To determine how much money Ron had initially, we need to backtrack through his purchases.
1. Final Amount: He was left with \[tex]$15.40 after all his expenditures.
2. Socks Purchase:
- Let \( x \) be the amount of money he had before buying the socks.
- He spent \(\frac{x}{2} + 2.80\) on socks.
- After buying socks, he had \$[/tex]15.40 left, so:
[tex]\[
x - \left(\frac{x}{2} + 2.80\right) = 15.40
\][/tex]
- Simplifying the above equation:
[tex]\[
x - \frac{x}{2} - 2.80 = 15.40
\][/tex]
[tex]\[
\frac{x}{2} - 2.80 = 15.40
\][/tex]
[tex]\[
\frac{x}{2} = 15.40 + 2.80
\][/tex]
[tex]\[
\frac{x}{2} = 18.20
\][/tex]
[tex]\[
x = 2 * 18.20
\][/tex]
[tex]\[
x = 36.40
\][/tex]
- So, before buying socks, he had \[tex]$36.40.
3. Stationery Purchase:
- Let \( y \) be the amount of money he had before buying the stationery.
- He spent \(\frac{y}{2} - 12.20\) on stationery.
- After buying stationery, he had \$[/tex]36.40 left, so:
[tex]\[
y - \left(\frac{y}{2} - 12.20\right) = 36.40
\][/tex]
- Simplifying the above equation:
[tex]\[
y - \frac{y}{2} + 12.20 = 36.40
\][/tex]
[tex]\[
\frac{y}{2} + 12.20 = 36.40
\][/tex]
[tex]\[
\frac{y}{2} = 36.40 - 12.20
\][/tex]
[tex]\[
\frac{y}{2} = 24.20
\][/tex]
[tex]\[
y = 2 * 24.20
\][/tex]
[tex]\[
y = 48.40
\][/tex]
- So, before buying stationery, he had \[tex]$48.40.
4. Textbooks Purchase:
- Let \( z \) be the amount of money he had before buying the textbooks.
- He spent \(\frac{z}{3} + 8.00\) on textbooks.
- After buying textbooks, he had \$[/tex]48.40 left, so:
[tex]\[
z - \left(\frac{z}{3} + 8.00\right) = 48.40
\][/tex]
- Simplifying the above equation:
[tex]\[
z - \frac{z}{3} - 8.00 = 48.40
\][/tex]
[tex]\[
\frac{2z}{3} - 8.00 = 48.40
\][/tex]
[tex]\[
\frac{2z}{3} = 48.40 + 8.00
\][/tex]
[tex]\[
\frac{2z}{3} = 56.40
\][/tex]
[tex]\[
2z = 56.40 * 3
\][/tex]
[tex]\[
2z = 169.20
\][/tex]
[tex]\[
z = \frac{169.20}{2}
\][/tex]
[tex]\[
z = 84.60
\][/tex]
So, Ron initially had \$84.60 before making any purchases.
