What must be added to each of the numbers 3, 7, 8 and 16 so that resulting numbers are in proportion ?

Options

A. 4
B. 3
C. 2
D. 1

Explanation

To solve the problem of what must be added to each of the numbers 3, 7, 8, and 16 so that the resulting numbers are in proportion, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Proportions**:
We need to find a number \( x \) that, when added to each of the numbers 3, 7, 8, and 16, results in the four numbers being in proportion. This means that the ratios of the numbers should be equal:
\[ \frac{3 + x}{7 + x} = \frac{8 + x}{16 + x} \]

2. **Cross-Multiplying**:
To eliminate the fractions, we can cross-multiply:
\[ (3 + x)(16 + x) = (7 + x)(8 + x) \]

3. **Expanding Both Sides**:
Now, we expand both sides of the equation:
- Left Side:
\[ 3 \cdot 16 + 3x + 16x + x^2 = 48 + 19x + x^2 \]
- Right Side:
\[ 7 \cdot 8 + 7x + 8x + x^2 = 56 + 15x + x^2 \]

4. **Setting Up the Equation**:
Now we set the two sides equal to each other:
\[ 48 + 19x + x^2 = 56 + 15x + x^2 \]

5. **Simplifying the Equation**:
We can subtract \( x^2 \) from both sides and simplify:
\[ 48 + 19x = 56 + 15x \]

6. **Isolating \( x \)**:
Next, we isolate \( x \) by moving all terms involving \( x \) to one side and constant terms to the other:
\[ 19x - 15x = 56 - 48 \]
\[ 4x = 8 \]

7. **Solving for \( x \)**:
Finally, we divide both sides by 4 to find \( x \):
\[ x = 2 \]

### Conclusion:
The number that must be added to each of the numbers 3, 7, 8, and 16 to make them proportional is **2**.

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