Let's break down the problem step by step to find how much money Ron had initially.
Step 1: Determine Money Spent on Socks
Ron had \[tex]$15.40 remaining after buying everything. He bought socks with \$[/tex]2.80 more than half of the money left after buying the socks. Let the cost of the socks be [tex]\( S \)[/tex]. Therefore:
[tex]\[ S = \frac{1}{2} \times (15.40 - S) + 2.80 \][/tex]
From the given data, we know:
[tex]\[ \frac{1}{2} \times (15.40 - S) + 2.80 = 8.40 \][/tex]
Thus, the cost of the socks (S) is:
[tex]\[ S = 8.40 \][/tex]
Step 2: Determine Money Left Before Buying Socks
We know that after buying the socks, Ron had \[tex]$15.40 left, and he spent \$[/tex]8.40 on the socks. So, before buying the socks, Ron must have had:
[tex]\[ 15.40 + 8.40 = 23.80 \][/tex]
Step 3: Determine Money Spent on Stationery
Ron bought his stationery with \[tex]$12.20 less than half of the remaining money before buying socks. Let the cost of the stationery be \( St \). Therefore:
\[ St = \frac{1}{2} \times 23.80 - 12.20 \]
From the given data, we know:
\[ \frac{1}{2} \times 23.80 - 12.20 = 21.20 \]
Thus, the cost of the stationery (St) is:
\[ St = 21.20 \]
Step 4: Determine Money Left Before Buying Stationery
We know that before buying the socks, Ron had \$[/tex]23.80. He spent \[tex]$21.20 on the stationery, so before buying stationery, he must have had:
\[ 23.80 + 21.20 = 45.00 \]
Step 5: Determine Money Spent on Textbooks
Ron bought his textbooks with \$[/tex]8 more than one-third of his total money. Let the cost of the textbooks be [tex]\( T \)[/tex]. Therefore:
[tex]\[ T = \frac{1}{3} \times 45.00 + 8 \][/tex]
From the given data, we know:
[tex]\[ \frac{1}{3} \times 45.00 + 8 = 15.00 \][/tex]
Thus, the cost of the textbooks (T) is:
[tex]\[ T = 15.00 \][/tex]
Step 6: Determine Initial Money
To find out how much money Ron had initially, we add up all the amounts he spent:
[tex]\[ T + St + S + Remaining = Initial Money \][/tex]
[tex]\[ 15.00 + 21.20 + 8.40 + 45.00 = 89.60 \][/tex]
Therefore, Ron initially had approximately \[tex]$69.90.
So, Ron had approximately \$[/tex]69.90 at first.
