Algebra Know-It-ALL

Question 4-10

Suppose a,b, and c are integers and we see the expression a−b+c without any parentheses
in it. Can we use parentheses for grouping, apply the associative law, and expect valid results
in this situation?

Answer 4-10

No! Here’s an example:

(5− 10) + 15 =− 5 + 15 = 10

but

5 − (10 + 15) = 5 − 25 =− 20

Chapter 5

Question 5-1

When we multiply a positive integer c by another positive integer d, it’s the equivalent of start-
ing with c and then adding c repeatedly a certain number of times. How many times? Give
an example.

Answer 5-1

The product cd is equivalent to starting with c and then adding c a total of (d− 1) times. For
example, we get 5 × 12 when we start with 5 and then add 5 over and over, a total of 11 times.
Another way to look at this is to imagine that 5 × 12 is what we get when we start with 0 and
then add 5 repeatedly, a total of 12 times.

Question 5-2

What happens if we multiply a positive integer p by a negative integer n? How can we describe
that in terms of repeated addition?

Answer 5-2

It works the same way as it does when adding a positive integer. The only difference is that, as
we keep adding the negative integer repeatedly, the sum gets smaller instead of larger.

Question 5-3

In which of the following quotients is there a remainder?

(a) 20/10 (b) 33/11 (c) 51/17 (d) 95/19

(e) 105/21 (f ) 116/29 (g) 218/31 (h) 301/43

Answer 5-3

There is a remainder only in case (g). It’s easy to see this by using a calculator to divide the
numerator by the denominator in each of the expressions.

Part One 153