Part One 157(c) The quotient is 9 with a remainder of 1 (out of 2). This is 9-1/2.
(d) The quotient is −3 with a remainder of −4 (out of 7). This is −3-4/7.
(e) The quotient is 3 with a remainder of 4 (out of 9). This is 3-4/9.
(f ) The quotient is −9 with a remainder of 1 (out of −11). This is −9-1/11.
Remember that the short dashes, separating the integers from the fractions, are not minus
signs! The minus signs are the longer dashes.
Question 6-2
Do the commutative and associative laws work for quotients or ratios?
Answer 6-2
Not in general. In most cases, we can’t apply the commutative or associative laws to quotients
or ratios and get valid results.
Question 6-3
What is the “brute force” method of reducing a ratio or fraction to its lowest terms?
Answer 6-3
We begin by factoring both the numerator and denominator into products of primes. If the
original numerator is negative, we attach an extra “factor” of −1, making sure all the prime
factors are positive. If the original denominator is negative, we do the same thing with it.
Next, we remove all the common prime factors from both the numerator and denominator.
Then we multiply all the factors in the numerator together, and do the same thing with the
factors in the denominator. If we end up with a negative denominator, we multiply both the
numerator and the denominator by −1.
Question 6-4
How can we use the “brute force” method to reduce 462/561 to lowest terms?
Answer 6-4
First, we factor the numerator and denominator into products of primes. This can take a little
while, but we eventually get
(2× 3 × 7 × 11)/(3 × 11 × 17)The common prime factors in the numerator and denominator are 3 and 11. When we
remove these factors, we get
(2× 7)/17Multiplying out the factors in the numerator gives us 14/17. That’s the lowest form of
462/561.