The sum of the digits of a three digit number is 16. If the ten's digit of the number is 3 times the unit's digit and the unit's digit is one-fourth of the hundredth digit, then what is the number? 

Options

A. 446
B. 561
C. 682
D. 862

Answer

To solve the problem step by step, we need to define the digits of the three-digit number. Let's denote:

- The hundreds digit as \( x \)
- The tens digit as \( y \)
- The units digit as \( z \)

According to the problem, we have the following conditions:

1. The sum of the digits is 16:
\[ x + y + z = 16 \]

2. The tens digit is three times the units digit:
\[ y = 3z \]

3. The units digit is one-fourth of the hundreds digit:
\[ z = \frac{x}{4} \]

Now, we can substitute the expressions for \( y \) and \( z \) into the first equation.

### Step 1: Substitute \( z \) in terms of \( x \)
From the third equation, we have:
\[ z = \frac{x}{4} \]

### Step 2: Substitute \( y \) in terms of \( z \)
From the second equation, we can express \( y \) as:
\[ y = 3z = 3 \left(\frac{x}{4}\right) = \frac{3x}{4} \]

### Step 3: Substitute \( y \) and \( z \) into the sum equation
Now, substitute \( y \) and \( z \) into the first equation:
\[ x + y + z = 16 \]
\[ x + \frac{3x}{4} + \frac{x}{4} = 16 \]

### Step 4: Combine the terms
Combine the terms on the left side:
\[ x + \frac{3x}{4} + \frac{x}{4} = x + \frac{4x}{4} = x + x = 2x \]
So, we have:
\[ 2x = 16 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 2:
\[ x = \frac{16}{2} = 8 \]

### Step 6: Find \( z \)
Now that we have \( x \), we can find \( z \):
\[ z = \frac{x}{4} = \frac{8}{4} = 2 \]

### Step 7: Find \( y \)
Now, we can find \( y \):
\[ y = 3z = 3 \times 2 = 6 \]

### Conclusion: Write the number
Now we have all the digits:
- Hundreds digit \( x = 8 \)
- Tens digit \( y = 6 \)
- Units digit \( z = 2 \)

Thus, the three-digit number is:
\[ 862 \]