AlgebraDifficulty 5.6AIME, harderFind the answerItaly
Problem:
Alberto, Barbara and Ciro get together one day to prepare ravioli for a charity dinner in support of the mathematical olympiads. As a first thing they decide to split the working hours equally between the morning and the afternoon, and of course they work simultaneously and for the same amount of time. Alberto is very reliable: he prepares 90 ravioli per hour for the whole working day. Barbara makes 110 ravioli per hour during the morning, but in the afternoon she is more distracted and prepares 70 ravioli per hour. Ciro makes 2/3 of his ravioli at a rate of 140 ravioli per hour and the last third at only 50 ravioli per hour. Who made the most ravioli by the end of the day?
Pick one
Solution
Solution:
The answer is (D). Let h be the number of working hours during the morning (and hence also during the afternoon). We know that Alberto prepares 2h⋅90=180h ravioli, while Barbara prepares h⋅110+h⋅70=180h of them. Let now t1 be the time Ciro takes to prepare the first 2/3 of his ravioli, and t2 the time he spends preparing the last third. If we call r the total number of ravioli prepared by Ciro we then have ⎩⎨⎧t1+t2=2ht1⋅140=32rt2⋅50=3r We can then write t1=3⋅1402r, t2=3⋅50r, and substituting into the first equation we find r(2101+1501)=2h, from which r=175h.
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Source: MathNet,
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