64 Addition and Subtraction
compared to the same month in the previous year. You look at the records for January
and find out that in Hoodopolis,- January 2005 averaged 5 degrees cooler than January 2004.
- January 2004 averaged 2 degrees warmer than January 2003.
- January 2003 averaged 1 degree cooler than January 2002.
- January 2002 averaged 7 degrees warmer than January 2001.
- January 2001 averaged the same temperature as January 2000.
- January 2000 averaged 6 degrees cooler than January 1999.
- January 1999 averaged 3 degrees warmer than January 1998.
What was the difference in the average temperature between January 2005 and
January 1998 in the town of Hoodopolis?- Show at least one situation where can you say that
a−b=b−a
Don’t use the trivial case where a and b are both equal to 0. - Show at least one situation where you can say that
(a−b)+c=a− (b+c)
where a, b, and c are integers. Don’t use the trivial case where a, b, andc are all 0. - Show at least one situation where you say that
(a−b)−c=a− (b−c)
where a,b, and c are integers. Don’t use the trivial case where a,b, and c are all 0. - Based on the commutative law for the sum of two integers and the associative law for
the sum of three integers, construct an S/R proof showing that for any four integers
a,b,c, and d
a+b+c+d=d+c+b+a
Here’s a hint: use the solution to the last “challenge” problem in this chapter as a
shortcut. A previously proved fact, when used to prove something new, is called a
lemma. - Based on the associative law for the sum of three integers, prove that for any four
integersa, b, c, and d
(a+b+c)+d=a+ (b+c+d)
Do this in narrative form. Don’t use the S/R table method. Here’s a hint: “zip up” the
sumb+c, and call it by another name. - Simplify and compare these expressions:
(a+b−c)+ (a−b+c)
and
a+ (b−c)+ (a−b)+c