Algebra Know-It-ALL

Suppose you want to describe how many people in Country X have green eyes, per 100,000 popula-
tion. You might discover that one out of every 10 people has green eyes. You could say that 1/10 of the
people have green eyes, or the ratio of people with green eyes to all the people in Country X is 1:10. But
if you use 100,000 population as a basis, you’ll have to say that the proportion of people with green eyes
is 10,000 per 100,000 population in Country X. That’s not even close to the lowest form, but it conveys
the intended meaning better.
In pure theory, it doesn’t matter if a fraction, ratio, or proportion is in lowest terms or not. These days,
nearly everyone uses computers in complicated calculations, and the machines don’t care about lowest terms.
You can input the numbers as they are, and the computer will output the data in any form you want.

Here’s a challenge!

Start with the general equation for adding two fractions:

a/b+c/d= (ad+bc)/(bd)

Based on this, and on the rules you already know, prove that

a/b−c/d= (ad−bc)/(bd)

as long as b≠ 0 and d≠ 0.

Solution

Table 6-2 shows how the subtraction formula is derived from the addition formula. Near the end of this
proof parentheses and brackets aren’t enough for grouping of an expression, so braces must be used! They
look exactly the same as they braces you use to enclose lists of symbols representing set elements, but the
purpose is different.

Table 6-2. Derivation of a general formula for the subtraction of one fraction

from another, based on the formula for addition of fractions. As you read down

the left-hand column, each statement is equal to all the statements above it.

Statements Reasons

a/b − c/d Begin here

a/b + [−(c/d)] Convert subtraction to addition of a negative

a/b + [−1(c/d)] Principle of the sign-changing element

a/b + (−1/1)(c/d) “Divisive identity element”: substitute −1/1 for −1

a/b + (−1)c/1d Multiplication of fractions to right of plus sign

a/b + (−1)c/d Multiplicative identity element: substitite d for 1d

[ad + b(−1)c]/bd Formula for addition of two fractions, considering (−1)c as a single quantity

[ad + (−1)bc]/bd Commutative law for multiplication

[ad + (−1)(bc)]/bd Group elements b and c with parentheses

{ad + [−(bc)]}/bd Principle of sign-changing element (the other way around)

(ad − bc)/bd Convert addition of a negative to subtraction

Q.E.D. Mission accomplished

Adding and Subtracting Fractions 93