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2014 AMC 12B Problem 6

Problem 6 of 25EasierAlgebra

Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50%50\% more than the regular. After both consume 34\tfrac34 of their drinks, Ann gives Ed a third of what she has left, and 22 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?

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Solution

Let a regular lemonade hold aa ounces, so Ann’s large holds 32a.\tfrac32 a. After each drinks 34,\tfrac34, Ann has 1432a=38a\tfrac14\cdot\tfrac32 a = \tfrac38 a left, and she gives Ed 1338a+2=18a+2\tfrac13\cdot\tfrac38 a + 2 = \tfrac18 a + 2 ounces. Ed drinks his full aa ounces plus that gift, and Ann drinks her 32a\tfrac32 a minus the gift. Setting these equal, a+(18a+2)=32a(18a+2), a + \left(\tfrac18 a + 2\right) = \tfrac32 a - \left(\tfrac18 a + 2\right), which gives 4=14a,4 = \tfrac14 a, so a=16.a = 16. Then Ed drank 1616 ounces and Ann drank 2424 ounces, for a total of 4040 ounces. Thus, the correct answer is D.

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Concepts: fraction · linear equation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.