A couple got married 9 years ago when the age of wife was `20%` less than her husband. 6 years from now the age of wife will be only `12.5%` less than her husband. Now they have six children including single, twins and triplets and the ratio of their ages is 2: 3: 4 respectively. What can be the maximum possible value for the present age of this family?

Options

A. 110 years
B. 103 years
C. 105 years
D. 83 years

Answer

To solve the problem step by step, we will analyze the information given about the couple's ages and their children's ages.

### Step 1: Define Variables
Let the husband's age 9 years ago be \( H \) and the wife's age 9 years ago be \( W \).

### Step 2: Set Up the First Equation
According to the problem, the wife was 20% less than her husband at the time of their marriage.
This can be expressed as:
\[ W = H - 0.2H = 0.8H \]

### Step 3: Express Current Ages
Since they got married 9 years ago, their current ages are:
- Husband's current age: \( H + 9 \)
- Wife's current age: \( W + 9 \)

Substituting \( W \) from Step 2:
\[ W + 9 = 0.8H + 9 \]

### Step 4: Set Up the Second Equation
In 6 years, the wife's age will be 12.5% less than the husband's age. This can be expressed as:
\[ W + 15 = H + 15 - 0.125(H + 15) \]
Simplifying the right side:
\[ W + 15 = H + 15 - 0.125H - 1.875 \]
\[ W + 15 = 0.875H + 13.125 \]

### Step 5: Substitute \( W \) in the Second Equation
Substituting \( W = 0.8H \) into the second equation:
\[ 0.8H + 15 = 0.875H + 13.125 \]

### Step 6: Solve for \( H \)
Rearranging the equation gives:
\[ 0.8H - 0.875H = 13.125 - 15 \]
\[ -0.075H = -1.875 \]
\[ H = \frac{-1.875}{-0.075} = 25 \]

### Step 7: Find \( W \)
Using \( H = 25 \):
\[ W = 0.8H = 0.8 \times 25 = 20 \]

### Step 8: Calculate Current Ages
Now, we can find their current ages:
- Husband's current age: \( 25 + 9 = 34 \)
- Wife's current age: \( 20 + 9 = 29 \)

### Step 9: Calculate Total Age of the Couple
The total age of the couple is:
\[ 34 + 29 = 63 \]

### Step 10: Analyze the Children’s Ages
The children’s ages are in the ratio of 2:3:4. Let the ages of the children be \( 2x, 3x, 4x \) respectively, and since there are six children, we can assume:
- 1 single child of age \( 2x \)
- 2 twins of age \( 3x \) each
- 3 triplets of age \( 4x \) each

### Step 11: Calculate Total Age of Children
The total age of the children can be calculated as:
\[ 2x + 2(3x) + 3(4x) = 2x + 6x + 12x = 20x \]

### Step 12: Total Age of the Family
The total age of the family is:
\[ 63 + 20x \]

### Step 13: Maximize the Age
To maximize the total age of the family, we need to find the maximum integer value of \( x \) that keeps the children's ages reasonable.

Assuming the children are all under 18 years old, we can set \( 4x < 18 \):
\[ x < 4.5 \implies x \leq 4 \]

### Step 14: Calculate Maximum Total Age
Substituting \( x = 4 \):
\[ \text{Total age of children} = 20 \times 4 = 80 \]
\[ \text{Total age of family} = 63 + 80 = 143 \]

### Final Answer
The maximum possible value for the present age of this family is **143 years**.

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