A couple got married 9 years ago when the age of wife was 20% less than her husband. 6 years from no 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 sum of the present age of the couple?

Options

A. a.65 years
B. b.63 years
C. c.56 years
D. d.83 years

Answer

To solve the problem step by step, we will break it down as follows:

### Step 1: Define the variables for the ages of the couple 9 years ago.
Let the husband's age 9 years ago be \( H \) and the wife's age be \( W \). According to the problem, the wife was 20% less than her husband at that time.

This can be expressed as:
\[ W = H - 0.2H = 0.8H \]

### Step 2: Express the ages in terms of a ratio.
We can express the ages in a ratio. If we let the husband's age be represented as 5 parts, then the wife's age can be represented as 4 parts (since 0.8 is equivalent to 4/5).

Thus, we can write:
\[ H = 5x \]
\[ W = 4x \]
where \( x \) is a common multiplier.

### Step 3: Set up the equation for ages 6 years from now.
According to the problem, 6 years from now, the wife's age will be 12.5% less than the husband's age. This can be expressed as:
\[ W + 6 = H + 6 - 0.125(H + 6) \]

This simplifies to:
\[ W + 6 = 0.875(H + 6) \]

### Step 4: Substitute the expressions for \( W \) and \( H \).
Substituting \( W \) and \( H \) in terms of \( x \):
\[ 4x + 6 = 0.875(5x + 6) \]

### Step 5: Solve the equation.
Expanding the right side:
\[ 4x + 6 = 4.375x + 5.25 \]

Now, rearranging the equation:
\[ 4x - 4.375x = 5.25 - 6 \]
\[ -0.375x = -0.75 \]
\[ x = 2 \]

### Step 6: Find the present ages of the couple.
Now we can find the ages of the husband and wife 9 years ago:
- Husband's age 9 years ago:
\[ H = 5x = 5 \times 2 = 10 \]
- Wife's age 9 years ago:
\[ W = 4x = 4 \times 2 = 8 \]

Now, adding 9 years to find their present ages:
- Present age of husband:
\[ 10 + 9 = 19 \]
- Present age of wife:
\[ 8 + 9 = 17 \]

### Step 7: Calculate the sum of their present ages.
Now, we can find the sum of their present ages:
\[ 19 + 17 = 36 \]

### Conclusion
The sum of the present ages of the couple is \( 36 \).

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