If 1 is added to each of the two given numbers then their ratio is 1 : 2. If 5 is subtracted from each the ratio is 5 : 11. Find the numbers.

Answer

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

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

\[ \frac{x + 1}{y + 1} = \frac{1}{2} \]

Cross-multiplying gives us:

\[ 2(x + 1) = 1(y + 1) \]

Expanding this, we get:

\[ 2x + 2 = y + 1 \]

Rearranging this equation leads to:

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

### Step 2: Set up the second equation
According to the second statement, if 5 is subtracted from each of the two numbers, their ratio becomes 5:11. This can be expressed as:

\[ \frac{x - 5}{y - 5} = \frac{5}{11} \]

Cross-multiplying gives us:

\[ 11(x - 5) = 5(y - 5) \]

Expanding this, we have:

\[ 11x - 55 = 5y - 25 \]

Rearranging this equation leads to:

\[ 11x - 5y - 30 = 0 \quad \text{(Equation 2)} \]

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

1. \( 2x - y + 1 = 0 \)
2. \( 11x - 5y - 30 = 0 \)

We can solve these equations using the substitution or elimination method. Here, we will use the substitution method.

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

\[ y = 2x + 1 \]

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

\[ 11x - 5(2x + 1) - 30 = 0 \]

Expanding this gives:

\[ 11x - 10x - 5 - 30 = 0 \]

Simplifying this leads to:

\[ x - 35 = 0 \]

Thus, we find:

\[ x = 35 \]

### Step 4: Find the value of \( y \)
Now that we have \( x \), we can substitute it back into the expression for \( y \):

\[ y = 2(35) + 1 = 70 + 1 = 71 \]

### Conclusion
The two numbers are:

\[ \boxed{35} \quad \text{and} \quad \boxed{71} \]