Meat related Algebra Problems!
When I showed this to my students, one comment was, "There is more than one kind of bacon?"
Thursday, March 29, 2012
Monday, March 19, 2012
Word Problem of the Week: Bad Pork
One of my fundamental premises is that the themes of algebra problems bear some relationship to the life of the times when they were written. That's why this one from 1857 really caught my eye. Fraudulent practices of this type must have been fairly common in the days before the government had the strength or inclination to regulate them.
As it happens, we know what Cincinnati looked like around that time, at least near the waterfront. There are newly restored daguerreotype images of the city taken in 1848 which have exceedingly high resolution. I mean, high-end digital camera level resolution.
Some of my favorite old problems are the ones that, like this one, combine very specific detail with bizarre contrivance. In this case, the contrivance comes toward the end:
But of course, I know why he did it. He needed the equations for solving it to be of a certain type, and having the problem make a little more sense would have upset that situation.
So imagine it said that the selling price p of the pork was 12 times as much as the price c of the chemical process, each in cents per pound. Then we have that the profit, 45000 = the gain - the expenditure. The gain is simply 10000p, and the expenditure is 10000 + 10000c.
So, 45000 = 10000p - 10000c - 10000. Clear the thousands to make it 45 = 10p - 10c - 10. But since p = 12c, This becomes 45 = 120c - 10c - 10 or 55 = 110c. This makes the cost of the chemical process half a cent and the price he charges for the pork 6 cents per pound.
This is a nice little linear system problem. But setting it up the way D. H. Hill did it, you end up having to clear a denominator, making the equation quadratic and introducing an extraneous solution.
In other words, this problem sacrifices some of its plausibility to make a different, harder solution necessary.
As it happens, we know what Cincinnati looked like around that time, at least near the waterfront. There are newly restored daguerreotype images of the city taken in 1848 which have exceedingly high resolution. I mean, high-end digital camera level resolution.
Some of my favorite old problems are the ones that, like this one, combine very specific detail with bizarre contrivance. In this case, the contrivance comes toward the end:
The price at which he sold the pork per pound multiplied by the cost per pound of the chemical process was 3 cents.So, what sort of quantity is this, anyway? The unit is "dollars-squared per pound-squared" which is more than a little weird. Especially since he could have just divided the two numbers to get a dimensionless ratio.
But of course, I know why he did it. He needed the equations for solving it to be of a certain type, and having the problem make a little more sense would have upset that situation.
So imagine it said that the selling price p of the pork was 12 times as much as the price c of the chemical process, each in cents per pound. Then we have that the profit, 45000 = the gain - the expenditure. The gain is simply 10000p, and the expenditure is 10000 + 10000c.
So, 45000 = 10000p - 10000c - 10000. Clear the thousands to make it 45 = 10p - 10c - 10. But since p = 12c, This becomes 45 = 120c - 10c - 10 or 55 = 110c. This makes the cost of the chemical process half a cent and the price he charges for the pork 6 cents per pound.
This is a nice little linear system problem. But setting it up the way D. H. Hill did it, you end up having to clear a denominator, making the equation quadratic and introducing an extraneous solution.
In other words, this problem sacrifices some of its plausibility to make a different, harder solution necessary.
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