The sum of four numbers is 48. When 5 and 1 are added to the first two and 3 & 7 are subtracted from the 3rd & 4th, the numbers will be equal. The numbers are

Options

A. ` 4, 12, 12, 20`
B. ` 5,11, 13, 19`
C. ` 6,10,14,18`
D. ` 9,7,15, 17`

Explanation

To solve the problem, let's denote the four numbers as \( a, b, c, \) and \( d \).

### Step 1: Set up the equations based on the problem statement.
From the problem, we know:
1. The sum of the four numbers is 48:
\[ a + b + c + d = 48 \]

2. When 5 is added to the first number \( a \) and 1 is added to the second number \( b \), and 3 is subtracted from the third number \( c \) and 7 is subtracted from the fourth number \( d \), all the numbers become equal:
\[ a + 5 = b + 1 = c - 3 = d - 7 \]

Let's denote the common value of these expressions as \( x \):
- From \( a + 5 = x \), we can express \( a \) as:
\[ a = x - 5 \]

- From \( b + 1 = x \), we can express \( b \) as:
\[ b = x - 1 \]

- From \( c - 3 = x \), we can express \( c \) as:
\[ c = x + 3 \]

- From \( d - 7 = x \), we can express \( d \) as:
\[ d = x + 7 \]

### Step 2: Substitute these expressions into the sum equation.
Now we substitute \( a, b, c, \) and \( d \) into the sum equation:
\[ (x - 5) + (x - 1) + (x + 3) + (x + 7) = 48 \]

### Step 3: Simplify the equation.
Combining like terms:
\[ x - 5 + x - 1 + x + 3 + x + 7 = 48 \]
\[ 4x + 4 = 48 \]

### Step 4: Solve for \( x \).
Subtract 4 from both sides:
\[ 4x = 48 - 4 \]
\[ 4x = 44 \]
Now divide by 4:
\[ x = 11 \]

### Step 5: Find the values of \( a, b, c, \) and \( d \).
Now we can substitute \( x \) back into the expressions for \( a, b, c, \) and \( d \):
- For \( a \):
\[ a = 11 - 5 = 6 \]

- For \( b \):
\[ b = 11 - 1 = 10 \]

- For \( c \):
\[ c = 11 + 3 = 14 \]

- For \( d \):
\[ d = 11 + 7 = 18 \]

### Final Answer:
The four numbers are \( a = 6, b = 10, c = 14, d = 18 \).