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Lesson 22

EQUIVALENT FRACTIONS



 1.   What are equivalent fractions?
 
  Equivalent fractions have different names but are at the same place on the number line. They have the same relationship to 1.
 

Here is an elementary example:

1
2
 =  2
4
.

1
2
 is 1 of 2 equal parts of 1.   2
4
 is 2 of 4 equal parts.  Each fraction is

one half of 1.

As fractions of a unit of measure, equivalent fractions are equal measurements.

1
2
 inch =  2
4
 inch.

 2.   How can we know when two fractions are equivalent?
 
 
  Both the numerator and denominator of one fraction have been multiplied by the same number, and those products are the numerator and denominator of the other fraction.
 
2
3
 and  10
15
 are equivalent fractions.

(Compare the theorem of the same multiple; for, a numerator has a ratio to the denominator.  Every property of ratios applies to fractions. From the viewpoint of comparing fractions, however, see Problem 2 of that lesson.)

  Example 1.   Name three fractions that are equivalent to  5
6
.

Answer.  For example,

10
12
  15
18
  50
60

To create them, we multiplied both 5 and 6 by the same number.  First by 2, then by 3, then by 10.


 3.   How do we convert a fraction to an equivalent fraction with a larger denominator?
 
 
  Multiply the original denominator so that it equals the larger denominator. Multiply the numerator by the same number.
 
3
4
 =  6 × 3
6 × 4
 =  18
24
.

(Compare Lesson 20, Problem 2c.)

Example 2.    Write the missing numerator:

6
7
=  ? 
28

Answer.   To make 7 into 28, we have to multiply it by 4.  Therefore, we must also multiply 6 by 4:

6
7
 =  24 4 × 6
28 4 × 7

In practice, to find the multiplier, mentally divide the original denominator into the new denominator, and then multiply the numerator by that quotient.  That is, say:

"7 goes into 28 four times. Four times 6 is 24."

The student who has studied ratio and proportion
will recognize this as the theorem of the same multiple (Lesson 18, Question 3). In fact, everything we know about ratios carries over into fractions; for, since numerators and denominators are natural numbers, each numerator has a ratio to its denominator.

Example 3.   Write the missing numerator:

5
8
=  ? 
48

Answer.   "8 goes into 48 six times.  Six times 5 is 30."

5
8
 =  30
48
.

In actual problems, we convert two (or more) fractions so that they have equal denominators.  When we do that, it is easy to compare them (see the next Lesson, Question 3), and equal denominators are necessary in order to add or subtract them (Lesson 25).  For we can only add or subtract quantities that have the same name, that is, that are units of the same kind; and it is the denominator of a fraction that names the unit. (Lesson 21.)

Now, since 15, for example, is a multiple of 5, we say that 5 is a divisor of 15.  (5 is not a divisor of 14, because 14 is not a multiple of 5.)

5 is also a divisor of 20.  5 is a common divisor of 15 and 20.

(15 and 14 have no common divisors, except 1, which is a divisor of every number.)

 

 4.   What denominator should we choose
when the denominators of two fractions have no common divisors,
2
3
  and   4
5
  and we want to convert them to equivalent fractions with equal denominators?
 
  Choose the product of the denominators.
 
  Example 4.   Convert   2
3
  and   4
5
  to equivalent fractions with equal

denominators.

Answer.   The denominators 3 and 5 have no common divisors (except 1).  Therefore, as a common denominator, choose 15.

2
3
 =  10
15
,    4
5
 =  12
15
.
To convert  2
3
 , we said, "3 goes into 15 five times. Five times 2 is 10."
To convert  4
5
 , we said, "5 goes into 15 three times. Three times 4 is 12."

Once we convert to a common denominator, we could then know

  that  4
5
 is greater than  2
3
.  Because when fractions have equal

denominators, then the larger the numerator, the larger the fraction. (Lesson 20, Question 11.)

Also, we could now add those fractions:

2
3
 +  4
5
 =  10
15
 +  12
15
 =  22
15
.

See Lesson 21, Example 3.

We can choose the product of denominators even when the denominators have a common divisor. But their product will not then be their lowest common multiple (Lesson 23). The student should prefer the lowest common multiple, because smaller numbers make for simpler calculations.

Same ratio

When fractions are equivalent, their numerators and denominators are in the same ratio.  That in fact is the best definition of equivalent fractions.

1
2
 =  2
4
.

1 is half of 2.  2 is half of 4.  In fact, any fraction where the numerator

is half of the denominator will be equivalent to  1
2
.
1
2
= 2
4
= 3
6
=  5 
10
=  8 
16
.

1 is half of 2.  2 is half of 4.  3 is half of 6.  5 is half of 10.  And so on.  These are all at the same place on the number line.

  Example 5.    4 
12
 and   5 
15
 are equivalent, because each numerator is a third

of its denominator.

Example 6.   Write the missing numerator:

 7 
28
=  ? 
16

Answer.  7 is a quarter of 28.  And a quarter of 16 is 4.

 7 
28
 =   4 
16
.

 7 is to 28  as  4 is to 16.

 7 
28
 and   4 
16
 are equivalent.

How to simplify, or reduce, a fraction

The numerator and denominator of a fraction are called its terms.  To simplify or reduce a fraction means to make the terms smaller.  To accomplish that, we divide both terms by a common divisor.

  Example 7.    24
36
 =  24 ÷ 4
36 ÷ 4
 =  6
9
 =  6 ÷ 3
9 ÷ 3
 =  2
3
.
24
36
 ,   6
9
 , and   2
3
  are equivalent fractions.  Of those three,
    2
3
 has the lowest terms -- we cannot divide any further.  We like to express

a fraction with its lowest terms, because it gives a better sense of its value, and it makes for simpler calculations.


 5.   How do we reduce a fraction to its lowest terms?
 
  Keep dividing both terms by a common divisor.
Or, take the same part of both terms.
 

  Example 8.   Reduce to lowest terms:   15
21
.

Answer.  15 and 21 have a common divisor, 3.

15
21
= 5 "3 goes into 15 five times."
7 "3 goes into 21 seven times."

Or, take a third of both 15 and 21.

  Example 9.   Reduce     200 
1200
.

Answer.  When the terms have the same number of 0's, we may ignore them.

  200
1200
 =  1
6

Effectively, we have divided 200 and 1200 by 100. (Lesson 2, Question 10.)


  Example 10.   Write as a mixed number:   20
 8 
.

Solution.  Divide 20 by 8.  "8 goes into 20 two (2) times (16) with 4 left over."

20
 8
  =  2 4
8
  =  2 1
2
.
4
8
  is equivalent to   1
2
.

Or, we could reduce first.  20 and 8 have a common divisor 4:

20
 8 
 =  5 "4 goes into 20 five times."
2 "4 goes into 8 two times."
 
   =  2½.

Notice that we are free to interpret the same symbol

20
 8
  in various ways.  It is the fraction  20
 8
.  It is 20 divided by 8.  And it signifies

"the ratio of 20 to 8."

  Example 11.   Reduce   5
5
.
  Answer.     5
5
 = 1.

Any fraction in which the numerator and denominator are equal, is equal to 1.

 
Summary
 

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section.


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