What number should be added to each of the four numbers 4. I, 26, and 12 so that the new numbers are in proportion?

Answer

To solve the problem of finding what number should be added to each of the four numbers 4, 1, 26, and 12 so that the new numbers 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. **Write the new numbers**: After adding \( x \) to each of the numbers, the new numbers will be:
- \( 4 + x \)
- \( 1 + x \)
- \( 26 + x \)
- \( 12 + x \)

3. **Set up the proportion**: For the numbers to be in proportion, the following condition must hold:
\[ \frac{4 + x}{1 + x} = \frac{26 + x}{12 + x} \]

4. **Cross-multiply**: To eliminate the fractions, we cross-multiply:
\[ (4 + x)(12 + x) = (1 + x)(26 + x) \]

5. **Expand both sides**:
- Left side:
\[ 4 \cdot 12 + 4x + 12x + x^2 = 48 + 16x + x^2 \]
- Right side:
\[ 1 \cdot 26 + 1x + 26x + x^2 = 26 + 27x + x^2 \]

6. **Set the equation**: Now, we can set the expanded forms equal to each other:
\[ 48 + 16x + x^2 = 26 + 27x + x^2 \]

7. **Cancel \( x^2 \)**: Since \( x^2 \) appears on both sides, we can cancel it out:
\[ 48 + 16x = 26 + 27x \]

8. **Rearrange the equation**: Move all terms involving \( x \) to one side and constant terms to the other side:
\[ 48 - 26 = 27x - 16x \]
\[ 22 = 11x \]

9. **Solve for \( x \)**: Divide both sides by 11:
\[ x = 2 \]

10. **Conclusion**: The number that should be added to each of the four numbers is \( 2 \).

11. **Verification**: To verify, we can check if the new numbers are in proportion:
- New numbers: \( 4 + 2 = 6 \), \( 1 + 2 = 3 \), \( 26 + 2 = 28 \), \( 12 + 2 = 14 \)
- Check the ratios:
\[ \frac{6}{3} = 2 \quad \text{and} \quad \frac{28}{14} = 2 \]
- Since both ratios are equal, the numbers are indeed in proportion.