What should be subtracted from each number 1, 2, 4, 7, so that remainder would be proportional ?

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. `(1)/(3)`
D. `(1)/(4)`

Answer

To solve the problem of what should be subtracted from each number 1, 2, 4, and 7 so that the remainders are proportional, we can follow these steps:

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

### Step 2: Write the expressions for the remainders
After subtracting \( x \), the new numbers will be:
- From 1: \( 1 - x \)
- From 2: \( 2 - x \)
- From 4: \( 4 - x \)
- From 7: \( 7 - x \)

### Step 3: Set up the proportion
We want the remainders to be proportional. This means we can set up the following proportion:
\[ \frac{1 - x}{2 - x} = \frac{4 - x}{7 - x} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (1 - x)(7 - x) = (2 - x)(4 - x) \]

### Step 5: Expand both sides
Expanding both sides:
- Left side:
\[ 1 \cdot 7 - 1 \cdot x - 7 \cdot x + x^2 = 7 - 8x + x^2 \]
- Right side:
\[ 2 \cdot 4 - 2 \cdot x - 4 \cdot x + x^2 = 8 - 6x + x^2 \]

### Step 6: Set the equation
Now we have:
\[ 7 - 8x + x^2 = 8 - 6x + x^2 \]

### Step 7: Simplify the equation
We can cancel \( x^2 \) from both sides:
\[ 7 - 8x = 8 - 6x \]

### Step 8: Rearrange the equation
Rearranging gives us:
\[ 7 - 8 = -6x + 8x \]
\[ -1 = 2x \]

### Step 9: Solve for \( x \)
Dividing both sides by 2:
\[ x = -\frac{1}{2} \]

### Conclusion
The number that should be subtracted from each of the numbers 1, 2, 4, and 7 is \( -\frac{1}{2} \).

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