**Problem:**
We are given two numbers such that the ratio between them is 3:5. Let's assume the two numbers as '3x' and '5x' respectively.

If each of these numbers is increased by 10, then the new numbers formed have a ratio of 5:7. Let's assume the new numbers as '3x + 10' and '5x + 10' respectively.

We need to find the original numbers.

**Solution:**

Let's solve the problem step by step.

**Step 1: Set up the ratios**

The given ratio between the original numbers is 3:5. We can write this as:

(3x) : (5x)

The ratio between the new numbers is 5:7. We can write this as:

(3x + 10) : (5x + 10)

**Step 2: Write the equation**

According to the given information, the ratio between the original numbers is equal to the ratio between the new numbers. So, we can set up an equation as:

(3x) : (5x) = (3x + 10) : (5x + 10)

**Step 3: Cross-multiply and simplify**

Cross-multiplying the equation, we get:

(3x) * (5x + 10) = (5x) * (3x + 10)

Simplifying both sides, we have:

15x^2 + 30x = 15x^2 + 50x

**Step 4: Eliminate like terms**

Subtracting 15x^2 from both sides of the equation, we get:

30x = 50x

**Step 5: Solve for x**

Subtracting 30x from both sides of the equation, we have:

30x - 30x = 50x - 30x

0 = 20x

Dividing both sides by 20, we get:

0/20 = 20x/20

0 = x

**Step 6: Find the original numbers**

Since we have found x = 0, we can substitute this value back into the equations for the original and new numbers.

Original numbers:
First number = 3x = 3(0) = 0
Second number = 5x = 5(0) = 0

Therefore, the original numbers are 0 and 0.

**Conclusion:**
The original numbers are 0 and 0.