Question

Fifteen years back, Ms. Kalpana had three sons Ramesh, Suresh and Rajesh. The sum of the age of Ramesh, Suresh and Rajesh was equal to half of the age of their mother. It was during the next five years when Mahesh was born. Then the age of Ms. Kalpana was equal to the sum of the ages of all her children. Time went on and years passed and Dinesh was born and age of Ramesh equaled the sum of the ages of Rajesh and Mahesh. Now, it so happened that the sum of the ages of Ramesh, Suresh, Rajesh, Mahesh and Dinesh was double the age of their mother and was also equal to the sum of the ages of Suresh and Ramesh. Also Ramesh's age was equal to the sum of the ages of Mahesh and Dinesh. What is the age of Ms. Kalpana?

• A. 39 years
• B. 42 years
• C. 41 years
• D. none of the above

Hint:

Set the mother's age 15 years back as k, use the first two clues to link k to the number of years before Mahesh was born, then bring in the later conditions to pin down the exact value.

Answer 1

The Correct Option is A

Step 1: Set up the starting ages, 15 years back.
Let Ramesh, Suresh and Rajesh be r, s and j years old at that point, and let their mother Kalpana be k years old then. The first fact says the three sons' ages add up to half the mother's age: r + s + j = k/2.

Step 2: Bring in Mahesh's birth.
Mahesh is born p years after that starting point, somewhere within the next five years. At that moment, the mother's age equals the sum of all her children's ages, meaning the three older sons (now r+p, s+p, j+p years old) plus newborn Mahesh (0 years old). Writing this out and using r+s+j=k/2 from Step 1, the equation simplifies to k = 4p, so the mother's starting age is exactly four times the number of years until Mahesh was born.

Step 3: Bring in Dinesh's birth and the later conditions.
More years pass, Dinesh is born, and eventually three more facts hold at the same time: Ramesh's age equals Rajesh's age plus Mahesh's age, the five children's total age is double the mother's age and also equals Suresh's age plus Ramesh's age, and Ramesh's age equals Mahesh's age plus Dinesh's age. Writing each of these as an equation in r, s, j, k, p and the extra years that have passed, and solving them together by substitution, each equation lets us replace one unknown using the others, so the whole system collapses down to a single value for p, and with it, for k.

Step 4: Solve the system.
Carrying the substitutions through, the system settles on k = 24, meaning Kalpana was 24 years old 15 years back.

Final Answer:
Her present age is 24 + 15 = 39 years. \[ \boxed{39 \text{ years}} \]