600 Worked-Out Solutions to Exercises: Chapters 1 to 9
- See Table A-5. Follow each statement and reason closely so you’re sure how it follows
from the statements before. This proves that for any four integers a,b,c, and d,
abcd=dcba- Here are the steps in the calculation process, using parentheses where needed:
− 15 × (−45)= 675
675 / (−25)=− 27
− 27 × (−9)= 243
243 / (−81)=− 3
− 3 × (−5)= 15- This always works if c= 1. Then a can be any integer, and b can be any integer except 0.
Here’s the original equation:
(a/b)/c=a/(b/c)Table A-4. Solution to Prob. 3 in Chap. 5. As you read down the
left-hand column, each statement is equal to every statement above it.
Statements Reasons
4 + 32 / 8 × (−2)+ 20 / 5 / 2 − 8 Begin here
4 + 32 / [8 × (−2)]+ 20 / 5 / 2 − 8 Group the multiplication
4 + 32 / (−16)+ 20 / 5 / 2 − 8 Do the multiplication
4 + [32/(−16)]+ [(20/5)/2)] − 8 Group the divisions
4 + (−2)+ [(20/5)/2] − 8 Do the division 32/(−16)=− 2
4 + (−2)+ (4/2) − 8 Do the division 20/5 = 4
4 + (−2)+ 2 − 8 Do the division 4/2 = 2
4 + (−2)+ 2 + (−8) Convert the subtraction to negative
addition
− 4 Do the additions from left to rightTable A-5. Solution to Prob. 4 in Chap. 5. As you read down the left-
hand column, each statement is equal to every statement above it.
Statements Reasons
abcd Begin here
a(bcd) Group the last three integers
a(dcb) Result of the “challenge” where it was proved that
you can reverse the order of a product of three
integers
(dcb)a Commutative law for the product of a and (dcb)
dcba Ungroup the first three integers
Q.E.D. Mission accomplished