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
= number of salads,
P
= number of pizzas,
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 Problems
Simple 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?
Units:
1 commisary = (200 men)·(8 provision-months).
The question is
1 commisary
= 200 ·8 man-provision-months
= 200 ·5 man-provision-months+ (200 − x)·8 man-provision-months.
Compound Proportions
5.
If 11 men build 45 rods of wall in 6 days of 10 hours each, how many men will be required to build 81 rods of wall in 12 days of 11 hours each?
Units
:
45 rods
= 11 men ·6 days·10  hours
 day
 ·
= 11 ·6 ·10 man-hours.
The question is
81 rods
= x men ·12 days·11  hours
 day
 ·
= x ·12 ·11 man-hours.
6.
(oral exercise) If 7 men can dig 32 rods of ditch in 1 day, how many men will be required to dig 92 rods in 3/4 day.


File translated from TEX by TTH, version 3.64.
On 08 Dec 2004, 17:17.