At the time of marriage of a son the average of age of mother, father and son is 42 yrs. After one year of marriage a baby is born when the baby turns 5 years old then the average age of whole family is 36 years. Find the age of their daughter-in-law at the time of her marriage.

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Determine the total age of the family at the time of marriage
Given that the average age of the mother, father, and son at the time of marriage is 42 years, we can calculate their total age.

Let the total age of the mother, father, and son be \( T \).

\[ \text{Average Age} = \frac{T}{3} = 42 \]

Multiplying both sides by 3 gives:

\[ T = 42 \times 3 = 126 \text{ years} \]

### Step 2: Calculate the ages after one year
After one year of marriage, the ages of the mother, father, and son will each increase by 1 year. Therefore, the new total age of the family (including the baby born after one year) will be:

\[ \text{Total Age after 1 year} = T + 3 + \text{(age of baby)} \]

At this point, the baby is born, and we will denote the age of the baby as 0 years.

So, the total age after one year is:

\[ \text{Total Age after 1 year} = 126 + 3 + 0 = 129 \text{ years} \]

### Step 3: Determine the total age when the baby turns 5
When the baby turns 5 years old, 5 years will have passed since the baby was born. This means that the mother, father, and son will each have aged 5 years more, and the baby will now be 5 years old.

The total age of the family at this point will be:

\[ \text{Total Age when baby is 5} = 126 + 3 + 5 \times 1 = 126 + 3 + 5 = 134 \text{ years} \]

### Step 4: Calculate the average age of the family when the baby is 5
At this point, the family consists of 5 members: mother, father, son, and the baby (now 5 years old). The average age of the family is given as 36 years.

Using the average formula:

\[ \text{Average Age} = \frac{\text{Total Age}}{5} = 36 \]

Multiplying both sides by 5 gives:

\[ \text{Total Age} = 36 \times 5 = 180 \text{ years} \]

### Step 5: Set up the equation
Now we have two expressions for total age when the baby is 5 years old:

1. From our calculation: \( 134 \) years
2. From the average: \( 180 \) years

This indicates that we need to account for the daughter-in-law's age at the time of marriage. Let \( x \) be the age of the daughter-in-law at the time of marriage.

So, we set up the equation:

\[ 134 + x = 180 \]

### Step 6: Solve for \( x \)
Now we can solve for \( x \):

\[ x = 180 - 134 = 46 \text{ years} \]

### Conclusion
The age of the daughter-in-law at the time of her marriage is **46 years**.

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