When `x` is subtracted from each of 55, 50, 23 and 22, the numbers so obtained in this order, are in proportion. What is the fourth proportional of 3, 7 and `x` ?

Options

A. 35
B. 27
C. 40
D. 30

Answer

To solve the problem step by step, we will first establish the relationship given in the question and then find the fourth proportional.

### Step 1: Set up the proportion
We know that when `x` is subtracted from each of the numbers 55, 50, 23, and 22, the resulting numbers are in proportion. This can be expressed mathematically as:

\[ \frac{55 - x}{50 - x} = \frac{23 - x}{22 - x} \]

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

\[ (55 - x)(22 - x) = (50 - x)(23 - x) \]

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

Left side:
\[ (55 - x)(22 - x) = 1210 - 55x - 22x + x^2 = x^2 - 77x + 1210 \]

Right side:
\[ (50 - x)(23 - x) = 1150 - 50x - 23x + x^2 = x^2 - 73x + 1150 \]

### Step 4: Set the equation to zero
Now we will set the equation to zero by equating both sides:

\[ x^2 - 77x + 1210 = x^2 - 73x + 1150 \]

Subtract \(x^2\) from both sides:

\[ -77x + 1210 = -73x + 1150 \]

### Step 5: Simplify the equation
Now, let's simplify the equation:

\[ -77x + 73x = 1150 - 1210 \]

This simplifies to:

\[ -4x = -60 \]

### Step 6: Solve for x
Now we can solve for `x`:

\[ x = \frac{60}{4} = 15 \]

### Step 7: Find the fourth proportional
Now we need to find the fourth proportional of 3, 7, and `x` (which is 15). We denote the fourth proportional as `y`. The relationship can be expressed as:

\[ \frac{3}{7} = \frac{15}{y} \]

### Step 8: Cross-multiply to find y
Cross-multiplying gives us:

\[ 3y = 15 \times 7 \]

### Step 9: Calculate y
Now we calculate `y`:

\[ 3y = 105 \implies y = \frac{105}{3} = 35 \]

### Conclusion
Thus, the fourth proportional of 3, 7, and `x` is:

\[ \boxed{35} \]