The ratio of two numbers is 2/3. If 2 is subtracted from the first and 8 from the second,the ratio becomes a reciprocal of the original ratio.find the numbers 

Let's solve the problem step by step.

### Step 1: Define the numbers
Let the two numbers be \( x \) and \( y \).

### Step 2: Use the given ratio
According to the problem, the ratio of \( x \) and \( y \) is \( \frac{2}{3} \). So, we can write the equation:

\[
\frac{x}{y} = \frac{2}{3}
\]

This implies:

\[
3x = 2y \quad \text{(Equation 1)}
\]

### Step 3: Adjust the numbers according to the problem
If 2 is subtracted from the first number \( x \), and 8 from the second number \( y \), the ratio becomes the reciprocal of the original ratio. The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

So, the equation becomes:

\[
\frac{x - 2}{y - 8} = \frac{3}{2}
\]

This implies:

\[
2(x - 2) = 3(y - 8) \quad \text{(Equation 2)}
\]

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

1. \( 3x = 2y \)
2. \( 2(x - 2) = 3(y - 8) \)

#### Solve Equation 1 for \( y \):

\[
y = \frac{3x}{2}
\]

#### Substitute \( y \) in Equation 2:

\[
2(x - 2) = 3\left(\frac{3x}{2} - 8\right)
\]

Expand and simplify:

\[
2x - 4 = \frac{9x}{2} - 24
\]

Multiply everything by 2 to eliminate the fraction:

\[
4x - 8 = 9x - 48
\]

Rearrange the terms:

\[
48 - 8 = 9x - 4x
\]

\[
40 = 5x
\]

\[
x = 8
\]

#### Find \( y \):

Substitute \( x = 8 \) back into \( y = \frac{3x}{2} \):

\[
y = \frac{3(8)}{2} = 12
\]

### Step 5: Conclusion
The two numbers are \( \boxed{8} \) and \( \boxed{12} \).

---

Would you like more details or have any questions? Here are some related problems you might find interesting:

1. Find two numbers whose ratio is \( \frac{3}{5} \) and whose difference is 10.
2. If the ratio of two numbers is \( \frac{4}{7} \) and 5 is added to each, the ratio becomes \( \frac{5}{7} \). Find the numbers.
3. The ratio of two numbers is \( \frac{5}{8} \). If each number is decreased by 3, the ratio becomes \( \frac{4}{7} \). Find the numbers.
4. If the sum of two numbers is 50 and their ratio is \( \frac{3}{2} \), find the numbers.
5. Two numbers are in the ratio 7:3. If 10 is added to the first number, the ratio becomes 2:1. What are the numbers?

**Tip:** When dealing with ratios and equations, always express one variable in terms of the other to simplify solving the system of equations.

Explore how to solve a ratio problem where the ratio of two numbers changes after specific adjustments. Detailed step-by-step solution provided, covering concepts of ratio and proportion, suitable for students in Grades 8-10.

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