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COMPARE FRACTIONS

Lesson 22  Section 2

The ratio of two fractions

In the previous Section, we saw that when two fractions have the same denominator, then the larger the numerator, the larger the fraction.

2
5
 is larger than  1
5
.
But what specifically is the ratio of  2
5
 to  1
5
?
2
5
 is to  1
5
  as  2 is to 1 --  2
5
 is two times  1
5
.

In other words,

Fractions with the same denominator are in the same ratio
as their numerators.

2
5
 is to  3
5
  as  2 is to 3 --  2
5
 is two thirds of  3
5
.

We now ask:


 6.   How can we know the ratio of any two fractions?
 
  By "cross-multiplying."
  Example 1.   2
3
 is to  5
8
 
  as    2 × 8  is to  3 × 5
 
  as  16  is to  15.

Why?  Because 16 and 15 are the numerators we would get if we

  expressed  2
3
 and  5
8
 with the common denominator 24: 

as

16
24
 is to  15
24
.

The numerators, 16 and 15, have been obtained by "cross-
multiplying."

(To change  2
3
 into  16
24
,  we multiplited both 2 and 3 by 8.  To change  5
8
,

we multiplied both terms by 3.)


  Example 2.    1
4
  is to   1
2
  as which whole numbers?

Answer.  On cross-multiplying,

1
4
  is to   1
2

as

2  is to  4.

That is,

1
4
  is half of   1
2
 .
  Example 3.    4
7
  is to   5
9
 as which two whole numbers?

Answer.  On cross-multiplying,

4
7
  is to   5
9

as

36  is to  35.

What is more, since 36 is larger than 35, this tells us that

4
7
 is larger than  5
9
.

Note:  We must begin multiplying with the numerator on the left:

4 × 9.

Example 4.   What ratio has 1½ to 2?

  Answer.   First, express 1½ as the improper fraction  3
2
.  Then, treat the

whole number 2 as a numerator, and cross-multiply:

3
2
 is to 2  as  3 is to 4.

1½ is three fourths of 2.

 Equivalently, since  2 =  4
2
  (Lesson 20, Question 2), then
3
2
  is to   4
2
  as  3 is to 4.

For an application of this, see Lesson 26:  Multiplying fractions.

Example 5.   What ratio has 2½ to 3?

   Answer. 2½ is to 3

as

5
2
  is to  3

as

5  is to  6.

2½ is five sixths of 3.


More than or less than ½


 7.   How can we know whether a fraction is more than
or less than ½?
 
  If the numerator is more than half of the denominator,
  then the fraction is more than ½. While if the numerator is less than half of the denominator,
 
  the fraction is less than ½.

4
8
 is equal  to  1
2
, because 4 is half of 8.  Therefore,  5
8
 is more than  1
2
,
  because 5 is more than half of 8;  while  3
8
 is less than  1
2
, because 3 is less 

than half of 8.


  Example 6.   Which is larger,    7 
12
 or   9 
20
?
  Answer.    7 
12
.  Because 7 is more than half of 12, while 9 is less than half

of 20.


  Example 7.   Which is larger,   11
21
 or  12
25
?
  Answer.   11
21
.  Because 11 is more than half of 21 (which is 10½); while 12 is

less than half of 25 (which is 12½).  (Lesson 15, Question 7.)

We could make these comparisons for any ratio of the terms.  For example, we could know that

 5 
15
 is larger than   6 
21
.

Because 5 is a third of 15, but 6 is less than a third of 21 (which is 7).

Example 8.   Which is the largest number?

 3 
10
  5
8
  1
2
  2
7
  5
9

Answer.  First, let us examine the list to see if there are numbers less than ½ or greater than ½.  We may eliminate any numbers less than (or equal to) ½.

Thus we may eliminate    3 
10
1
2
, and  2
7
.
We are left with   5
8
 and  5
9
.

Since the numerators are the same (Section 1, Question 1), we

   conclude that the largest number is  5
8

Example 9.   Which is the largest number?

5
9
  2
5
   6 
11
  Answer.  We may eliminate  2
5
 because it is less than ½, while the others

are greater. Which is larger, then,

5
9
 or   6 
11
 ?

On cross-multiplying, we have 5 × 11 versus 9 × 6.  And

55 is greater than 54.

Therfore,

5
9
 is greater than   6 
11
.


Please "turn" the page and do some Problems.

or


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