A number is larger than half of 100. It is more than 6 tens and less than 8 tens. The sum of its digits is 9. The tens digit is the double of the ones digit. What is the number?

Options

A. 72
B. 63
C. 54
D. 81

Answer

To solve the problem step by step, we will break down the information provided in the question and use algebra to find the number.

### Step 1: Define the digits
Let:
- \( x \) = ones digit
- \( y \) = tens digit

### Step 2: Express the number
The number can be expressed as:
\[ \text{Number} = 10y + x \]

### Step 3: Set up the equations based on the problem statement
1. The tens digit is double the ones digit:
\[ y = 2x \]

2. The sum of the digits is 9:
\[ x + y = 9 \]

### Step 4: Substitute the first equation into the second
Substituting \( y = 2x \) into \( x + y = 9 \):
\[ x + 2x = 9 \]
\[ 3x = 9 \]

### Step 5: Solve for \( x \)
Dividing both sides by 3:
\[ x = 3 \]

### Step 6: Find \( y \) using the first equation
Substituting \( x = 3 \) back into \( y = 2x \):
\[ y = 2(3) = 6 \]

### Step 7: Form the number
Now, substituting \( x \) and \( y \) back into the expression for the number:
\[ \text{Number} = 10y + x = 10(6) + 3 = 60 + 3 = 63 \]

### Step 8: Verify the conditions
1. Is 63 larger than half of 100? Yes, \( 63 > 50 \).
2. Is it more than 6 tens (60) and less than 8 tens (80)? Yes, \( 63 \) is between \( 60 \) and \( 80 \).
3. Does the sum of the digits equal 9? Yes, \( 6 + 3 = 9 \).
4. Is the tens digit double the ones digit? Yes, \( 6 = 2 \times 3 \).

### Conclusion
The number that satisfies all the conditions is:
\[ \boxed{63} \]