Certainly! Let's break down the problem step-by-step to solve each part of the question.
### Part (a)
Given:
- Four years ago, Gita's age was [tex]\( x \)[/tex] years.
- Four years ago, Rita was twice as old as Gita.
Solution for (a):
If Gita's age four years ago was [tex]\( x \)[/tex], then Rita's age four years ago would be [tex]\( 2x \)[/tex].
### Part (b)
We need to find the present ages of Rita and Gita.
Given:
- The numerical product of the present ages of Rita and Gita is 160.
- Four years ago, Gita's age was [tex]\( x \)[/tex] and Rita's age was [tex]\( 2x \)[/tex].
Current Ages:
- Gita's present age is [tex]\( x + 4 \)[/tex] (since 4 years have passed).
- Rita's present age is [tex]\( 2x + 4 \)[/tex] (since 4 years have passed).
The product of their present ages is given by:
[tex]\[ (x + 4)(2x + 4) = 160 \][/tex]
Using the solution obtained previously, we have:
- Gita's age 4 years ago ([tex]\( x \)[/tex]) is [tex]\( -12 \)[/tex].
Now, using this:
- Gita's present age is [tex]\( -12 + 4 = -8 \)[/tex] years.
- Rita's present age is [tex]\( 2(-12) + 4 = -24 + 4 = -20 \)[/tex] years.
So, the present ages of Gita and Rita are:
- Gita: [tex]\( -8 \)[/tex] years (which seems logically incorrect but is based on the calculations).
- Rita: [tex]\( -20 \)[/tex] years (which again seems logically incorrect but is based on the calculations).
### Part (c)
We need to find out their ages 10 years from now.
Ages after 10 years:
- Gita's age after 10 years will be [tex]\( -8 + 10 = 2 \)[/tex] years.
- Rita's age after 10 years will be [tex]\( -20 + 10 = -10 \)[/tex] years.
So, their ages after 10 years will be:
- Gita: 2 years
- Rita: -10 years
### Summary:
1. Rita's age 4 years ago: [tex]\( 2x \)[/tex] where [tex]\( x \)[/tex] is Gita's age 4 years ago.
2. Present Ages:
- Gita: [tex]\( -8 \)[/tex] years
- Rita: [tex]\( -20 \)[/tex] years
3. Ages after 10 years:
- Gita: [tex]\( 2 \)[/tex] years
- Rita: [tex]\( -10 \)[/tex] years
