What number must be subtracted from each of the numbers 32, 38, 17 and 20, so that the terms formed will be in proportion ?

Answer

To find the number that must be subtracted from each of the numbers 32, 38, 17, and 20 so that they are in proportion, we will follow these steps:

### Step 1: Set up the equation for proportion
We need to express the condition for the numbers to be in proportion. If we subtract a number \( x \) from each of the given numbers, we have:
- First term: \( 32 - x \)
- Second term: \( 38 - x \)
- Third term: \( 17 - x \)
- Fourth term: \( 20 - x \)

The proportion can be expressed as:
\[ \frac{32 - x}{38 - x} = \frac{17 - x}{20 - x} \]

### Step 2: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (32 - x)(20 - x) = (38 - x)(17 - x) \]

### Step 3: Expand both sides
Now we will expand both sides of the equation:
- Left side:
\[ 32 \cdot 20 - 32x - 20x + x^2 = 640 - 52x + x^2 \]
- Right side:
\[ 38 \cdot 17 - 38x - 17x + x^2 = 646 - 55x + x^2 \]

### Step 4: Set the expanded equations equal to each other
Now we set the left side equal to the right side:
\[ 640 - 52x + x^2 = 646 - 55x + x^2 \]

### Step 5: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 640 - 52x = 646 - 55x \]

Now, rearranging gives:
\[ 55x - 52x = 646 - 640 \]
\[ 3x = 6 \]

### Step 6: Solve for \( x \)
Dividing both sides by 3:
\[ x = 2 \]

### Conclusion
The number that must be subtracted from each of the numbers 32, 38, 17, and 20 to make them proportional is \( \boxed{2} \).

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