At first, Paul had some money. He spent 20% of it on some books and of the remainder on food. After that, his parents gave him $90. The ratio of the total amount of money he had in the end to the amount of money he had at first was 3 : 2. How much money did Paul have at first?
Question
At first, Paul had some money. He spent 20% of it on some books and of the remainder on food. After that, his parents gave him $90. The ratio of the total amount of money he had in the end to the amount of money he had at first was 3 : 2. How much money did Paul have at first?
Solution
Let's denote the amount of money Paul had at first as X.
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Paul spent 20% of his money on books. This means he spent 0.2X on books.
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The remainder of his money after buying books is X - 0.2X = 0.8X.
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The problem states that Paul spent half of the remainder on food. This means he spent 0.5 * 0.8X = 0.4X on food.
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After buying food, Paul has 0.8X - 0.4X = 0.4X left.
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His parents then gave him $90, so the total amount of money he has in the end is 0.4X + 90.
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The problem states that the ratio of the total amount of money he had in the end to the amount of money he had at first was 3 : 2. This means that (0.4X + 90) / X = 3 / 2.
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Solving this equation for X gives X = (2 * (0.4X + 90)) / 3.
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Simplifying the equation gives X = (0.8X + 180) / 3.
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Subtracting 0.8X from both sides gives -0.4X = 180 / 3.
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Dividing both sides by -0.4 gives X = -180 / (3 * -0.4) = $150.
So, Paul had $150 at first.
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