If `5{x}=x+[x]` and `[x]-{x}=(1)/(2)` when `{x}` and `[x]` are fractional and integral part of x then x is

Options

A. `(1)/(2)`
B. `(3)/(2)`
C. `(5)/(2)`
D. `(7)/(2)`

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Define the Variables
Let:
- \([x]\) = integral part of \(x\) (denote it as \(n\))
- \(\{x\}\) = fractional part of \(x\) (denote it as \(f\))

From the definition of \(x\):
\[ x = [x] + \{x\} = n + f \]

### Step 2: Rewrite the First Equation
The first equation given is:
\[ 5\{x\} = x + [x] \]
Substituting \(x\) and \([x]\):
\[ 5f = (n + f) + n \]
This simplifies to:
\[ 5f = 2n + f \]

### Step 3: Rearranging the First Equation
Now, rearranging the equation:
\[ 5f - f = 2n \]
\[ 4f = 2n \]
Dividing both sides by 2:
\[ 2f = n \]
Thus, we have our first equation:
\[ n = 2f \] (Equation 1)

### Step 4: Rewrite the Second Equation
The second equation given is:
\[ [x] - \{x\} = \frac{1}{2} \]
Substituting \(n\) and \(f\):
\[ n - f = \frac{1}{2} \]

### Step 5: Substitute Equation 1 into the Second Equation
Now, substitute \(n\) from Equation 1 into the second equation:
\[ 2f - f = \frac{1}{2} \]
This simplifies to:
\[ f = \frac{1}{2} \]

### Step 6: Find the Integral Part
Now, substitute \(f\) back into Equation 1 to find \(n\):
\[ n = 2f = 2 \times \frac{1}{2} = 1 \]

### Step 7: Find \(x\)
Now we can find \(x\):
\[ x = n + f = 1 + \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{\frac{3}{2}} \]

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