Maths Olympiad Prep

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Algebra Difficulty 5.6 AIME, harder Find the answer Italy

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 2/32 / 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 hh be the number of working hours during the morning (and hence also during the afternoon). We know that Alberto prepares 2h90=180h2 h \cdot 90 = 180 h ravioli, while Barbara prepares h110+h70=180hh \cdot 110 + h \cdot 70 = 180 h of them. Let now t1t_1 be the time Ciro takes to prepare the first 2/32 / 3 of his ravioli, and t2t_2 the time he spends preparing the last third. If we call rr the total number of ravioli prepared by Ciro we then have
{t1+t2=2ht1140=2r3t250=r3 \left\{ \begin{array}{l} t_1 + t_2 = 2 h \\ t_1 \cdot 140 = \frac{2 r}{3} \\ t_2 \cdot 50 = \frac{r}{3} \end{array} \right.
We can then write t1=2r3140t_1 = \frac{2 r}{3 \cdot 140}, t2=r350t_2 = \frac{r}{3 \cdot 50}, and substituting into the first equation we find r(1210+1150)=2hr \left( \frac{1}{210} + \frac{1}{150} \right) = 2 h, from which r=175hr = 175 h.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.