If 2 is added to each of two given numbers, their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers.

Answer

To solve the problem, we need to set up equations based on the information provided. Let's denote the two numbers as \( A \) and \( B \).

### Step 1: Set up the first equation based on the first condition
According to the problem, if 2 is added to each of the two numbers, their ratio becomes 1:2. This can be expressed as:

\[ \frac{A + 2}{B + 2} = \frac{1}{2} \]

Cross-multiplying gives us:

\[ 2(A + 2) = 1(B + 2) \]

Expanding this, we have:

\[ 2A + 4 = B + 2 \]

Rearranging the equation gives us:

\[ 2A - B + 2 = 0 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation based on the second condition
The problem also states that if 4 is subtracted from each of the numbers, the ratio becomes 5:11. This can be expressed as:

\[ \frac{A - 4}{B - 4} = \frac{5}{11} \]

Cross-multiplying gives us:

\[ 11(A - 4) = 5(B - 4) \]

Expanding this, we have:

\[ 11A - 44 = 5B - 20 \]

Rearranging the equation gives us:

\[ 11A - 5B - 24 = 0 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:

1. \( 2A - B + 2 = 0 \)
2. \( 11A - 5B - 24 = 0 \)

From Equation 1, we can express \( B \) in terms of \( A \):

\[ B = 2A + 2 \]

Now, substitute this expression for \( B \) into Equation 2:

\[ 11A - 5(2A + 2) - 24 = 0 \]

Expanding this gives:

\[ 11A - 10A - 10 - 24 = 0 \]

Simplifying, we have:

\[ A - 34 = 0 \]

Thus, we find:

\[ A = 34 \]

### Step 4: Find \( B \)
Now that we have \( A \), we can substitute it back into the expression for \( B \):

\[ B = 2(34) + 2 = 68 + 2 = 70 \]

### Conclusion
The two numbers are:

\[ A = 34 \quad \text{and} \quad B = 70 \]