Here’s a challenge!
Suppose you have bought a new car and you want to go for a test drive. You live on a flat plain that seems
to stretch forever in all directions. You start driving on a straight highway that runs north and south for
hundreds of miles on either side of your home town. You drive 25 miles north, then turn around and
drive 45 miles south. Then you turn around again, driving 50 miles north. Then you go 7 miles south,
12 miles north, 49 miles south, and finally 5 more miles south. How far from your home town, and it
what direction, will you finish? Solve this problem in two different ways.
Solution
First, consider driving north as “positive mileage” and driving south as “negative mileage.” Then you drive
the following distances in miles, and in this order:
25,−45, 50, −7, 12, −49,− 5
Add these all up:
25 + (−45)+ 50 + (−7)+ 12 + (−49)+ (−5)=− 19
That’s 19 miles south of your home town. Now imagine that driving south is “positive mileage” and driv-
ing north is “negative mileage.” Then you your trip is a sum of displacements like this:
− 25 + 45 + (−50)+ 7 + (−12)+ 49 + 5 = 19
Again, that’s 19 miles south.
The Associative Law for Addition
Another major rule that applies to addition involves how the addends are grouped when you
have three or more numbers. You can lump the addends together any way you want, and you’ll
always end up with the same result. The simplest case of this rule, called the associative law for
addition, involves sums of three integers.
It works when you add
Here’s how a mathematician would state the associative law for three addends. For any three
integersa, b, and c
(a+b)+c=a+ (b+c)
For instance:
(3 + 5) + 7 = 8 + 7 = 15
and
3 + (5 + 7) = 3 + 12 = 15
The Associative Law for Addition 59
