What must be subtracted from 19, 28, 55, 91 so that they are in the same proportion

Options

A. 4
B. 9
C. 7
D. 8

Answer

To solve the problem step by step, we need to find the value that must be subtracted from the numbers 19, 28, 55, and 91 so that they are in the same proportion.

### Step-by-Step Solution:

1. **Define the variable**: Let \( x \) be the number that we need to subtract from each of the four numbers.

2. **Set up the proportions**: After subtracting \( x \), the numbers will be \( 19 - x \), \( 28 - x \), \( 55 - x \), and \( 91 - x \). We need to set up the proportion:
\[ \frac{19 - x}{28 - x} = \frac{55 - x}{91 - x} \]

3. **Cross-multiply**: To eliminate the fractions, we cross-multiply:
\[ (19 - x)(91 - x) = (28 - x)(55 - x) \]

4. **Expand both sides**:
- Left side:
\[ 19 \cdot 91 - 19x - 91x + x^2 = 1729 - 110x + x^2 \]
- Right side:
\[ 28 \cdot 55 - 28x - 55x + x^2 = 1540 - 83x + x^2 \]

5. **Set the equation**: Now we have:
\[ 1729 - 110x + x^2 = 1540 - 83x + x^2 \]

6. **Cancel \( x^2 \) from both sides**:
\[ 1729 - 110x = 1540 - 83x \]

7. **Rearrange the equation**: Move all terms involving \( x \) to one side and constant terms to the other:
\[ 1729 - 1540 = 110x - 83x \]
\[ 189 = 27x \]

8. **Solve for \( x \)**:
\[ x = \frac{189}{27} = 7 \]

9. **Conclusion**: The number that must be subtracted from 19, 28, 55, and 91 so that they are in the same proportion is \( 7 \).

### Final Answer:
The answer is \( 7 \).