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Ron was getting some items for the new school year. He first bought some textbooks with \[tex]$8 more than \(\frac{1}{3}\) of his money. Then, he bought his stationery with \$[/tex]12.20 less than [tex]\(\frac{1}{2}\)[/tex] of his remaining money. Lastly, he bought some socks with \[tex]$2.80 more than \(\frac{1}{2}\) of the money left. He had \$[/tex]15.40 with him. How much money did he have at first?

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GinnyAnsweridn
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.
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