What number must be added to each of the numbers 10, 18, 22, 38 to get the numbers which are in proportion ?

Answer

To solve the problem of finding the number \( x \) that must be added to each of the numbers 10, 18, 22, and 38 so that the resulting numbers are in proportion, we can follow these steps:

### Step 1: Set up the equation
We need to add \( x \) to each of the numbers. The new numbers will be:
- \( 10 + x \)
- \( 18 + x \)
- \( 22 + x \)
- \( 38 + x \)

We want these numbers to be in proportion, which means:
\[ \frac{10 + x}{18 + x} = \frac{22 + x}{38 + x} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ (10 + x)(38 + x) = (18 + x)(22 + x) \]

### Step 3: Expand both sides
Expanding both sides:
- Left side:
\[ 10 \cdot 38 + 10x + 38x + x^2 = 380 + 48x + x^2 \]
- Right side:
\[ 18 \cdot 22 + 18x + 22x + x^2 = 396 + 40x + x^2 \]

### Step 4: Set the equation
Now we have:
\[ 380 + 48x + x^2 = 396 + 40x + x^2 \]

### Step 5: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 380 + 48x = 396 + 40x \]

### Step 6: Rearrange the equation
Rearranging gives us:
\[ 48x - 40x = 396 - 380 \]
\[ 8x = 16 \]

### Step 7: Solve for \( x \)
Dividing both sides by 8:
\[ x = 2 \]

### Conclusion
The number that must be added to each of the numbers 10, 18, 22, and 38 to get the numbers which are in proportion is \( \boxed{2} \).

---