A number consists of two digits. The digit in the ten's place exceeds the digit in the unit's place by 4. The sum of the digits is `1/7` of the number. The number is :

Options

A. `27`
B. `72`
C. `48`
D. `84`

Answer

To solve the problem step by step, we need to find a two-digit number based on the given conditions. Let's break it down:

### Step 1: Define the Digits
Let the digit in the unit's place be \( b \). According to the problem, the digit in the ten's place exceeds the digit in the unit's place by 4. Therefore, we can express the digit in the ten's place as:
\[ a = b + 4 \]

### Step 2: Express the Number
A two-digit number can be expressed as:
\[ \text{Number} = 10a + b \]
Substituting the value of \( a \) from Step 1:
\[ \text{Number} = 10(b + 4) + b = 10b + 40 + b = 11b + 40 \]

### Step 3: Set Up the Equation for the Sum of the Digits
The sum of the digits is given as \( 1/7 \) of the number. The sum of the digits (the ten's place and the unit's place) is:
\[ \text{Sum of digits} = a + b = (b + 4) + b = 2b + 4 \]

According to the problem:
\[ 2b + 4 = \frac{1}{7} \times (11b + 40) \]

### Step 4: Cross Multiply to Eliminate the Fraction
To eliminate the fraction, we can cross-multiply:
\[ 7(2b + 4) = 11b + 40 \]

### Step 5: Expand and Simplify the Equation
Expanding the left side:
\[ 14b + 28 = 11b + 40 \]

### Step 6: Rearrange the Equation
Now, let's move all terms involving \( b \) to one side and constant terms to the other side:
\[ 14b - 11b = 40 - 28 \]
This simplifies to:
\[ 3b = 12 \]

### Step 7: Solve for \( b \)
Now, divide both sides by 3:
\[ b = 4 \]

### Step 8: Find the Value of \( a \)
Using the value of \( b \) to find \( a \):
\[ a = b + 4 = 4 + 4 = 8 \]

### Step 9: Form the Number
Now we can form the two-digit number:
\[ \text{Number} = 10a + b = 10(8) + 4 = 80 + 4 = 84 \]

### Conclusion
Thus, the two-digit number is:
\[ \boxed{84} \]