To solve the problem regarding Asha and Nisha's ages, we can set up a couple of equations based on the information provided. Let's break it down step by step.
Defining Variables
First, we need to define the ages of Asha and Nisha. Let's denote:
- A = Asha's current age
- N = Nisha's current age
Setting Up the Equations
According to the problem, we have two key pieces of information:
- Asha’s age is 2 more than the square of Nisha’s age.
- When Nisha reaches Asha’s current age, Asha will be one year less than 10 times Nisha’s current age.
From the first piece of information, we can write the equation:
A = N² + 2
For the second piece of information, we need to determine how many years it will take for Nisha to reach Asha's current age. This will be:
A - N years.
At that time, Asha’s age will be:
A + (A - N) = 2A - N
According to the problem, this age will be one year less than 10 times Nisha's current age:
2A - N = 10N - 1
Solving the Equations
Now we have a system of two equations:
- 1. A = N² + 2
- 2. 2A - N = 10N - 1
Let's substitute the first equation into the second equation:
2(N² + 2) - N = 10N - 1
Expanding this gives:
2N² + 4 - N = 10N - 1
Rearranging the equation leads to:
2N² - 11N + 5 = 0
Applying the Quadratic Formula
This is a quadratic equation in standard form, which we can solve using the quadratic formula:
N = (-b ± √(b² - 4ac)) / 2a
Here, a = 2, b = -11, and c = 5.
Calculating the discriminant:
b² - 4ac = (-11)² - 4(2)(5) = 121 - 40 = 81
Now substituting back into the formula:
N = (11 ± √81) / 4 = (11 ± 9) / 4
This gives us two potential solutions for N:
- N = (20) / 4 = 5
- N = (2) / 4 = 0.5 (not a valid age)
Finding Asha's Age
Since Nisha's age must be a whole number, we take N = 5. Now we can find Asha's age:
A = N² + 2 = 5² + 2 = 25 + 2 = 27
Calculating the Sum of Their Ages
Now that we have both ages:
- Asha's age (A) = 27 years
- Nisha's age (N) = 5 years
The sum of their present ages is:
27 + 5 = 32
Final Answer
Thus, the sum of Asha and Nisha's present ages is 32 years.
