Finding Numbers: Sum 69, Quotient 7, Remainder 5

Hey guys! Today, we're diving into a fun math problem where we need to find two natural numbers that fit specific criteria. This kind of problem is not just about crunching numbers; it’s about understanding the relationships between numbers and applying some clever problem-solving techniques. So, let’s jump right in and figure out how to tackle this! We will break down the problem step by step, ensuring that you not only get the answer but also understand the method behind it. This approach will help you in solving similar problems in the future, making your math skills even stronger. Remember, the key to mastering math is practice and understanding, not just memorizing formulas.

Problem Statement

The problem states: The sum of two natural numbers is 69. If we divide the first number by the second, we get a quotient of 7 and a remainder of 5. Our mission is to find these two elusive numbers. Sounds like a bit of a puzzle, right? Well, that’s because it is! But don’t worry, we’re going to solve it together. The beauty of math problems like this is that they often have a logical structure that, once understood, makes the solution quite straightforward. It's all about translating the words into mathematical expressions and then using those expressions to solve for the unknowns. So, let's get started and see how we can unravel this numerical mystery!

Breaking Down the Problem

To solve this, let's break it down into smaller, more manageable parts. This is a crucial strategy in problem-solving, especially in mathematics. By breaking down complex problems, we can focus on individual components and understand how they fit together. First, we have two unknowns, so let's call them x and y. We know two important things about these numbers:

1. Their sum is 69.
2. When x is divided by y, the quotient is 7, and the remainder is 5.

These two pieces of information are the keys to unlocking our solution. The first statement gives us a simple equation, while the second involves the concept of division and remainders, which we will need to express mathematically. By translating these statements into equations, we can use algebraic methods to find the values of x and y. So, let's move on to the next step and see how we can transform these words into mathematical expressions that we can work with.

Forming the Equations

Now, let's translate the given information into mathematical equations. This is a crucial step in solving word problems. It allows us to move from a verbal description to a symbolic representation, which is easier to manipulate and solve. The first statement, "The sum of two natural numbers is 69," can be written as:

x + y = 69

The second statement is a bit trickier, but we can use the division algorithm, which states that any number (x) can be expressed as the divisor (y) times the quotient (7) plus the remainder (5). This can be written as:

x = 7y + 5

So, now we have a system of two equations with two variables. This is a classic setup for solving simultaneous equations. We can use methods like substitution or elimination to find the values of x and y. In the next section, we will explore how to use these equations to solve for our unknowns. This is where the real problem-solving fun begins!

Solving the System of Equations

We now have two equations:

1. x + y = 69
2. x = 7y + 5

Let's use the substitution method to solve this system. Since we already have x expressed in terms of y in the second equation, we can substitute this expression into the first equation. This will give us a single equation with one variable, which we can easily solve. Substituting the second equation into the first, we get:

(7y + 5) + y = 69

Now, let's simplify and solve for y. Combining like terms, we have:

8y + 5 = 69

Subtracting 5 from both sides gives:

8y = 64

Dividing both sides by 8, we find:

y = 8

Great! We've found the value of y. Now, we can use this value to find x. In the next section, we'll plug y back into one of our original equations to solve for x. We're getting closer to cracking this problem!

Finding the Value of x

Now that we know y = 8, we can substitute this value into either of our original equations to find x. Let's use the first equation, which is simpler:

x + y = 69

Substituting y = 8, we get:

x + 8 = 69

Subtracting 8 from both sides, we find:

x = 61

So, we've found that x = 61. Now, we have both x and y, but it's always a good idea to check our solution to make sure it satisfies the original conditions of the problem. In the next section, we'll verify our solution to ensure we haven't made any mistakes along the way. This is a crucial step in problem-solving, as it helps us catch any errors and build confidence in our answer.

Verifying the Solution

Let's check if our solution, x = 61 and y = 8, satisfies the conditions of the problem. First, we need to verify that their sum is 69:

61 + 8 = 69

This checks out! Now, let's verify the second condition: when 61 is divided by 8, the quotient should be 7, and the remainder should be 5. Let's perform the division:

61 ÷ 8 = 7 with a remainder of 5

This also checks out! Both conditions are satisfied, so we can confidently say that our solution is correct. It's always satisfying when everything falls into place like this. In the next section, we'll summarize our solution and reflect on the problem-solving process.

Conclusion

So, the two natural numbers are x = 61 and y = 8. We successfully found the solution by breaking down the problem, forming equations, solving the system of equations, and verifying our answer. This problem demonstrates how we can use algebraic techniques to solve real-world problems. Remember, the key to success in math is understanding the underlying concepts and practicing regularly. I hope this explanation has helped you understand the problem-solving process better. Keep practicing, and you'll become a math whiz in no time!

If you guys have any questions or want to explore more problems like this, feel free to ask. Keep up the great work, and I'll see you in the next math adventure! Remember, every problem you solve is a step forward in your mathematical journey. So, keep challenging yourself and keep learning!