Fractions of Whole Numbers
Equivalent or Equal Fractions
Equivalent fractions are fractions that have the same value or represent
the same part of an object. If a pie is cut into two pieces , each piece is also
one-half of the pie. If a pie is cut into 4 pieces, then two pieces represent the
same amount of pie that 1/2 did. We say that 1/2 is equivalent or equal
to 2/4 and we write
The Fundamental Law of Fractions describes the general relationship be-
tween equivalent fractions.
The Fundamental Law of Fractions
For any fraction a/b and any nonzero whole number c we have
Show that the fraction 6/14 is equivalent to 9/21 .
The following theorem shows that two fractions are
equivalent if and only if
their cross-products are equal.
if and only if ad = bc.
Suppose first that Since we must have
But this is true only when ad = bc.
Conversely, if ad = bc then By the Fundamental Law of Fractions
Find a value for x so that
By the above theorem we must have 42*x = 210*12.
But 210*12 = 60*42
When a fraction is replaced with we say that has been simplified.
We say that a fraction is in simplest form (or lowest terms ) if a and b
have no common divisor greater than 1. For example, the fraction
We write a fraction in simplest form by dividing both a and b by the
Find the simplest form of each of the following fractions.
(a) First, we find GCF(240,72). Since and we nd
GCF(240, 72) = = 24.
(b) Since 399 = and
483 = we have GCF(399, 483) = 3*7 = 21.
Simplify the fraction 54/72 .
Since GCF(54, 72) = 18 then
Use drawings of fractions strips to show that 3/4 , 6/8 , and 9/12 are equivalent.
Write each fraction in simplest form.
Two companies conduct surveys asking people if they favor stronger controls
on air pollution. The first company asks 1,500 people, and the second asks
2,000 people. In the first group, 1,200 say yes. Make up results for the second
group that would be considered equivalent.
Find four different fractions equivalent to 4/9 .
Fill in the missing number to make the fractions equivalent.
Rewrite the following fractions in simplest form.
Find the prime factorizations of the numerators and denominators of these
fractions and use them to express the fractions in simplest form.
Determine if each of the following is correct.
If what must be true?
Solve for x.
Rewrite as a mixed number in simplest form.
We say that a/b is less than c/b , and we write
if and only if a < c.
The above definition compares fractions with the same denominator. What
about fractions with unlike denominators ? To compare fractions with unlike
denominators, say with we compare the fractions
since By the above definition, it follows that if and
only if ad < bc.
This establishes a proof of the following theorem.
If a, b, c, d are whole numbers with b ≠ 0, d ≠ 0 then if and only if
ad < bc.
Compare the fractions 7/8 and 9/11 .
Since we have
We conclude this section with the following question.
Given two fractions
Is there a fraction between these two fractions?
The answer is a rmative according to the following theorem.
Theorem 18.3 (Density Property)
Since by Theorem 18.2 we have ad < bc.
Add cd to both sides to
obtain ad + cd < bc + cd.
That is, (a + c)d < (b + d)c or
Similarly, if we add ab to both sides of ad < bc we find ad + ab < bc + ab or
a(b + d) < b(a + c).
Find a fraction between the fractions 9/13 and 12/17 .
Since 9*17 < 12*13 we have By the previous theorem we have
Compare the pairs of fractions.
You have two different recipes for making orange juice from concentrate. The
first says to mix 2 cups of concentrate with 6 cups of water. The second says
to mix 3 cups of concentrate with 8 cups of water. Which recipe will have a
stronger orange flavor?
A third grader says that 1/4 is less than 1/5 because 4 is less than 5. What would
you tell the child?
Find a fraction between 3/4 and 7/8 .
Order the following fractions from least to greatest .
If , compare the fractions
Find a fraction between