94 Fractions Built of Integers
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not
represent the only way a problem can be figured out. If you think you can solve a particular
problem in a quicker or better way than you see there, by all means try it!- The highest wind speed in a “category 1” hurricane is 95 miles per hour. The highest
wind speed in a “category 4” hurricane is 155 miles per hour. What is the ratio of these
wind speeds, expressed in lowest terms between two integers? - On the absolute temperature scale, the coldest possible temperature is 0, defined in
units called kelvins. On this scale, pure water at sea level freezes at 273 kelvins and boils
at 373 kelvins (to the nearest kelvin). Based on this information, what is the ratio of the
absolute temperature of the boiling point to the absolute temperature of the freezing
point, expressed in lowest terms between two integers? - Is the fraction 231/230 in lowest terms? If not, reduce it to lowest terms.
- Is the fraction −154/165 in lowest terms? If not, reduce it to lowest terms.
- If two fractions are in lowest terms and they are multiplied by each other, is the product
always in lowest terms? If so, prove it. If not, provide an example. If the product is
sometimes in lowest terms but not always, provide examples of both situations. - If two fractions are in lowest terms and one is divided by the other, is the quotient
always in lowest terms? If so, prove it. If not, provide an example. If the quotient is
sometimes in lowest terms but not always, provide examples of both situations. - In one of the “challenge” problems, you found a general expression for
[(a/b)/(c/d)]/(e/f )
when a, b, c, d, e, and f are all nonzero integers. You divided a/b by c/d, and then
divided the result by e/f. Now find a general expression for
(a/b)/[(c/d)/(e/f )]
Once you’ve done this, you’ll know what happens when you divide c/d by e/f first, and
then divide a/b by the result. Compare this with the formula you got when you solved
the “challenge” problem. - Imagine that you have a fraction of the form a/b and another of the form c/d, where
a and c are integers, and b and d are positive integers. Show that the commutative
law works for multiplication of these fractions, based on your knowledge of the
commutative law for integers. - Imagine that you have fraction of the form a/b, another of the form c/d, and a third of
the form e/f, where a, c, and e are integers, and b, d, and f are positive integers. Show
that the associative law works for multiplication of these fractions, based on your
knowledge of the associative law for integers. - Give at least one example of a situation in which the following equation is true:
(a/b)/(c/d)= (c/d)/(a/b)
where a, b, c, andd are integers. Don’t use any of the trivial cases where all four of the
integers have absolute values of 1.