A curious reader asked:
Pizza Plus sold a total of 580 pizzas ($6 each) and calzones ($8 each) during Sunday’s football game. How many of each item did Pizza Plus sell if the total sales were $3,734?
I’ve never been good at these things. Please help me to understand this. Thanks!!
Professor Homunculus replies:
OK, let me just say that not being a mathematician, or “official” math teacher, the answer did not just pop out to me.
So I did what I usually do when faced with problems like this, for which I don’t have a formula at hand – I looked at it empirically.
What that basically means, is I looked at the situation presented in concrete terms, with my senses.
I imagined 580 items sold. But since I didn’t know the “mix” ( how many were pizzas and how many were calzones) I started by imagining that they were all one sort. I chose the cheap one – the pizzas. (I could have chosen either – it was just a starting point. I had no plan – just wanted to see where I might go.)
From there reasoning went something like this (I guess rationalism was starting to take over):
I have 580 of the $6 pizzas. 580 x $6 = $3,480.
The total sales were $3,734, so I have to account for the missing $254 (because $3,734 – $3,480 = 254).
We know that the difference between a pizza and a calzone is $2. How many groups of $2 are there in $254? There are 127 groups. So 127 of the items must have cost $2 more than the $6 items.
Therefore, 127 of the items must have cost $8 (calzones).
If there were a total of 580 items, and 127 were $8 calzones, then the rest (580 – 127) must be $6 pizzas, making 453 $6 pizzas.
So there were:
453 $6 pizzas and 127 $8 calzones.
Let’s check it (always check – it works your brain again, and it makes sure you don’t do anything as dumb as I’d normally do):
453 pizzas at $6 each = $2,718
127 calzones at $8 each = $1,016
$2,718 + $1,016 = $3,734
Looks good to me. Let’s eat!
Of course there are easier ways. You can always figure out your own. By just studying what a book says, without thinking up your own ways, you only learn school math.
When you start empirically, you learn how to think. Imagine that!
Once you become confident of your thinking skills, you can rely more on rationalism and book knowledge, because you will understand how things work, and your natural curiosity combined with your empirical thinking skills will keep you on the right track. In other words, you’ll never just be able to do a problem because, “That is the way the teacher told us to do it.”
I use the terms rationalism and empiricism very loosely. If you want a more scholarly and historical explanation of them, check out the blog at:
http://cantseetheforest.org/2007/10/07/rationalism-and-empiricism/
There is a very lucid discussion of it there.
I hope this helped.
Brian (a.k.a. Professor Homunculus )
There are many variations on the approach you illustrate, all of which can be summarized as Guess & Check. However, that particular problem-solving strategy is somewhat of a misnomer, and in any event has come under unfair attack from the anti-reform forces (who insist upon using it to bash what you’re calling the empirical approach). The reasons they do so are the usual political ones coupled with the fact that most mathematicians see themselves as working deductively and with algorithms. Here, the “official” approach would be, as you likely know and as any basic algebra book would no doubt show, to set up some algebraic expressions that result in a simple solvable system of linear equations. For example, call the number of $6 pizzas x and the number of $8 pizzas y. Then x + y = 580. The other equation would be something like $6x + $8y = $3734. You can use substitution, elimination, graphing, or matrix methods to solve this system for x and y.
Of course, I agree that your approach is more thoughtful and creative. A variant would be to assume that there are an equal number of $6 & $8 pizzas, giving 290*6 + 290*8 = 290*14= 4060. You’re $326 too high, so by a similar argument to the one you made, you need to increase the number of $6 pizzas so as to reduce your income by $326 @ $2 per cheaper pizza. That’s 163 more $6 pies and 163 fewer $8 pies. This gives the same result as yours: 290 + 163 = 453 @ $6 and 127 @ $8.
The point is that you can start with ANY guess. But what follows is not simply “checking.” It’s a guess (perhaps an ‘educated’ one) followed by a TEST of the results, followed by THINKING, and then ADJUSTING. So this is mathematical, not just a bunch of random trial-and-error. The nay-sayers prefer to misrepresent the method as all guessing, no thinking. And unfortunately, some teachers teach it that way. More’s the pity in both cases, because as you correctly suggest, this “empirical” approach involves definite thought. And it is just as mathematically sound as the standard approach.
I am having trouble reading math questions. Here’s my question on your math problem with the pizza. Where and how did you come up with the 127? I see that you divided the 254 but how did you know to do that. It wasn’t explained or I’m just not getting it. Please let me know. Thanks. Cathy
When I first looked at this problem, before reading your procedure, my first reaction was to whip out the old 2*2 matrix and solve. This worked of course but is of course the uninteresting way to do it. A variation of the semi guess and check method you used would be the following.
Consider the limits.
Pizza’s Sold Calzones Sold Income
580 0 6*580=3480
0 580 8*580=4640
Now lets focus on one product. I like calzones so lets go with that. We know that the number of calzones sold is between 0 and 580, and we know that these numbers correlate to profits of 3480 and 4640 respectively. The key to this is that we know the actual profit was 3734. This allows us to interpolate.
Interpolating (for those who don’t know)- Is finding a third set of data using two sets of data by assuming all three sets of data are related linearly. Very usefull for finiding values between numbers given in tables and for speeding up any iterative (guess and check) problem solving process.
In this problem the increase in profit is actually linearly related to the number of calzones sold making interpolation as viable as any other math option.
#calzones = slope of linear relationship*distance along line in known value
Slope = (Max # calzones – Min # calzones)/(Max profit – Min profit)
Distance in known value = Known profit – min profit
#clazones = ((580-0)/(4640-3480))*(3734-3480) = 127
And finaly for our pizzas
#pizzas = 580 – #calzones = 453
I often have to deal with problems that ultimately require iteration and I found that once you have created two sets of data (guess and checked twice) interpolating to find your next guess is far better than a shot in the dark.
Just another way of looking at the problem.
Professor Homunculus sez:
Jonathan,
Thanks for a great description of a great solution.
I’m sure this will help many readers.
There’s actually no guessing at all the empirical method described. You start with the lowest possibility (nothing random). Then you apply the method described above.
This is not the “textbook” method, and that’s the point. Most people who have problems with math really only have problems with “textbook” math. Math is unlimited. I can’t think of any cases of any “only one way” to solve anything. Basically, textbooks teach you the “standard codes” for math. Those are helpful, and probably the best way to teach some portion of the population.
On the other hand, those codes are really just a roadmap. Very often people become dependent on roadmaps, and never visit the territory. Empiricism starts with the territory, and gives you a greater “feel” for your experiences. It keeps you more aware of “reality” (such as it is).
One of the problems of both camps of the “math wars” is their closed-mindedness to each other.
By doing that, they will always alienate some portion of the learning population. This is so unfortunate. Dependency on any one method is always a crime against the mind.
That’s why MathMojo tends to explore the empirical method as the first line of exploration. The rational method is generally the only one the traditional method explores. There’s nothing wrong with going with rationalism first. But learners should learn both methods, so the choice is THEIRS, not dictated by some over-paid administrator in a bad suit.