What number must be subtracted from each of the numbers 10, 12, 19, 24 to get the numbers which are in proportion ?

Answer

To solve the problem, we need to find the number \( x \) that must be subtracted from each of the numbers 10, 12, 19, and 24 so that the resulting numbers are in proportion. Here’s the step-by-step solution:

### Step 1: Set up the equation
Let \( x \) be the number that we need to subtract from each of the numbers. After subtracting \( x \), the numbers become:
- \( 10 - x \)
- \( 12 - x \)
- \( 19 - x \)
- \( 24 - x \)

We want these numbers to be in proportion, which means:
\[ \frac{10 - x}{12 - x} = \frac{19 - x}{24 - x} \]

### Step 2: Cross-multiply
To eliminate the fractions, we cross-multiply:
\[ (10 - x)(24 - x) = (12 - x)(19 - x) \]

### Step 3: Expand both sides
Now, we expand both sides of the equation:
- Left side:
\[ 10 \cdot 24 - 10x - 24x + x^2 = 240 - 34x + x^2 \]
- Right side:
\[ 12 \cdot 19 - 12x - 19x + x^2 = 228 - 31x + x^2 \]

### Step 4: Set the equation
Now we have:
\[ 240 - 34x + x^2 = 228 - 31x + x^2 \]

### Step 5: Simplify the equation
We can subtract \( x^2 \) from both sides:
\[ 240 - 34x = 228 - 31x \]

Now, rearranging gives:
\[ 240 - 228 = -31x + 34x \]
\[ 12 = 3x \]

### Step 6: Solve for \( x \)
Now, divide both sides by 3:
\[ x = \frac{12}{3} = 4 \]

### Conclusion
The number that must be subtracted from each of the numbers 10, 12, 19, and 24 to make them proportional is \( \boxed{4} \).

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