The difference of two number is seven. when the larger number is decreased by five, then doubled, it is five more than the small number. Find the numbers.

let the larger number be x

then the smaller is x-7

"the larger number is decreased by five, then doubled" ---> 2(x-5)

2(x-5) = x-7 +5
2x - 10 = x - 2
x = 8

Finish it up, and check

a = larger number

b = smaller number

The difference of two number is seven mean:

a - b = 7

a - b = 7 Add b to both sides

a - b + b = 7 + b

a = 7 + b

a = b + 7

When the larger number is decreased by five, then doubled, it is five more than the small number mean:

( a - 5 ) * 2 = b + 5

Replace a = b + 7 in this equation

( b + 7 - 5 ) * 2 = b + 5

( b + 2 ) * 2 = b + 5

2 * b + 2 * 2 = b + 5

2 b + 4 = b + 5 Subtract b to both sides

2 b + 4 - b = b + 5 - b

b + 4 = 5 Subtract 4 to both sides

b + 4 - 4 = 5 - 4

b = 1

a = b + 7 = 1 + 7 = 8

a = 8

Proof:

a - b = 7

8 - 1 = 7

( a - 5 ) * 2 = b + 5

( 8 - 5 ) * 2 = 1 + 5

3 * 2 = 6

6 = 6

To solve this problem, we need to set up equations based on the given information.

Let's assume the larger number is represented by 'x' and the smaller number is represented by 'y'.

According to the problem, the difference of the two numbers is seven:

x - y = 7 <-- Equation 1

Furthermore, it is given that when the larger number is decreased by five, then doubled, it is five more than the small number:

2(x - 5) = y + 5 <-- Equation 2

Now, we have a system of two equations with two variables. We can solve it using the concept of simultaneous equations.

First, let's simplify Equation 2:

2x - 10 = y + 5

Next, rearrange Equation 1 to express x in terms of y:

x = y + 7

Now, substitute the value of x in Equation 2:

2(y + 7) - 10 = y + 5

Expand and simplify:

2y + 14 - 10 = y + 5
2y + 4 = y + 5

Subtract y from both sides:

2y - y + 4 = 5
y + 4 = 5

Subtract 4 from both sides:

y = 1

Now, substitute the value of y back into Equation 1 to find x:

x - 1 = 7
x = 8

Therefore, the larger number is 8 and the smaller number is 1.