What number should be subtracted from each of 50, 61, 92, 117 so that the numbers, so obtained in this order, are in proportion?

Options

A. 14
B. 17
C. 19
D. 23

Answer

To solve the problem of what number should be subtracted from each of the numbers 50, 61, 92, and 117 so that the resulting numbers are in proportion, we can follow these steps:

### Step 1: Set up the equation
Let the number to be subtracted be \( x \). The new numbers will be:
- \( 50 - x \)
- \( 61 - x \)
- \( 92 - x \)
- \( 117 - x \)

We want these numbers to be in proportion, which means:
\[ \frac{50 - x}{61 - x} = \frac{92 - x}{117 - x} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ (50 - x)(117 - x) = (61 - x)(92 - x) \]

### Step 3: Expand both sides
Now we expand both sides of the equation:
- Left side:
\[ 50 \cdot 117 - 50x - 117x + x^2 = 5850 - 167x + x^2 \]
- Right side:
\[ 61 \cdot 92 - 61x - 92x + x^2 = 5592 - 153x + x^2 \]

### Step 4: Set the equation
Now we set the two expanded sides equal to each other:
\[ 5850 - 167x + x^2 = 5592 - 153x + x^2 \]

### Step 5: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 5850 - 167x = 5592 - 153x \]

### Step 6: Rearrange the equation
Rearranging gives us:
\[ 5850 - 5592 = 167x - 153x \]
\[ 258 = 14x \]

### Step 7: Solve for \( x \)
Now, divide both sides by 14:
\[ x = \frac{258}{14} = 18.42857 \approx 18.43 \]

### Step 8: Conclusion
The number that should be subtracted from each of the numbers 50, 61, 92, and 117 is approximately \( 18.43 \).

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