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

Options

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

Answer

To solve the problem step by step, we will denote the four numbers as \( a, b, c, \) and \( d \).

### Step 1: Set up the equations
We know from the problem that the sum of the four numbers is 48. Therefore, we can write the first equation as:
\[ a + b + c + d = 48 \]

### Step 2: Establish relationships based on the conditions given
According to the problem, when we add 5 to the first number \( a \) and 1 to the second number \( b \), and subtract 3 from the third number \( c \) and 7 from the fourth number \( d \), all resulting numbers will be equal. This gives us the following relationships:
\[ a + 5 = b + 1 = c - 3 = d - 7 \]

### Step 3: Express \( b, c, \) and \( d \) in terms of \( a \)
From the equation \( a + 5 = b + 1 \), we can express \( b \) in terms of \( a \):
\[ b = a + 5 - 1 = a + 4 \]

From the equation \( a + 5 = c - 3 \), we can express \( c \) in terms of \( a \):
\[ c = a + 5 + 3 = a + 8 \]

From the equation \( a + 5 = d - 7 \), we can express \( d \) in terms of \( a \):
\[ d = a + 5 + 7 = a + 12 \]

### Step 4: Substitute \( b, c, \) and \( d \) back into the sum equation
Now we substitute \( b, c, \) and \( d \) into the first equation:
\[ a + (a + 4) + (a + 8) + (a + 12) = 48 \]

### Step 5: Simplify the equation
Combining like terms, we have:
\[ 4a + 24 = 48 \]

### Step 6: Solve for \( a \)
Now, we can isolate \( a \):
\[ 4a = 48 - 24 \]
\[ 4a = 24 \]
\[ a = \frac{24}{4} = 6 \]

### Step 7: Find the values of \( b, c, \) and \( d \)
Now that we have \( a \), we can find \( b, c, \) and \( d \):
\[ b = a + 4 = 6 + 4 = 10 \]
\[ c = a + 8 = 6 + 8 = 14 \]
\[ d = a + 12 = 6 + 12 = 18 \]

### Final Answer
The four numbers are:
\[ \boxed{6, 10, 14, 18} \]