Let's solve this problem step-by-step.
1. Define Variables:
- Let [tex]\( x \)[/tex] be the initial amount of money Ron had.
- Ron's final amount of money is \[tex]$15.40.
2. First Purchase - Textbooks:
- Ron spent \( \frac{1}{3}x + 8 \) on textbooks.
- Remaining money after buying textbooks:
\[
x - \left( \frac{1}{3}x + 8 \right) = \frac{2}{3}x - 8
\]
3. Second Purchase - Stationery:
- Ron spent \( \frac{1}{2} \) of his remaining money minus \$[/tex]12.20 on stationery.
- Remaining money after buying textbooks is [tex]\( \frac{2}{3}x - 8 \)[/tex].
- Amount spent on stationery is [tex]\( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) - 12.20 \)[/tex].
- Remaining money after buying stationery:
[tex]\[
\left( \frac{2}{3}x - 8 \right) - \left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) - 12.20 \right)
\][/tex]
- Simplifying this:
[tex]\[
\frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 = \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20
\][/tex]
4. Third Purchase - Socks:
- Ron spent [tex]\( \frac{1}{2} \)[/tex] of the remaining money plus \[tex]$2.80 on socks.
- Remaining money after buying stationery is:
\[
\frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20
\]
- Amount spent on socks is \( \frac{1}{2} \left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 \right) + 2.80 \).
- So, the remaining money after buying socks:
\[
\left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 \right) - \left( \frac{1}{2} \left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 \right) + 2.80 \right)
\]
- Simplifying this:
\[
\frac{1}{2} \left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 \right) - 2.80
\]
Given Ron ended with \$[/tex]15.40, we can set up the equation:
[tex]\[
\frac{1}{2} \left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 \right) - 2.80 = 15.40
\][/tex]
Now let's simplify:
1. Remaining Money After Buying Socks:
[tex]\[
\frac{1}{2} \left( \frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 \right) - 2.80
\][/tex]
Given the result:
[tex]\[
\frac{1}{2} \left( \frac{2}{3}x - 8 \right) + 12.20 = 0.333333333333333x + 8.20
\][/tex]
Thus, equating the simplified amount with the final amount, we get:
[tex]\[
0.166666666666667x + 1.3 = 15.40
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
0.166666666666667x = 15.40 - 1.3
\][/tex]
[tex]\[
0.166666666666667x = 14.10
\][/tex]
[tex]\[
x = \frac{14.10}{0.166666666666667}
\][/tex]
[tex]\[
x = 84.60
\][/tex]
Ron initially had $84.60.
