skill

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

COMPARING FRACTIONS

Lesson 23 Section 2

Back to Section 1

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

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

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.

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

5
9

Answer. On cross-multiplying,

4
7

is to

5
9

36 is to 35.

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

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 3.

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

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 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

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.

Continue on to the next Lesson.

Section 1 on Comparing fractions

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