ARITHMETIC OF FRACTIONS
Objectives: To recall how fractions are added or multiplied to other fractions or real numbers.
Alternative Reading: Section 1.2 of Beginning Algebra, by K. Elayn Martin-Gay. Available on reserve in the library.
Recall 1: Basic definitions
A natural number is a number from the set
1, 2, 3, 4, 5, 6, 7, 8, 9, . . .
where the three dots . . . on the right mean that these numbers continue forever.
A whole number is a natural number or 0.
An integer is a number from the set
. . . -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6,. . .
A fractional number is a number of the form m/n , where both m and n are whole numbers, and n is different from 0.
So, 1/2, 3/5, 2/13, 4/1 and 0/1 are examples of frational numbers, but 1/0 and (-1)/2 are not.
A rational number is a number of the form m/n, where both m and n are integers, and n is different from 0.
So, 9/(-5), (-1)/2, 2/13 and 0/1 are examples of rational numbers, but 1/0 is not.
You may have forgotten the basic rules for multiplying and adding fractions. Here are some general rules.
1.1 , if a, b, c, d are integers, with b and d different form 0.
if a, b, c, d are integers, with b and d
different form 0.
It is easy to convince yourself that
the two rules above are
valid. Take a few sample numbers -- let a/b=1/2
and c/d=3/4. The question
is whether or not the equalities
hold. The validity of these equations does not prove the general equations 1.1 and 1.2, but it does make 1.1 and 1.2 more believable. How can we test the validity of these equalities?
Notice that 3/4 may be represented by the picture
If the whole rectangle is divided into 4 equal pieces, then three of those (lightly shaded) pieces represent 3/4 of the whole rectangle. Multiplying 3/4 by 1/2 is the same thing as dividing by 2. Pictorially, this is represented by dividing the lightly shaded pieces in two. The darkly shaded areas therefore represent 1/2 of 3/4.
This picture has 8 parts. Six of them represent 3/4 of the whole; three of them represent 1/2 of 3/4 of the whole. This means that 1/2 of 3/4 equals 3/8.
Now we can see that the equality 1.1 is valid.
A similar argument would convince you of the validity of 1.2.
Recall 2: If a/b is a fractional or rational number, then b (the lower number) is called the denominator and a (the upper number) is called the numerator.
If we are adding two fractional or rational numbers such as a/b and c/b, where the denominators are equal, then the result is
a/b+c/b =(a+c)/b .
In other words -- in this case -- we add the numerators and keep the denominator the same.
Keep in mind that adding the numerators is valid only in the special case where the denominators are equal.
(a) Do you recall that adding two numbers, a and b is the same as adding b and a? In other words, a + b = b + a. It is the same with multiplying a and b. In other words axb = bxa. The laws that permit you to flip a and b are called the commutative laws.
(b) Do you recall that adding two numbers, a and b and then a third number c is the same as adding a to the sum of b and c? In other words, (a + b) + c = a + (b + c). The laws that permit you to move the parentheses around are called the associative laws.
(c) Do you recall that multiplying a number a by the sum of two other numbers, b and c is the same as adding axb and axc? In other words, ax(b + c) = axb + axc. The law that permits you to combine the multiplication and addition in this way is called the distributive law.
Recall 4 When the
and denominator of a rational number are equal, then
1. For example, the following rational numbers are all
equivalent to the
1/1, (-1)/(-1), 17/17, x/x (x different from 0).
Recall 6 An expression is a symbol, or sequence of symbols, that represents a quantity. For example, the following are expressions:
An expression of the form of a single fraction is called a rational expression. For example,
x^2/y^2, (x+1)y/2y and (x+1)/2
are rational expressions.
7 Do you recall how to
divide two fractional expressions? Suppose that
you are faced with the
Do you know how to evaluate such an
The general rule is that if you are faced with a rational expression
multiply the numerator by
the flip or inverse of the denominator; that is
(a/b)*(d/c) . Therefore
Certain ambiguities may arise through incorrect use of notation, as in the following case:
Does this rational number stand for 1/4 of 2/3, or 2 divided by 3/4? It is not at all clear. The ambiguity is the fault of the notation. If it stands for 1/4 of 2/3, then it should be written as
Notice the difference between the long and short lines.
rational numbers 9/18, 3/6, 1/2
are all equivalent representations of 1/2.
We say that 1/2 is the reduced
form of the rational numbers of the form
(1*a)/(2*a) , where a stands
for any integer. In Recall 7 we
equals 18/28. The reduced form of 18/28 is 9/14.
Example 1. Simplify (x/y)(x/y)
Solution: From Rule 1 we know that the expression may be written as (xx)/(yy). We often denote xx as x^2; so this last expression may be written as x^2/y^2.
Example 2. Simplify [(x+1)/y](y/2).
Solution: From Rule 1, we know
If we now use the commutative law
3 a), we see that the expression on the right may be
Using Rule 1, once again, we may write
this last expression as
(y/y)(x+1)/2. By Recall 4, we know that
(y/y)=1. So we have
1(x+1)/2. By Recall 5, we may write this last
Example 3. Write 3/y+x/4 as a rational expression.
Solution: The rational number 3/y is equivalent to (4*3)/(4*y) and the rational number x/4 is equivalent to (x*y)/(4*y).
(4*3)/(4*y) and (x*y)/(4*y) have
the same denominator; so we may add the
numerators and keep the denominator
4*y. We have
Example 4. Write (x/y)[3/y+x/4] as a rational expression.
Solution: From Example 3, we know that 3/y+x/4 may be represented in its equivalent form as (12+xy)/4y. Therefore, we may start with (x/y)(12+xy)/4y. Using Rule 1, together with Recall 4, we may write this last expression as (12x+x^2*y)/(4*y^2), which is the desired rational expression.
1) 0/1= 0 is not considered to be a rational number.
2) 0/0 is meaningless for now. (More about this later.)
Write each expression as a fractional expression
5 (a) (2/3)/4 (b)2/(3/4)