Comparing fractions -- A complete course in arithmetic

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

Lesson 19 Section 2

The Relative Sizes of
Fractions

11.	What is the relative size of fractions that have equal numerators?

	The larger the denominator, the smaller the fraction.

In each of these, the numerator is the same: 1. As the number of equal parts -- the denominator -- gets larger, and since we are taking only

1 of them, then the size of each part gets smaller.

1
6

is smaller than

1
5

What is more, since one-sixth is smaller than one-fifth, then two will be smaller than two:

2
6

is smaller than

2
5

Three will be smaller than three:

3
6

is smaller than

3
5

And so on.

In this sequence, then,

2
3

2
4

2
5

2
6

the fractions are getting smaller. (The number of equal parts is getting larger, but there are always 2 of them.)

In terms of ratios, the ratio of 1 to 2, for example, is greater than the ratio of 1 to 3:

When we compare 1 with 2, it appears greater than when we compare it with 3.

12.	What is the relative size of fractions that have equal denominators?

	The larger the numerator, the larger the fraction.

In this sequence,

2
6

3
6

4
6

5
6

the fractions are getting larger. Each one is one more of the 6 equal parts into which 1 has been divided.

As for ratios, we say that the ratio of 2 to 5 is smaller than the ratio of 3 to 5:

2, when compared with 5, appears smaller than 3 when compared with 5.

Example 1. Arrange these from smallest to largest:

5
7

4
9

4
7

Solution. We must compare them in pairs.

4
9

and

4
7

have the same

numerator; therefore

4
9

is smaller than

4
7

5
7

and

4
7

have the same denominator; therefore

5
7

is larger than

4
7

The sequence is

4
9

4
7

5
7

Example 2. Which is smaller,

1
10

2
9

Answer. Since

1
10

is smaller than

1
9

, then it is surely smaller than

2
9

In Lesson 22 we will see how to compare fractions with different numerators and denominators.

The following questions will lead to what we call equivalent, or equal, fractions.

13.	How can we make a fraction 2, 3, 4, etc. times larger?

	Multiply the numerator by 2, 3, 4, etc., without changing the denominator. Or take half, a third, a fourth, etc. of the denominator, without changing the numerator.

For, if we make the numerator larger, then we are increasing how many of the equal parts.

While if we make the denominator -- the number of equal parts --

smaller and do not change how many, then each part will be larger.

6
8

3
4

Note: To take half, a third, a fourth, etc., of a number, divide by the cardinal number that corresponds to the part. Lesson 14, Question 4.

14.	How can we make a fraction 2, 3, 4, etc. times smaller?

	Take half, a third, a fourth, etc. of the numerator, without changing the denominator. Or multiply the denominator by 2, 3, 4, etc., without changing the numerator.

For, by changing the numerator, we will be taking half, a third, a fourth, etc., of the equal parts.

While if we increase the number of equal parts, but keep the same

number of them, then each part will be smaller.

Either way,

1
4

3
12

(See Lesson 26: Parts of Fractions.)

Finally:

15.	What will happen if we make both terms of a fraction 2, 3, 4, etc., times larger or smaller?

	The value of the fraction will not change, because the change in the numerator will make up for the change in the denominator, and vice-versa.
	When we change both terms in the same way, we have created an equivalent fraction.

4
6

4 × 2
6 × 2

8
12


4 6	=	4 ÷ 2 6 ÷ 2	=	2 3

4
6

8
12

, and

2
3

are equivalent fractions. We will have much more to

say about them in Lesson 21.

Problem. What is the relationship with respect to relative size -- the ratio -- between each pair of fractions?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

2
3

and

4
3

6
8

and

2
8

3
10

and

3
5

and

6
10

5
6

and

15
18

Please "turn" the page and do some Problems.

Continue on to the Section 3.

Section 1 on Fractions

Introduction | Home | Table of Contents

Please make a donation to keep TheMathPage online.
Even $1 will help.

Questions or comments?

E-mail: themathpage@nyc.rr.com