Eight years ago there were 5 members in the Arthur's family and then the average age of the family was 36 years. Mean while Arthur got married and gave birth to a child. Still the average age of his family is same now.
The present age of his wife is

Options

A. a) 25 years
B. b) 26 years
C. c) 32 years
D. d) Data insufficient

Answer

To solve the problem step by step, we can follow this reasoning:

### Step 1: Understand the information given
Eight years ago, there were 5 members in Arthur's family, and the average age of the family was 36 years.

### Step 2: Calculate the total age of the family 8 years ago
The average age is calculated as:
\[ \text{Average Age} = \frac{\text{Total Age}}{\text{Number of Members}} \]
From the information given:
\[ 36 = \frac{\text{Total Age}}{5} \]
To find the total age of the family 8 years ago, we can rearrange the formula:
\[ \text{Total Age} = 36 \times 5 = 180 \text{ years} \]

### Step 3: Calculate the total age of the family now
Since this was 8 years ago, we need to add 8 years to each of the 5 members:
\[ \text{Total Age Now} = 180 + (5 \times 8) = 180 + 40 = 220 \text{ years} \]

### Step 4: Determine the new family composition
After 8 years, Arthur got married and had a child, making the total number of family members 7 (5 original members + 1 wife + 1 child).

### Step 5: Calculate the average age now
According to the problem, the average age of the family remains the same (36 years). Therefore:
\[ \text{Average Age Now} = \frac{\text{Total Age Now}}{\text{Number of Members Now}} = 36 \]
Substituting the known values:
\[ 36 = \frac{220 + \text{Wife's Age} + \text{Child's Age}}{7} \]

### Step 6: Solve for the total age of the wife and child
Multiplying both sides by 7 gives:
\[ 220 + \text{Wife's Age} + \text{Child's Age} = 252 \]
Now, we can isolate the sum of the wife's age and the child's age:
\[ \text{Wife's Age} + \text{Child's Age} = 252 - 220 = 32 \]

### Step 7: Analyze the information
At this point, we have the equation:
\[ \text{Wife's Age} + \text{Child's Age} = 32 \]
However, we do not have enough information to determine the individual ages of the wife and the child.

### Conclusion
Since we cannot uniquely determine the present age of Arthur's wife without additional information about either the wife's age or the child's age, we conclude that the data is insufficient.

### Final Answer
The present age of Arthur's wife cannot be determined with the given information.

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