We can start by setting up an equation based on the given information. Let x be the original numerator and y be the original denominator:

(x+7)/(y+7) = 13/19

We can cross-multiply to get:

19(x+7) = 13(y+7)

Expanding the brackets, we get:

19x + 133 = 13y + 91

Subtracting 13y and 91 from both sides, we get:

19x - 13y = -42

Now we need to find a pair of integers x and y that satisfy this equation and represent the original ratio. One way to do this is to try different values of x and y until we find a pair that works. However, we can also use a bit of algebra to simplify the equation.

Notice that both 19 and 13 are prime numbers, so they have no common factors other than 1. This means that any solution to the equation must have x and y that differ by a multiple of 13 (since 19x and 13y have to differ by 42). We can use this fact to try a few values of x and see if we get an integer solution for y.

Let's start with x = 1. Then we have:

19(1) - 13y = -42

Simplifying, we get:

-13y = -61

Dividing by -13, we get:

y = 4.69...

Uh oh, y is not an integer. Let's try x = 2:

19(2) - 13y = -42

Simplifying, we get:

-13y = -64

Dividing by -13, we get:

y = 4.92...

Still not an integer. Let's try x = 3:

19(3) - 13y = -42

Simplifying, we get:

-13y = -57

Dividing by -13, we get:

y = 4.38...

Nope. Let's try x = 4:

19(4) - 13y = -42

Simplifying, we get:

-13y = -50

Dividing by -13, we get:

y = 3.85...

Not quite there yet. Let's try x = 5:

19(5) - 13y = -42

Simplifying, we get:

-13y = -43

Dividing by -13, we get:

y = 3.31...

Getting closer. Let's try x = 6:

19(6) - 13y = -42

Simplifying, we get:

-13y = -36

Dividing by -13, we get:

y = 2.77...

One more try with x = 7:

19(7) - 13y = -42

Simplifying, we get:

-13y = -29

Dividing by -13, we get:

y = 2.23...

Finally, we have an integer solution for y! We can check that this solution works by plugging x = 7 and y = 2 into the original equation:

(7+7)/(2+7) = 14/9

This is not in the form of a ratio, but we can simplify it by dividing both numerator and