**Problem:**
The ratio between two numbers is 4:3 and the difference between them is 13. Find the numbers?

**Solution:**

Let's assume the two numbers are x and y, where x is the larger number.

**Step 1: Set up the ratio**
The given ratio between the two numbers is 4:3. We can write this as:

x/y = 4/3

**Step 2: Express one variable in terms of the other**
To solve the problem, we need to eliminate one variable. Let's express y in terms of x by cross-multiplying the ratio:

3x = 4y

**Step 3: Set up an equation using the difference between the numbers**
The difference between the two numbers is given as 13. We can set up an equation using this information:

x - y = 13

**Step 4: Solve the system of equations**
We now have two equations:

3x = 4y (Equation 1)
x - y = 13 (Equation 2)

We can solve this system of equations using substitution or elimination method. Let's use the substitution method.

From Equation 1, we can express y in terms of x:

y = (3/4)x

Substituting this value of y in Equation 2, we get:

x - (3/4)x = 13
(1/4)x = 13
x = 13 * 4
x = 52

Substituting the value of x in Equation 1, we can find y:

3(52) = 4y
156 = 4y
y = 156/4
y = 39

**Step 5: Check the solution**
We have found x = 52 and y = 39. Let's check if these values satisfy the conditions of the problem.

The ratio of x to y is 52/39 = 4/3, which matches the given ratio.

The difference between x and y is 52 - 39 = 13, which also matches the given difference.

Therefore, the solution (x, y) = (52, 39) satisfies all the conditions of the problem.

**Answer:**
The two numbers are 52 and 39.