The sum of the ages of a mother and a daughter is 50 years. Also,5 years ago, the mother’s age was 7 times of the age of her daughter. The present ages of the mother and the daughter respectively are-

Options

A. 35 years,15 years
B. 38 years,15 years
C. 40 years,10 years
D. 42 years,8 years

Answer

To solve the problem, we will use algebraic equations based on the information given in the question.

### Step-by-Step Solution

1. **Define Variables**:
Let the present age of the mother be \( X \) years and the present age of the daughter be \( Y \) years.

2. **Set Up the First Equation**:
According to the problem, the sum of their ages is 50 years. Therefore, we can write the first equation as:
\[ X + Y = 50 \quad \text{(1)} \]

3. **Set Up the Second Equation**:
The problem states that 5 years ago, the mother’s age was 7 times the age of her daughter. Five years ago, the mother’s age would be \( X - 5 \) and the daughter’s age would be \( Y - 5 \). This gives us the second equation:
\[ X - 5 = 7(Y - 5) \quad \text{(2)} \]

4. **Expand and Rearrange Equation (2)**:
Expanding equation (2):
\[ X - 5 = 7Y - 35 \]
Rearranging gives:
\[ X = 7Y - 30 \quad \text{(3)} \]

5. **Substitute Equation (3) into Equation (1)**:
Now, we substitute equation (3) into equation (1):
\[ (7Y - 30) + Y = 50 \]
Simplifying this:
\[ 8Y - 30 = 50 \]
Adding 30 to both sides:
\[ 8Y = 80 \]
Dividing by 8:
\[ Y = 10 \quad \text{(4)} \]

6. **Find the Mother's Age Using Equation (1)**:
Now that we have \( Y \), we can find \( X \) using equation (1):
\[ X + 10 = 50 \]
Subtracting 10 from both sides:
\[ X = 40 \quad \text{(5)} \]

7. **Conclusion**:
The present ages of the mother and daughter are:
- Mother: \( X = 40 \) years
- Daughter: \( Y = 10 \) years

### Final Answer:
The present ages of the mother and the daughter are 40 years and 10 years, respectively.

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