What should be added to each of the numbers 8, 32,and 1,11 so that the new numbers are in proportion?

Answer

To solve the problem of what should be added to each of the numbers 8, 32, 1, and 11 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 to each of the numbers be \( x \).

2. **Form New Numbers**:
The new numbers will be:
- \( 8 + x \)
- \( 32 + x \)
- \( 1 + x \)
- \( 11 + x \)

3. **Set Up the Proportion**:
Since the new numbers are in proportion, we can set up the equation:
\[ \frac{8 + x}{32 + x} = \frac{1 + x}{11 + x} \]

4. **Cross Multiply**:
Cross multiplying gives us:
\[ (8 + x)(11 + x) = (1 + x)(32 + x) \]

5. **Expand Both Sides**:
Expanding both sides:
- Left side:
\[ 8 \cdot 11 + 8x + 11x + x^2 = 88 + 19x + x^2 \]
- Right side:
\[ 1 \cdot 32 + 1x + 32x + x^2 = 32 + 33x + x^2 \]

6. **Simplify the Equation**:
Now we have:
\[ 88 + 19x + x^2 = 32 + 33x + x^2 \]
We can cancel \( x^2 \) from both sides:
\[ 88 + 19x = 32 + 33x \]

7. **Rearrange the Equation**:
Rearranging gives:
\[ 88 - 32 = 33x - 19x \]
Simplifying further:
\[ 56 = 14x \]

8. **Solve for \( x \)**:
Dividing both sides by 14:
\[ x = \frac{56}{14} = 4 \]

9. **Find the New Numbers**:
Now, we can find the new numbers by adding \( x \) to each original number:
- \( 8 + 4 = 12 \)
- \( 32 + 4 = 36 \)
- \( 1 + 4 = 5 \)
- \( 11 + 4 = 15 \)

10. **Check the Proportion**:
We need to check if the ratios are equal:
\[ \frac{12}{36} = \frac{1}{3} \quad \text{and} \quad \frac{5}{15} = \frac{1}{3} \]
Since both ratios are equal, the new numbers are indeed in proportion.

### Conclusion:
The value of \( x \) that should be added to each number is **4**.