Algebra Know-It-ALL

88 Fractions Built of Integers

Getting the lowest form

Even after we have reduced a fraction to a lower form, we might not have it in lowest terms.

We can always get the lowest form if we are willing to go through a rather tedious process. A

mathematician might call this the “brute-force approach.” It isn’t elegant but it always works,

and we don’t need any intuition to grind it out.

We start by factoring both the numerator and denominator into products of primes. If

the original numerator is negative, we attach an extra “factor” of −1 to its product of primes,

making sure all the prime factors are positive. We do the same thing with the denominator if

it is negative. Once we have factored both the numerator and the denominator into products

of primes, we look at those products closely. If the same prime appears in both the numerator

and the denominator, then that prime is a common prime factor. We remove all the common

prime factors from both the numerator and denominator. That leaves us with a smaller prod-

uct of primes in the numerator, and a completely different product of primes in the denomi-

nator. 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.

Let’s reduce the fraction 210/(−390) to lowest terms according to this set of rules. First,

we convert both the numerator and the denominator into products of primes, and attach an

extra “factor” of −1 to the denominator. The numerator then becomes

210 = 2 × 3 × 5 × 7

and the denominator becomes

390 =− 1 × 2 × 3 × 5 × 13

Next, we use these products to build a fraction in which both the numerator and the denomi-

nator consist of prime factors, and the denominator has the extra “factor” −1:

(2× 3 × 5 × 7) / (− 1 × 2 × 3 × 5 × 13)

The common prime factors are 2, 3, and 5. We remove these from both the numerator and

the denominator, getting

7/(− 1 × 13)

That’s 7/(−13). We finish up by multiplying both the numerator and the denominator by − 1

to obtain −7/13. This is the lowest form.

You can check to see that −7/13= 210/(−390) by dividing 210 by −7 and −390 by 13.

You’ll get the same integer, −30, in either case.

Are you confused?

When you try to factor the numerator and denominator of a fraction into products of primes, you might

find that one or the other is already prime or is the negative of a prime. Maybe both are like that! This

means the original fraction is in lowest form, except when the denominator is negative. If the denominator

is negative, you simply change both the numerator and the denominator to their additive inverses.