My sister was born 26 yrs after my mother was born and I was born 28 yrs after my father was born. The present average age of my family is 32.5 yrs.After 3 yrs from now my sister will get married and willleave the family, then average age of family will become 41-years. What is the present age of my mother? (in yrs)

Options

A. A)40
B. B)50
C. C)45
D. D)42

Answer

To find the present age of the mother, let's break down the problem step by step.

### Step 1: Define the variables
Let:
- D = Present age of my sister
- M = Present age of my mother
- S = Present age of my brother (son)
- F = Present age of my father

From the information given:
- My sister was born 26 years after my mother: \( D = M - 26 \)
- I was born 28 years after my father: \( S = F - 28 \)

### Step 2: Calculate the total age of the family
The average age of the family is given as 32.5 years. Since there are 4 members in the family (mother, father, son, and sister), we can calculate the total age of the family:
\[ \text{Total Age} = \text{Average Age} \times \text{Number of Family Members} = 32.5 \times 4 = 130 \]

### Step 3: Set up the equation for total age
The total age of the family can also be expressed as:
\[ M + F + S + D = 130 \]
Substituting \( D \) and \( S \) from the earlier definitions:
\[ M + (S + 28) + S + (M - 26) = 130 \]
This simplifies to:
\[ 2M + 2S + 2 = 130 \]
\[ 2M + 2S = 128 \]
\[ M + S = 64 \quad \text{(Equation 1)} \]

### Step 4: Calculate the average age after 3 years
After 3 years, my sister will leave the family. The ages will then be:
- Mother: \( M + 3 \)
- Father: \( F + 3 \)
- Son: \( S + 3 \)

The new average age of the remaining family members (mother, father, son) is given as 41 years. Therefore:
\[ \frac{(M + 3) + (F + 3) + (S + 3)}{3} = 41 \]
Multiplying through by 3 gives:
\[ M + F + S + 9 = 123 \]
\[ M + F + S = 114 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
From Equation 1, we know:
\[ F = 114 - (M + S) \]
Substituting \( M + S = 64 \) from Equation 1 into Equation 2:
\[ F = 114 - 64 = 50 \]

### Step 6: Find the age of the son
Using \( F = S + 28 \):
\[ 50 = S + 28 \]
\[ S = 50 - 28 = 22 \]

### Step 7: Find the age of the sister
Using Equation 1:
\[ M + 22 = 64 \]
\[ M = 64 - 22 = 42 \]

### Conclusion
The present age of my mother is **42 years**.