Question

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.

Answer 1

Step-by-step explanation:

When Nisha grows up to her mother's present age. So we can find that time by subtracting Nisha's age at that time and present age. But age should be a whole number. So Nisha's present age is 5 years.

Answer 2

Appropriate Question:

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 the present age of Nisha and Asha.

Answer:

$\begin{gathered}\begin{gathered}\bf\: \begin{cases}&\sf{Nisha\:present\:age = 5 \: years} \\ \\ &\sf{Asha\:present\:age = 27\: years} \end{cases}\end{gathered}\end{gathered}$

Step-by-step explanation:

$\begin{gathered}\begin{gathered}\bf\: Let\:assume\:that : \begin{cases} &\sf{Nisha\:present\:age = x \: years} \\ \\ &\sf{Asha\:present\:age = y \: years} \end{cases}\end{gathered}\end{gathered}$

Case :- 1

According to statement, At present Asha's age (in years) is 2 more than the square of her daughter Nisha's age.

$\implies\sf \: \boxed{\sf \: y = {x}^{2} + 2 \: } - - - (1)$

Case :- 2

When Nisha grows to her mother's present age, it means when Nisha attains the age of her mother present age, i.e. y years, it can be achieved after y - x years.

So,

$\begin{gathered}\begin{gathered}\bf\: After\:(y - x)\: years\:\begin{cases} &\sf{Nisha\:age = y \: years} \\ \\ &\sf{Asha\:age = y + y - x = 2y-x \: years} \end{cases}\end{gathered}\end{gathered}$

According to statement, it is given that Asha's age would be one year less than 10 times the present age of Nisha.

$\sf \: 2y - x = 10x - 1$

On substituting the value of y, we get

$\sf \: 2( {x}^{2} + 2) - x = 10x - 1$

$\sf \: 2{x}^{2} + 4 - x - 10x + 1 = 0$

$\sf \: 2{x}^{2} - 11x + 5 = 0$

$\sf \: 2{x}^{2} - 10x - x + 5 = 0$

$\sf \: 2x(x - 5) - (x - 5) = 0$

$\sf \: (x - 5)(2x - 1) = 0$

$\implies\sf \: x = 5 \: \: or \: \: x = \dfrac{1}{2} \: \{rejected \}$

On substituting the value of x in equation (1), we get

$\implies\sf \: y = 25 + 2 = 27 \: years$

Hence,

$\begin{gathered}\begin{gathered}\bf\: \implies \begin{cases} &\sf{Nisha\:present\:age = 5 \: years} \\ \\ &\sf{Asha\:present\:age = 27\: years} \end{cases}\end{gathered}\end{gathered}$