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Thread: (SAT) Proportionality

  1. #1
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    (SAT) Proportionality

    If 500 pounds of mush will feed 20 pigs for a week, for how many days will 200 pounds of mush feed 14 pigs?

    Inversely proportional.

    $\displaystyle \displaystyle 500\ \times\ \frac{20}{7} = \frac{10 000}{7} $
    $\displaystyle \displaystyle = 1428\frac{4}{7} \implies 1428\frac{4}{7}$ pig-days

    $\displaystyle \displaystyle 200\ \times\ y\ days = 1428\frac{4}{7}$
    $\displaystyle \displaystyle y = \frac{1428\frac{4}{7}}{200} = 7\frac{1}{7}$
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  2. #2
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    $\displaystyle n_{pigs} \cdot n_{days} \cdot n_{lb/(day \cdot pig)} = n_{lb}$

    Since the amount of mush eaten per pig per day is assumed to be constant then $\displaystyle 20 \cdot 7 \cdot n_{lb/(day \cdot pig)} = 500$

    $\displaystyle n_{lb/(day \cdot pig)} = \dfrac{500}{20 \cdot 7} = \dfrac{25}{7}$

    Using this info:

    $\displaystyle 14 \cdot n_{days} \cdot \dfrac{25}{7} = 200$


    Solve for n. I get 4 days which is around what we'd expect from intuition (not like 0.4, 40 and 25 which I got when trying to solve!)
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  3. #3
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    First, you need to know how much a pig will eat each day.

    If 500 lbs feeds 20 pigs for 7 days, then if we set up the ratio mush : day

    $\displaystyle \displaystyle 500 : 7$

    $\displaystyle \displaystyle \frac{500}{7} : 1$.


    So 20 pigs will eat $\displaystyle \displaystyle \frac{500}{7}$ lbs of mush per day.

    Now we can set up another ratio of mush : pigs

    $\displaystyle \displaystyle \frac{500}{7} : 20$

    $\displaystyle \displaystyle \frac{25}{7} : 1$

    $\displaystyle \displaystyle 50:14$.

    So 14 pigs will eat 50 lbs of mush per day.


    Finally, setting up the ratio mush : days

    $\displaystyle \displaystyle 50 : 1$

    $\displaystyle \displaystyle 200 : 4$.


    So 14 pigs will eat 200 lbs of mush in 4 days.
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  4. #4
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    This is a tough question for the SAT. I would do it with a ratio (direct proportion) followed by an inverse proportion as follows:

    First I'll answer this question: "If 500 pounds of mush will feed 20 pigs for a week, then 200 pounds of mush will feed how many pigs for a week?"

    $\displaystyle \rm{mush}\ \ 500 \ 200$

    $\displaystyle \rm{pigs} \ \ \ \ 20 \ \ \ x$

    So $\displaystyle \frac{500}{20}=\frac{200}{x}$

    $\displaystyle 500x=4000$

    $\displaystyle x=8$

    Now I'll answer this question: "If you can feed 8 pigs for 7 days, then for how many days can you feed 14 pigs?"

    $\displaystyle 8\cdot 7=14x$

    $\displaystyle 56=14x$

    $\displaystyle x=4$

    The problem is particularly confusing because there are three active variables here. I don't believe I've ever seen this in an actual SAT problem (if it ever did occur you could probably get it wrong and still get an 800). I have seen an additional inactive variable by which I mean the third variable wasn't needed to solve the problem (it was just there to confuse you). I'm curious what book this problem came from.
    Last edited by DrSteve; Jan 11th 2011 at 05:54 AM.
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