The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the two numbers.

X = Smaller number.

(2x+6) = Larger number.

(2x+6) + x = 2((2x+6) - x)
3x+6 = 2(x+6)
3x+6 = 2x+12
3x-2x = 12-6

X = 6
2x+6 = 2*6 + 6 = 18.

Oh, this is a classic math problem! Let me put on my funny hat and give it a shot.

Let's call the smaller number "x" and the larger number "y." According to the problem, we have two equations.

First, the sum of the two numbers is twice their difference. So, we have x + y = 2(x - y).

Second, the larger number is 6 more than twice the smaller. That gives us y = 2x + 6.

Now, let's use our funny math skills to solve this.

We'll start with the second equation, and substitute y with its value from the first equation. Here we go!

x + (2x + 6) = 2(x - (2x + 6))

Now, simplify and solve for x:

3x + 6 = 2(x - 2x - 6)
3x + 6 = 2(-x - 6)
3x + 6 = -2x - 12

With some more math acrobatics, we end up with:

5x = -18

Hmm, that's not as funny as I thought. Anyway, let's finish solving:

x = -18 / 5

So, the smaller number is x = -18 / 5. Oh dear, that's a tricky one.

Plugging this back into the second equation, we can find y:

y = 2(-18 / 5) + 6

y = -36 / 5 + 6

y = -36 / 5 + 30 / 5

y = -6 / 5

Oh no, it seems like these numbers are not as funny as they could be. But hey, they are the solution to the problem. The smaller number is approximately -3.6, and the larger number is approximately -1.2.

Let's assume the smaller number as x.

According to the given conditions:
The larger number is 6 more than twice the smaller.
Therefore, the larger number is 2x + 6.

The sum of two numbers is twice their difference.
This can be expressed as (x + (2x + 6)) = 2((2x + 6) - x).

Simplifying it:
(x + 2x + 6) = 2(2x + 6 - x).
(3x + 6) = 2(x + 6).
3x + 6 = 2x + 12.

Bringing the x terms to one side and the constant terms to the other side:
3x - 2x = 12 - 6.
x = 6.

Therefore, the smaller number is 6.

To find the larger number:
The larger number = 2x + 6 = 2 * 6 + 6 = 18.

Hence, the two numbers are 6 and 18.

To solve this problem, let's assign variables to the unknown numbers. Let's call the larger number "x" and the smaller number "y".

The first piece of information given is that the sum of the two numbers is twice their difference. Mathematically, we can represent this as:

x + y = 2(x - y)

The second piece of information states that the larger number (x) is 6 more than twice the smaller number (y). In equation form, this can be written as:

x = 2y + 6

We now have a system of two equations:

1) x + y = 2(x - y)
2) x = 2y + 6

To solve this system, we can use the substitution method or the elimination method. Let's solve it by substitution:

From equation 2), we can express x in terms of y: x = 2y + 6.

Now we substitute the expression for x in equation 1):

(2y + 6) + y = 2((2y + 6) - y)

Simplifying:

3y + 6 = 2(2y + 6 - y)
3y + 6 = 2(2y + 6 - y)

Next, distribute the 2 on the right side:

3y + 6 = 4y + 12 - 2y

Combine like terms:

3y + 6 = 2y + 12

Subtract 2y from both sides:

3y - 2y + 6 = 2y + 12 - 2y

Simplify:

y + 6 = 12

Subtract 6 from both sides:

y + 6 - 6 = 12 - 6

Simplify:

y = 6

Now that we have the value of y, we can substitute it back into equation 2) to find x:

x = 2y + 6
x = 2(6) + 6
x = 12 + 6
x = 18

So, the two numbers are x = 18 and y = 6.