go 150 miles per hour under the same conditions. You can then write the ratio of the car’s top speed to
the truck’s top speed as 150/100 or 150:100. Conversely, the ratio of the truck’s top speed to the car’s top
speed is 100/150 or 100:150. In these cases, a= 50 in the above equations:
c/t= (3 × 50)/(2 × 50)
and
t/c= (2 × 50)/(3 × 50)
“Reducing” a Fraction or Ratio
Any fraction or ratio can be expressed in countless ways, but one form is considered the most
“elegant.” That’s the form in which the absolute values of the numerator and denominator are
both as small as possible, and the denominator is positive. A fraction or ratio in this form is
said to be in lowest terms or lowest form.
Negative denominators
A fraction with a negative denominator is okay in theory, but it’s hard to think about. You can
probably imagine “negative three-fifths” without much trouble, but how about “three negative
fifths”? That’s tough for almost everybody. Why bother with such ugly fractions? Both of the
fractions or ratios −3/5 and 3/(−5) have the same numerical value. Why not use the one that
makes more sense?
When you see a fraction or ratio with a negative denominator, you can multiply both the
numerator and the denominator by −1. That will turn the denominator positive while mul-
tiplying the value of the entire fraction by −1/(−1). Of course, −1/(−1) is equal to 1, which
is the multiplicative identity element. So you end up with the same number in a form that is
easier to comprehend.
Finding common factors
When a fraction is not in lowest terms, it means that the numerator and denominator can
both be divided by at least one integer, called a common factor, and no remainder will be left in
either case. For example, 6/10 is not in lowest terms. We can divide both the numerator and
denominator by 2 and get 3/5:
(6/2)/(10/2)= 3/5
If we start with −6/(−10), we can divide both the numerator and the denominator by −2 and
get 3/5 again:
[−6/(−2)]/[−10/(−2)]= 3/5
If we divide both the numerator and the denominator of a fraction by the same integer and
get integers in both places, we have the same fraction in a lower form.
“Reducing” a Fraction or Ratio 87
