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COMPARING FRACTIONS

Lesson 23  Section 2

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The ratio of two fractions

To know the ratio of fractions, is to compare them. We are about to see:

Fractions have the same ratio to one another as natural numbers.

If you knew that

2
3
 is to  5
8
  as   16 is to 15,
  then since 16 is larger than 15, you would know that   2
3
 is larger than  5
8
.

Now we saw in Lesson 20 that when two fractions have equal denominators, 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 equal denominators are in the same ratio
as their numerators

2
5
 is to  3
5
  as  2 is to 3.

When fractions do not have equal denominators, then we can know their ratio -- we can compare them -- by cross-multiplying.  Because that gives the numerators if we had expressed them with equal denominators.


 4.   How can we compare fractions by cross-multiplying?
 
  Cross-multiply and compare the numerators.
 
  Example 1.   2
3
 is to  5
8
 
  as    2 × 8  is to  3 × 5
 
  as  16  is to  15.

  16 and 15 are the numerators we would get if we expressed  2
3
 and  5
8

with the common denominator 24.

cross multiplying

as

16
24
 is to  15
24
.
And since 16 is larger than 15, we would know that  2
3
 is larger than  5
8
.
  Example 2.   Which is larger,   4
7
  or   5
9
 ?

Answer.  On cross-multiplying,

4
7
  is to   5
9

as

36  is to  35.

Now, 36 is larger than 35.  Therefore,

4
7
 is larger than  5
9
.

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

4 × 9.

  Example 3.    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 4.   What ratio has 2½ to 3?

  Answer.   First, express 2½ as the improper fraction  5
2
.  Then, treat the

whole number 3 as a numerator, and cross-multiply:

5
2
 is to 3  as  5 is to 6.

2½ is five sixths of 6.

 Equivalently, since  3 =  6
2
  (Lesson 21, Question 2), then
5
2
  is to   6
2
  as  5 is to 6.

In general:

To express the ratio of a fraction to a whole number,
multiply the whole number by the denominator.

6
7
 is to 3  as  6 is to 21.

For an application of this, see Lesson 26.

   Example 5.   On a map,  3
4
 of an inch represents 60 miles.  How many

miles does 2 inches represent?

 Solution.  Proportionally,

3
4
 of an inch  is to  2 inches  as  60 miles  is to  ? miles.
What ratio has  3
4
 to 2?
3
4
 is to 2  as  3 is to 8.

Therefore:

3 is to 8  as  60 miles  is to  ? miles.

Since  20 × 3 = 60,  then 20 × 8 = 160 miles.

The theorem of the same multiple.

Or, inversely:

8 is to 3  as  ? miles is to 60 miles.

Now,

8 is two and two thirds times 3.

(Lesson 18, Example 5.)  Therefore, the missing term will be

Two and two thirds times 60  =  Two times 60 + two thirds of 60
(Lesson 16)
   =  120 + 40
 
   =  160 miles.

More than or less than ½


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

compare fractions
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 16, Question 8.)

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.

And so 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 (Lesson 20, Question 11), 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.

Therefore,

5
9
 is greater than   6 
11
.

Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

Section 1 on Comparing fractions

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