Here’s a challenge!
Suppose that a, b, c, d, e, and f are all primes, and no two of them are the same. Reduce the following ratio
to its lowest form:
−ab^3 cde/(−ab^2 ce^3 f )
An exponent of 2 after any factor means that the factor appears twice. An exponent of 3 after any factor
means that the factor appears three times. So, for example, b^2 means bb, andb^3 means bbb.
Solution
This problem is “messy” but not difficult. Let’s expand the numerator and the denominator out so we
don’t have any exponents, and let’s put small elevated dots (they can represent multiplication, remember!)
between the factors to make them easy to tell apart. Let’s write the numerator out in full, scratch a long
dashed line underneath it, and then below that, write the denominator out in full. We get
−1 · a · b · b · b · c · d · e
------------------------------------------
−1 · a · b · b · c · e · e · e · f
Now it’s easy to see which factors are duplicated in the numerator and the denominator. They are −1,a,
b (twice),c, and e.When we remove these factors from the expressions above and below the dashed line,
we get
b · d
------------------------------------------
e · e · f
This can be written as bd/e^2 f. The parentheses in the denominator are no longer needed because the minus
sign is gone. This ratio is in lowest form because both the numerator and the denominator are products of
primes, and none of the primes in the numerator and denominator are duplicates.
Multiplying and Dividing Fractions
When you want to multiply two fractions, the process is simple. Dividing one fraction by
another is a little more complicated, but not much. Let’s look at how the processes work, using
variables rather than numerical examples.
A fraction times a fraction
To multiply a fraction by another fraction, first be sure they are both “pure fractions.” That
means they should both consist of an integer divided by a positive integer. The absolute value
of the numerator doesn’t have to be smaller than the absolute value of the denominator, but
one fraction should be of the form a/b and the other should be of the form c/d, where a and c
are integers (positive, negative, or 0), and b and d are positive integers. If either fraction has a
negative denominator, you know what to do!
Multiplying and Dividing Fractions 89
