The Algebra Symposium: Discussion of Variables and Units
The Algebra Symposium: Discussion of Variables and Units
1.
I went to Pompeii and bought the same number of salads and
small pizzas. Salads cost two dollars each and pizzas cost six
dollars each. I spent $40 all together. Assuming that the
equation 2S +6P = 40 is correct. Then
2S +6P = 40.
Since S = P, I can write
2P +6P = 40.
So
8P = 40.
The last
equation says 8 pizzas is equal to $40 so each pizza costs $5.
What is wrong with the above reasoning? Be as detailed as
possible. How would you try to help a student who made this
mistake.
Discussion
The paradox is that the data told us that pizzas cost six dollars
each but the calculation seems to show that each pizza costs $5.
Let's examine the units of variables and constants in the equation
2S +6P = 40.
S
= numberofsalads,
P
= numberofpizzas,
2
= 2
dollars
salad
,
6
= 6
dollars
pizza
,
40
= 40 dollars.
Thus the equation reads
2
dollars
salad
S salads + 6
dollars
pizza
P pizzas = 40 dollars.
Using that S = P,
2P dollars +6P dollars
= 40 dollars,
P
= 5,
and P = 5, where the variable P represents the number of
pizzas , not the price per pizza , in
[dollars/pizza].
Old Proportion ProblemsSimple Proportion
2.
If the interest upon a sum of money for 9 months is 318.69, what
will be the interest for 11 1/2 months?
3.
If 15 men can do a piece of work in 36 days, in how many days
can they perform the same work with the assistance of 9 men more?
Units:
1 workpiece = (15 men) ·(36 days) = 15 ·36 man-days.
The question is
1 workpiece
= 15 ·36 man-days
= (24 men) ·(x days).
4.
If a garrison of 200 men has provisions for 8 months, how many men
must leave at the end of 5 months that the provisions remaining
may last the rest 8 months longer?