The sum of the digits of a three-digit number is 16. If the tens digit of the number is 3 times the units digit and the units 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 will define the digits of the three-digit number and set up equations based on the conditions given in the question.

### Step 1: Define the digits
Let:
- The hundreds digit be \( x \)
- The units digit be \( y \)
- The tens digit be \( z \)

### Step 2: Set up the equations based on the conditions
1. The sum of the digits is 16:
\[ x + y + z = 16 \quad \text{(Equation 1)} \]

2. The tens digit is 3 times the units digit:
\[ z = 3y \quad \text{(Equation 2)} \]

3. The units digit is one-fourth of the hundreds digit:
\[ y = \frac{x}{4} \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equations 1 and 2
From Equation 3, substitute \( y \) into Equation 1:
\[ x + \frac{x}{4} + z = 16 \]

Now, substitute \( z \) from Equation 2 into the equation:
\[ x + \frac{x}{4} + 3y = 16 \]

### Step 4: Substitute \( y \) in terms of \( x \) into the equation
Substituting \( y = \frac{x}{4} \) into the equation:
\[ x + \frac{x}{4} + 3\left(\frac{x}{4}\right) = 16 \]
This simplifies to:
\[ x + \frac{x}{4} + \frac{3x}{4} = 16 \]

### Step 5: Combine like terms
Combine the terms on the left side:
\[ x + \frac{4x}{4} = 16 \]
\[ x + x = 16 \]
\[ 2x = 16 \]

### Step 6: Solve for \( x \)
Divide both sides by 2:
\[ x = 8 \]

### Step 7: Find \( y \) and \( z \)
Now substitute \( x \) back into Equation 3 to find \( y \):
\[ y = \frac{x}{4} = \frac{8}{4} = 2 \]

Now substitute \( y \) into Equation 2 to find \( z \):
\[ z = 3y = 3 \times 2 = 6 \]

### Step 8: Write the number
The digits of the three-digit number are:
- Hundreds digit \( x = 8 \)
- Tens digit \( z = 6 \)
- Units digit \( y = 2 \)

Thus, the three-digit number is \( 862 \).

### Final Answer
The three-digit number is **862**.

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