What number must be added to each of the numbers 5, 9, 7, 12 to get the numbers which are in proportion?

Answer

To solve the problem of finding the number that must be added to each of the numbers 5, 9, 7, and 12 so that they are in proportion, we can follow these steps:

### Step-by-Step Solution:

1. **Define the variable**: Let the number to be added be \( x \).

2. **Set up the ratios**: After adding \( x \) to each number, the new numbers will be \( 5 + x \), \( 9 + x \), \( 7 + x \), and \( 12 + x \). We want these numbers to be in proportion. This means:
\[ \frac{5 + x}{9 + x} = \frac{7 + x}{12 + x} \]

3. **Cross-multiply**: To eliminate the fractions, we cross-multiply:
\[ (5 + x)(12 + x) = (9 + x)(7 + x) \]

4. **Expand both sides**:
- Left side:
\[ 5 \cdot 12 + 5x + 12x + x^2 = 60 + 17x + x^2 \]
- Right side:
\[ 9 \cdot 7 + 9x + 7x + x^2 = 63 + 16x + x^2 \]

5. **Set the equation**: Now we have:
\[ 60 + 17x + x^2 = 63 + 16x + x^2 \]

6. **Simplify the equation**: We can subtract \( x^2 \) from both sides:
\[ 60 + 17x = 63 + 16x \]

7. **Rearrange the equation**: Move all terms involving \( x \) to one side and constant terms to the other:
\[ 17x - 16x = 63 - 60 \]
\[ x = 3 \]

8. **Conclusion**: The number that must be added to each of the numbers 5, 9, 7, and 12 to make them proportional is \( 3 \).