Find Two Numbers: Sum Is 63, Complex Condition

Hey guys! Let's dive into this fun math problem that might seem a bit tricky at first, but don't worry, we'll break it down step-by-step. We're on a mission to find two numbers based on some clues. Ready? Let's get started!

Decoding the Problem

Okay, so the first thing we know is that if we add two mystery numbers together, we get 63. Think of it like having two piles of candy, and when you put them together, you have 63 candies in total. Mathematically, we can express this as:

x + y = 63

Where 'x' is our first number and 'y' is our second number. Easy peasy, right?

Now, here comes the twist! If we take 5 away from the first number (x) and give it to the second number (y), something cool happens. The new number we get for the second number becomes 6 times bigger than the first number before we took away the 5. Woah, that's a mouthful! Let's write this down as an equation to make it clearer:

y + 5 = 6 * x

See? Not so scary when we write it down. The second equation tells us that if we transfer 5 units from the first number to the second one, the second number balloons up to be six times the size of the original first number. This sets up a relationship that we can leverage to solve for our unknowns.

Solving the Puzzle

Alright, now we have two equations and two unknowns (x and y). This means we can solve for our mystery numbers! There are a couple of ways to do this, but let's use the substitution method. It's like a cool detective trick.

First, let's rearrange our first equation (x + y = 63) to solve for y. We can do this by subtracting x from both sides:

y = 63 - x

Now we know that 'y' is the same as '63 - x'. So, wherever we see 'y' in our second equation, we can replace it with '63 - x'. Let's do that:

(63 - x) + 5 = 6x

See what we did? We swapped 'y' for '63 - x' in the second equation. Now we only have one variable, 'x', which makes the equation much easier to solve.

Let's simplify this equation by combining the numbers on the left side:

68 - x = 6x

Now, let's get all the 'x' terms on one side of the equation. We can do this by adding 'x' to both sides:

68 = 7x

Finally, to solve for 'x', we need to divide both sides by 7:

x = 68 / 7

This means our first number, x, is approximately 9.71.

Now that we know 'x', we can plug it back into our equation for 'y' (y = 63 - x):

y = 63 - 9.71

y = 53.29

So, our second number, y, is approximately 53.29.

Therefore, the first number is approximately 9.71 and the second number is approximately 53.29.

Double-Checking Our Work

Before we declare victory, let's make sure our answers actually work. This is like the detective making sure they have the right suspect!

First, let's check if the two numbers add up to 63:

9.71 + 53.29 = 63 (approximately)

Yep, that checks out!

Now, let's see if subtracting 5 from the first number and adding it to the second number gives us a number 6 times larger than the first number:

(53.29 + 5) = 6 * 9.71

58.29 = 58.26 (approximately)

That's super close! The slight difference is just because we rounded our numbers earlier. So, we can be confident that our answers are correct.

Wrapping Up

So, there you have it! The two numbers are approximately 9.71 and 53.29. We solved this problem by setting up two equations based on the information given and then using the substitution method to solve for our unknowns. Great job, everyone! Keep practicing, and you'll become a math whiz in no time!

Remember: Practice makes perfect, and even the trickiest problems become easier with a little bit of logical thinking!

Tackling a Similar Problem with Integer Constraints

Let's assume that the numbers we are looking for should be integers. That changes the game quite a bit, doesn't it? The initial problem description tells us that the sum of two numbers is 63. We can still write it as:

x + y = 63

And the second condition states that if we subtract 5 from the first number and add it to the second number, we obtain a number that is 6 times larger than the first number. The equation is still:

y + 5 = 6 * (x - 5)

Notice the subtle change here? We are subtracting 5 from x before multiplying by 6. This is a crucial adjustment based on a slightly different reading of the original problem. Let's solve this system of equations.

Solving with the Integer Constraint

Again, from the first equation, let's express y in terms of x:

y = 63 - x

Substitute this into the second equation:

(63 - x) + 5 = 6 * (x - 5)

Simplify and expand:

68 - x = 6x - 30

Move the x terms to one side and constants to the other:

68 + 30 = 6x + x

98 = 7x

Divide by 7 to solve for x:

x = 14

Now, substitute x = 14 back into the equation for y:

y = 63 - 14

y = 49

So, our two numbers are x = 14 and y = 49.

Verification

Let's verify if our solution is correct:

1. Sum Check: 14 + 49 = 63. Correct!
2. Second Condition Check:
   • Subtract 5 from the first number: 14 - 5 = 9
   • Add 5 to the second number: 49 + 5 = 54
   • Is 54 six times 9? 6 * 9 = 54. Correct!

Therefore, the two numbers are indeed 14 and 49.

Key Takeaways and Strategy

Breaking down a complex word problem into manageable equations is crucial. The steps we followed highlight a systematic approach to problem-solving:

1. Translate: Convert the word problem into mathematical equations.
2. Simplify: Rearrange the equations to isolate variables.
3. Solve: Use substitution or elimination methods to find the values of the variables.
4. Verify: Plug the values back into the original equations to check for accuracy.

Important: Always double-check the problem statement to ensure you've captured all the conditions accurately. A slight misinterpretation can lead to a completely different solution.

Whether dealing with real numbers or integers, the core strategy remains the same. Mastering these steps will empower you to tackle a wide range of mathematical challenges. Keep practicing, and happy problem-solving!