The ratio of a 2-digit number to the sum of its digits is 7: 1. If the digit in the ten's place is 1 more than the digit in the one's place, then the number is

Options

A. 65
B. 43
C. 32
D. 21

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the digits
Let the digit in the unit's place be \( x \). According to the problem, the digit in the ten's place is 1 more than the digit in the unit's place. Therefore, the digit in the ten's place can be expressed as \( x + 1 \).

### Step 2: Formulate the two-digit number
The two-digit number can be represented as:
\[ 10 \times (\text{ten's place}) + (\text{unit's place}) = 10(x + 1) + x = 10x + 10 + x = 11x + 10 \]

### Step 3: Calculate the sum of the digits
The sum of the digits of the number is:
\[ (\text{ten's place}) + (\text{unit's place}) = (x + 1) + x = 2x + 1 \]

### Step 4: Set up the ratio
According to the problem, the ratio of the two-digit number to the sum of its digits is given as 7:1. We can express this as:
\[ \frac{11x + 10}{2x + 1} = 7 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 11x + 10 = 7(2x + 1) \]

### Step 6: Expand the equation
Expanding the right-hand side:
\[ 11x + 10 = 14x + 7 \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate \( x \):
\[ 11x + 10 - 7 = 14x \]
\[ 11x + 3 = 14x \]
\[ 3 = 14x - 11x \]
\[ 3 = 3x \]

### Step 8: Solve for \( x \)
Dividing both sides by 3:
\[ x = 1 \]

### Step 9: Find the digits
Now that we have \( x \), we can find the digits:
- The digit in the unit's place is \( x = 1 \).
- The digit in the ten's place is \( x + 1 = 1 + 1 = 2 \).

### Step 10: Form the two-digit number
Thus, the two-digit number is:
\[ 10 \times (\text{ten's place}) + (\text{unit's place}) = 10 \times 2 + 1 = 20 + 1 = 21 \]

### Final Answer
The required two-digit number is **21**.

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