What number should be added to each of 4,16,18 and 58 so that the resulting numbers will be in proportion in the given order?

Options

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

Answer

To solve the problem of finding what number should be added to each of the numbers 4, 16, 18, and 58 so that the resulting numbers are in proportion, we can follow these steps:

### Step 1: Set up the equation for proportion
We want the numbers \( (4 + x), (16 + x), (18 + x), (58 + x) \) to be in proportion. This means that the ratio of the first two numbers should be equal to the ratio of the last two numbers. In mathematical terms, this can be expressed as:

\[ \frac{4 + x}{16 + x} = \frac{18 + x}{58 + x} \]

### Step 2: Cross-multiply the proportions
Cross-multiplying gives us:

\[ (4 + x)(58 + x) = (16 + x)(18 + x) \]

### Step 3: Expand both sides
Now, we expand both sides of the equation:

Left side:
\[ 4 \cdot 58 + 4x + 58x + x^2 = 232 + 62x + x^2 \]

Right side:
\[ 16 \cdot 18 + 16x + 18x + x^2 = 288 + 34x + x^2 \]

### Step 4: Set the equation
Now we have:

\[ 232 + 62x + x^2 = 288 + 34x + x^2 \]

### Step 5: Simplify the equation
We can eliminate \( x^2 \) from both sides:

\[ 232 + 62x = 288 + 34x \]

### Step 6: Rearrange the equation
Rearranging gives:

\[ 62x - 34x = 288 - 232 \]

This simplifies to:

\[ 28x = 56 \]

### Step 7: Solve for \( x \)
Dividing both sides by 28:

\[ x = \frac{56}{28} = 2 \]

### Conclusion
The number that should be added to each of 4, 16, 18, and 58 is **2**.