What must be subtracted from each of the numbers 23, 40, 57 and 108 so that the remainders are in proportion ?

Answer

To solve the problem of what must be subtracted from each of the numbers 23, 40, 57, and 108 so that the remainders are in proportion, we can follow these steps:

### Step 1: Set up the equation
Let \( x \) be the number that needs to be subtracted from each of the numbers. The condition for the remainders to be in proportion can be expressed as:

\[ \frac{23 - x}{40 - x} = \frac{57 - x}{108 - x} \]

### Step 2: Cross-multiply
To eliminate the fractions, we cross-multiply:

\[ (23 - x)(108 - x) = (40 - x)(57 - x) \]

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

Left side:
\[ 23 \cdot 108 - 23x - 108x + x^2 = 2484 - 131x + x^2 \]

Right side:
\[ 40 \cdot 57 - 40x - 57x + x^2 = 2280 - 97x + x^2 \]

### Step 4: Set the equation
Now, we can set the expanded forms equal to each other:

\[ 2484 - 131x + x^2 = 2280 - 97x + x^2 \]

### Step 5: Simplify the equation
We can simplify the equation by canceling out \( x^2 \) from both sides:

\[ 2484 - 131x = 2280 - 97x \]

### Step 6: Rearrange the equation
Now, let's rearrange the equation to isolate \( x \):

\[ 2484 - 2280 = 131x - 97x \]

This simplifies to:

\[ 204 = 34x \]

### Step 7: Solve for \( x \)
Now, divide both sides by 34 to find \( x \):

\[ x = \frac{204}{34} = 6 \]

### Conclusion
Thus, the number that must be subtracted from each of the numbers 23, 40, 57, and 108 is \( 6 \).

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