Correct Option (3)
Let Manisha's present age be m years and her mother's present age be m₁ years.
Statement-1 provides the relation: m = m₁ - 24. This is a single linear equation with two variables. It establishes a difference in ages but does not provide a unique value for either m or m₁. Therefore, Statement-1 alone is not sufficient to answer the question.
Statement-2 provides the relation for ages 5 years later: (m + 5) / (m₁ + 5) = 3 / 5. This is also a single linear equation with two variables (which simplifies to 5m + 25 = 3m₁ + 15 or 5m - 3m₁ = -10). It establishes a ratio between their future ages but does not provide a unique value for either m or m₁. Therefore, Statement-2 alone is not sufficient to answer the question.
When both Statement-1 and Statement-2 are considered together, we have a system of two linear equations with two variables:
- m = m₁ - 24
- (m + 5) / (m₁ + 5) = 3 / 5
Substitute the expression for m from equation (1) into equation (2):
((m₁ - 24) + 5) / (m₁ + 5) = 3 / 5
(m₁ - 19) / (m₁ + 5) = 3 / 5
Cross-multiply to solve for m₁:
5(m₁ - 19) = 3(m₁ + 5)
5m₁ - 95 = 3m₁ + 15
2m₁ = 110
m₁ = 55
Now, substitute the value of m₁ back into equation (1) to find m:
m = 55 - 24
m = 31
Since a unique value for Manisha's age (m = 31 years) can be determined by combining both statements, both Statement-1 and Statement-2 are sufficient to answer the question.
Incorrect Options:
Option 1: Statement-1 alone is sufficient to answer the question.
Statement-1, m = m₁ - 24, provides a relationship between Manisha's age (m) and her mother's age (m₁). As it is a single equation with two unknown variables, it yields an infinite number of possible age combinations and thus cannot determine a unique age for Manisha.
Option 2: Statement-2 alone is sufficient to answer the question.
Statement-2, (m + 5) / (m₁ + 5) = 3 / 5, also represents a single equation with two unknown variables (m and m₁). This equation alone is insufficient to determine a unique value for Manisha's age, as it would also lead to an infinite set of possible age pairs.
Option 4: Both statement-1 and statement-2 are not sufficient to answer the question.
As demonstrated in the explanation for the correct option, combining both statements forms a system of two independent linear equations with two variables. This system can be uniquely solved to find Manisha's age. Therefore, the assertion that both statements are not sufficient is incorrect.
