The sum of the ages of a daughter and mother is 56 years. After 4 years, the age of the mother will be 3 times that of the daughter. Their respective ages are.

Options

A. (a) 10 years and 46 years
B. (b) 12 years and 44 years
C. (c) 11 years and 45 years
D. (d) 13 years and 43 years

Explanation

To solve the problem step by step, we can set up equations based on the information given.

### Step 1: Define the variables
Let:
- \( D \) = age of the daughter
- \( M \) = age of the mother

### Step 2: Set up the first equation
According to the problem, the sum of the ages of the daughter and mother is 56 years. This gives us our first equation:
\[ D + M = 56 \]

### Step 3: Set up the second equation
The problem states that after 4 years, the age of the mother will be 3 times that of the daughter. In 4 years, the daughter will be \( D + 4 \) and the mother will be \( M + 4 \). This leads to our second equation:
\[ M + 4 = 3(D + 4) \]

### Step 4: Simplify the second equation
Expanding the second equation:
\[ M + 4 = 3D + 12 \]
Now, rearranging it gives:
\[ M = 3D + 12 - 4 \]
\[ M = 3D + 8 \]

### Step 5: Substitute the value of M in the first equation
Now, we can substitute \( M \) from the second equation into the first equation:
\[ D + (3D + 8) = 56 \]
Combining like terms:
\[ 4D + 8 = 56 \]

### Step 6: Solve for D
Subtract 8 from both sides:
\[ 4D = 48 \]
Now, divide by 4:
\[ D = 12 \]

### Step 7: Find the age of the mother
Now that we have the age of the daughter, we can find the age of the mother using the first equation:
\[ M = 56 - D \]
\[ M = 56 - 12 \]
\[ M = 44 \]

### Conclusion
The ages are:
- Daughter's age \( D = 12 \) years
- Mother's age \( M = 44 \) years