Comparing fractions, Section 2 -- A complete course in arithmetic

S k i l l
i n
A R I T H M E T I C

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

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:

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

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

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

5
2

is to 3

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

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

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

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.

Continue on to the next Lesson.

Section 1 on Comparing Fractions

Introduction | Home | Table of Contents

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