Each of the questions given below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements sufficient to answer the question. Read both the statements and give answer
What is A' s mother’s present age?
I. The respective’ ratio between A’s, father’s present age and A’s present age is `3 : 1`.
II The combined present age of A’s father and A is 8 years more than the combined present age of A’s mother and A. 12 years ago, A’s fathers age that time was `1 2/7` times A’s mother’s age that time.

Options

A. if the data in statement I alone are sufficient to answer the question, while the data in statement II alone are not sufficient .to answer the question.
B. if the data in statement II alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question
C. if the data 'in statement I alone or in statement II alone are sufficient to answer the question.
D. if the data in both the statements I and II are not sufficient to answer the question.

Answer

To solve the question regarding A's mother's present age, we will analyze the two statements provided and determine if they are sufficient to answer the question.

### Step 1: Analyze Statement I
Statement I states that the ratio between A's father's present age and A's present age is 3:1.

Let:
- A's present age = A
- A's father's present age = 3A (since the ratio is 3:1)

This statement alone does not provide any information about A's mother's age. Therefore, we cannot determine A's mother's present age from this statement alone.

**Hint for Step 1:** Check if the statement provides direct information about A's mother's age.

### Step 2: Analyze Statement II
Statement II provides two pieces of information:
1. The combined present age of A's father and A is 8 years more than the combined present age of A's mother and A.
2. 12 years ago, A's father's age was \( \frac{12}{7} \) times A's mother's age.

From the first part of Statement II:
- A's father's age + A's age = A's mother's age + A's age + 8
- This simplifies to: \( 3A + A = Y + A + 8 \) (where Y is A's mother's age)
- Thus, \( 4A = Y + A + 8 \)
- Rearranging gives us: \( 3A = Y + 8 \) (Equation 1)

From the second part of Statement II:
- 12 years ago, A's father's age = \( 3A - 12 \)
- 12 years ago, A's mother's age = \( Y - 12 \)
- According to the statement: \( 3A - 12 = \frac{12}{7}(Y - 12) \)

Multiplying through by 7 to eliminate the fraction:
- \( 7(3A - 12) = 12(Y - 12) \)
- This expands to: \( 21A - 84 = 12Y - 144 \)
- Rearranging gives us: \( 21A - 12Y = -60 \) (Equation 2)

### Step 3: Solve the Equations
Now we have two equations:
1. \( 3A = Y + 8 \) (Equation 1)
2. \( 21A - 12Y = -60 \) (Equation 2)

From Equation 1, we can express Y in terms of A:
- \( Y = 3A - 8 \)

Substituting \( Y \) into Equation 2:
- \( 21A - 12(3A - 8) = -60 \)
- This simplifies to: \( 21A - 36A + 96 = -60 \)
- Combining like terms gives: \( -15A + 96 = -60 \)
- Rearranging gives: \( -15A = -156 \)
- Thus, \( A = 10.4 \)

Now substituting back to find Y:
- \( Y = 3(10.4) - 8 = 31.2 - 8 = 23.2 \)

### Conclusion
A's mother's present age is 23.2 years.

### Final Answer
- The data in Statement II alone is sufficient to determine A's mother's present age, while Statement I alone is not sufficient.