What number should be subtracted fram each of the numbers `23`, `30`, `57` and `78`, so that the remainders are in proportion?

Answer

To solve the problem of finding the number that should be subtracted from each of the numbers 23, 30, 57, and 78 so that the remainders are in proportion, we can follow these steps:

### Step 1: Define the variable
Let \( x \) be the number that we need to subtract from each of the given numbers.

### Step 2: Write the expressions for the remainders
After subtracting \( x \), the remainders will be:
- From 23: \( 23 - x \)
- From 30: \( 30 - x \)
- From 57: \( 57 - x \)
- From 78: \( 78 - x \)

### Step 3: Set up the proportion
According to the problem, the remainders should be in proportion. Therefore, we can set up the equation:
\[ \frac{23 - x}{30 - x} = \frac{57 - x}{78 - x} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (23 - x)(78 - x) = (30 - x)(57 - x) \]

### Step 5: Expand both sides
Now, we will expand both sides of the equation:
- Left side:
\[ 23 \times 78 - 23x - 78x + x^2 = 1794 - 101x + x^2 \]
- Right side:
\[ 30 \times 57 - 30x - 57x + x^2 = 1710 - 87x + x^2 \]

### Step 6: Set the equation
Now we have:
\[ 1794 - 101x + x^2 = 1710 - 87x + x^2 \]

### Step 7: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 1794 - 101x = 1710 - 87x \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 1794 - 1710 = 101x - 87x \]
\[ 84 = 14x \]

### Step 9: Solve for \( x \)
Dividing both sides by 14:
\[ x = \frac{84}{14} = 6 \]

### Conclusion
The number that should be subtracted from each of the numbers 23, 30, 57, and 78 is \( 6 \).

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