The age of Manish is half of that of his mother. His mother is 9 years younger…

Concept

Age word-problems are solved by translating each relationship into a linear equation in a single chosen variable, then substituting one equation into another until one unknown remains. The phrase "A is younger than B by k" means B = A + k, and "A is k times B" means A = k·B. Pin every quantity to one base variable so the system collapses to a single equation.

Setting up the equations

Let Manish's age be M. Translate each statement, expressing everything in terms of M:

1. Manish is half his mother's age, so mother = 2M.

2. Mother is 9 years younger than father, so father = mother + 9 = 2M + 9.

3. Manish is 7 years older than his sister, so sister = M − 7.

4. Father's age is three times the sister's age, so father = 3 × (M − 7).

Solving

Equate the two expressions for the father's age:

1. 2M + 9 = 3(M − 7)

2. 2M + 9 = 3M − 21

3. 9 + 21 = 3M − 2M

4. M = 30

Cross-check

With M = 30: mother = 60, father = 2(30) + 9 = 69, sister = 30 − 7 = 23. Verify the last condition: 3 × 23 = 69 = father. Every relationship holds, so Manish is 30 years old.