At present Asha’s age (in years) is \(2\) more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than \(10\) times the present age of Nisha. Find their present ages.
Nisha: 5 years; Asha: 27 years
Step 1: Suppose Nisha’s present age is \(n\) years.
Step 2: We are told Asha’s present age is 2 more than the square of Nisha’s age. So Asha’s age = \(n^2 + 2\). Let this be \(a\).
Step 3: The difference in their ages is \(a - n\). This is the number of years after which Nisha will reach her mother’s current age.
Step 4: At that time, Asha’s age will also increase by \(a - n\) years. So Asha’s age will then be \(a + (a - n) = 2a - n\).
Step 5: The question says that when Nisha is as old as her mother is now, Asha’s age will be one less than 10 times Nisha’s current age. So, \(2a - n = 10n - 1\).
Step 6: Substitute \(a = n^2 + 2\) into this equation: \(2(n^2 + 2) - n = 10n - 1\).
Step 7: Simplify: \(2n^2 + 4 - n = 10n - 1\) \(2n^2 - 11n + 5 = 0\).
Step 8: Solve this quadratic equation. Possible values of \(n\) are 5 or 0.5. Since age must be a whole number, \(n = 5\).
Step 9: Now find Asha’s age: \(a = n^2 + 2 = 5^2 + 2 = 27\).
Final Answer: Nisha is 5 years old and Asha is 27 years old.