What must be added to each of the four numbers 6, 9, 12, and 17 such that the new numbers are in proportion?

Answer

To solve the problem of what must be added to each of the four numbers 6, 9, 12, and 17 such that the new numbers are in proportion, we can follow these steps:

### Step 1: Define the variable
Let \( x \) be the number that we need to add to each of the four numbers.

### Step 2: Write the new numbers
After adding \( x \), the new numbers will be:
- First number: \( 6 + x \)
- Second number: \( 9 + x \)
- Third number: \( 12 + x \)
- Fourth number: \( 17 + x \)

### Step 3: Set up the proportion
For the numbers to be in proportion, the following relationship must hold:
\[ \frac{6 + x}{9 + x} = \frac{12 + x}{17 + x} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ (6 + x)(17 + x) = (9 + x)(12 + x) \]

### Step 5: Expand both sides
Expanding both sides:
- Left side:
\[ 6 \cdot 17 + 6x + 17x + x^2 = 102 + 23x + x^2 \]
- Right side:
\[ 9 \cdot 12 + 9x + 12x + x^2 = 108 + 21x + x^2 \]

### Step 6: Simplify the equation
Now, we can simplify the equation by removing \( x^2 \) from both sides:
\[ 102 + 23x = 108 + 21x \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 23x - 21x = 108 - 102 \]
\[ 2x = 6 \]

### Step 8: Solve for \( x \)
Dividing both sides by 2:
\[ x = \frac{6}{2} = 3 \]

### Step 9: Conclusion
Thus, the number that must be added to each of the four numbers is \( \boxed{3} \).