To find out how much money Aravinda starts with, let's denote Aravinda's initial amount of money as [tex]\( x \)[/tex].
1. Since Ravindra has three times as much money as Aravinda, his initial amount is [tex]\( 3x \)[/tex].
2. Ravindra gives a quarter of his money to Aravinda. A quarter of Ravindra's money is:
[tex]\[
\frac{3x}{4}
\][/tex]
3. After giving this amount to Aravinda:
- Ravindra's new amount of money is:
[tex]\[
3x - \frac{3x}{4}
\][/tex]
- Aravinda's new amount of money is:
[tex]\[
x + \frac{3x}{4}
\][/tex]
4. According to the question, after the transaction, Aravinda has Rs 80 less than Ravindra. So, we can set up the following equation:
[tex]\[
x + \frac{3x}{4} = \left( 3x - \frac{3x}{4} \right) - 80
\][/tex]
5. Simplify the left side of the equation:
[tex]\[
x + \frac{3x}{4} = \frac{4x}{4} + \frac{3x}{4} = \frac{7x}{4}
\][/tex]
Simplify the right side of the equation:
[tex]\[
3x - \frac{3x}{4} = \frac{12x}{4} - \frac{3x}{4} = \frac{9x}{4}
\][/tex]
and
[tex]\[
\frac{9x}{4} - 80
\][/tex]
6. Now equate both sides:
[tex]\[
\frac{7x}{4} = \frac{9x}{4} - 80
\][/tex]
7. To eliminate the fractions, multiply every term by 4:
[tex]\[
7x = 9x - 320
\][/tex]
8. Bring like terms together:
[tex]\[
7x - 9x = -320
\][/tex]
9. Simplify the left-hand side:
[tex]\[
-2x = -320
\][/tex]
10. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 160
\][/tex]
Therefore, Aravinda starts with Rs 160.
