2013 AMC 10A Problems/Problem 5

Problem

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?

Solution 1

The total amount paid is . To get how much each should have paid, we do .

Thus, we know that Tom needs to give Sammy 30 dollars, and Dorothy 10 dollars. This means that .

Solution 2

The difference in the money that Tom paid and Dorothy paid is . In order for them both to have paid the same amount, Tom must pay more than Dorothy. The answer is .

Solution 3

The meaning of sharing costs equally is meaning that, after the vacation, they are equally dividing the money in a way such that, each person would have the same amount left. As each person spends an amount of money, greater than 100, let it be that they all had dollars to spend. This means that after the vacation we want the amount of money, they currently have. After the trip, Tom would've dollars, Dorothy would've dollars, and Sammy had dollars. This gives us a total of dollars.

We want to equally split this money, as that is what happens after splitting the cost equally. This means that we want Dorothy, Tom, and Sammy to each have dollars. We know that Tom gave Sammy dollars meaning that we want to split this money first. As Tom gives money to no one else, we want him to reach dollars in this trade, meaning that as Tom has dollars and Sammy has dollars, we can do a trade of so . After this trade, we get that Tom has dollars, Sammy has dollars, and Dorothy has dollars.

Next trade is where Dorothy gives dollars to Sammy. Dorothy has dollars and Sammy has dollars. As both of these don't have dollars and this is the last trade, we need to make sure both have dollars at the end. This is possible if

We want to find

See Also

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