The average of two number is 6 and four times and difference between them is 16. Find the numbers.

Answer

To solve the problem step by step, we will define the two numbers and set up equations based on the information given.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the First Equation
The average of the two numbers is given as 6. The formula for the average of two numbers is:
\[ \text{Average} = \frac{x + y}{2} \]
Setting this equal to 6, we have:
\[ \frac{x + y}{2} = 6 \]
Multiplying both sides by 2 to eliminate the fraction:
\[ x + y = 12 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem states that four times the difference between the two numbers is 16. The difference between the two numbers can be expressed as \( x - y \). Therefore, we can write:
\[ 4(x - y) = 16 \]
Dividing both sides by 4 to simplify:
\[ x - y = 4 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x + y = 12 \)
2. \( x - y = 4 \)

We can add these two equations together to eliminate \( y \):
\[ (x + y) + (x - y) = 12 + 4 \]
This simplifies to:
\[ 2x = 16 \]
Dividing both sides by 2 gives:
\[ x = 8 \]

### Step 5: Find the Value of \( y \)
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). Using Equation 1:
\[ 8 + y = 12 \]
Subtracting 8 from both sides:
\[ y = 12 - 8 \]
Thus, we find:
\[ y = 4 \]

### Final Answer
The two numbers are \( x = 8 \) and \( y = 4 \).

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