f(x) is defined to be the sum of all the digits in a number x. If x is a three-digit number, what is the difference between the largest and the smallest values of x such that f(x)=11?

Options

A. 81
B. 101
C. 791
D. 801

Answer

To solve the problem, we need to find the largest and smallest three-digit numbers \( x \) such that the sum of the digits \( f(x) = 11 \).

### Step 1: Understanding the function \( f(x) \)
The function \( f(x) \) is defined as the sum of the digits of the number \( x \). For a three-digit number \( x \), we can express it as \( x = 100a + 10b + c \), where \( a \), \( b \), and \( c \) are the digits of \( x \). Here, \( a \) is the hundreds digit, \( b \) is the tens digit, and \( c \) is the units digit. Since \( x \) is a three-digit number, \( a \) must be between 1 and 9, while \( b \) and \( c \) can be between 0 and 9.

### Step 2: Finding the largest three-digit number
To find the largest three-digit number such that \( f(x) = 11 \), we want to maximize \( x \). This means we should start with the largest possible digit for \( a \), which is 9.

- Let \( a = 9 \). Then we have:
\[ f(x) = 9 + b + c = 11 \]
This simplifies to:
\[ b + c = 2 \]

To maximize \( x \), we can choose the largest possible value for \( b \):
- If \( b = 2 \), then \( c = 0 \). This gives us the number:
\[ x = 100 \cdot 9 + 10 \cdot 2 + 0 = 920 \]

### Step 3: Finding the smallest three-digit number
Next, we need to find the smallest three-digit number such that \( f(x) = 11 \). To minimize \( x \), we should start with the smallest possible digit for \( a \), which is 1.

- Let \( a = 1 \). Then we have:
\[ f(x) = 1 + b + c = 11 \]
This simplifies to:
\[ b + c = 10 \]

To minimize \( x \), we should choose the smallest possible value for \( b \):
- If \( b = 0 \), then \( c = 10 \) is not valid since \( c \) must be a single digit. So we try \( b = 1 \):
- If \( b = 1 \), then \( c = 9 \). This gives us the number:
\[ x = 100 \cdot 1 + 10 \cdot 1 + 9 = 119 \]

### Step 4: Calculating the difference
Now we have the largest three-digit number \( x = 920 \) and the smallest three-digit number \( x = 119 \).

To find the difference:
\[ \text{Difference} = 920 - 119 = 801 \]

### Final Answer
The difference between the largest and smallest three-digit numbers such that \( f(x) = 11 \) is \( 801 \).