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Chapter 7: Fractions

    Introduction

    A Fraction

    Fraction on the Number Line

    Proper Fractions

    Improper and Mixed Fractions

    Equivalent Fractions

    Like Fractions and Unlike Fractions

    Comparing Fractions

    Addition and Subtraction of Fractions

    Exercise 7.1

    Exercise 7.2

    Exercise 7.3

    Exercise 7.4

    Exercise 7.5

    Exercise 7.6

    What Have We Discussed ?

    Enhanced Curriculum Support

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Chapter 7: Fractions > Comparing Fractions

Comparing Fractions

Sohni has 3 x 12 rotis in her plate and Rita has 2 x 34 rotis in her plate.

Who has more rotis in her plate?

Clearly, Sohni has 3 full rotis and more and Rita has less than 3 rotis. So, Sohni has more rotis.

Try representing the fractions: 12 and 13 below. We see that, the portion corresponding to 12 is clearly larger than the portion corresponding to 13.

So 12 than 13.

But often it is not easy to say which one out of a pair of fractions is larger.

For example, which is greater, 14 or 310 ?

For this, we may wish to show the fractions using figures but drawing figures may not be easy especially with denominators like 13.

Therefore, we need to have a systematic procedure to compare fractions.

On the other hand, it is particularly easy to compare like fractions.

You get one-fourth of a bottle of juice and your sister gets one third of the same size of a bottle of juice. Who gets more?

Instruction

Given that a person gets one-fourth of a bottle of juice and his sister gets one-third of a bottle of juice.
One-fourth of a bottle of juice can be written as .
Similarly, one-third of a bottle of juice can be written as .
Here, the numerators of both fractions are , but the denominators are .
Both 14 and 13 are fractions, because their numerators are than denominators.
So, the fraction which has denominator has greater value.
Thus, 13 14.
Hence, we conclude that sister gets more juice.

Like fractions are fractions with the same denominator.

Which of these are like fractions?

25
47
15
72
35
45
34
Like fractions
Unlike fractions

Let us compare two like fractions: 38 and 58.

Instruction

In both the fractions the whole is divided into equal parts.
For 38 and 58, we take and parts respectively out of the 8 equal parts.
Clearly, out of 8 equal parts, the portion corresponding to parts is larger than the portion corresponding to parts.
Hence, 58 38 . Note the number of the parts taken is given by the numerator.
It is, therefore, clear that for two fractions with the same denominator, the fraction with the greater numerator is greater.

Between 45 and 35 , 45 is .

Between 1120 and 1320, 1320 is .

1. Which is the larger fraction?

(i) 710 810

(ii) 1124 1324

(iii) 17102 12102

Why are these comparisons easy to make?

Comparisons are when the fractions have the denominator because you only need to look at the numerators.

2. Write these in ascending and also in descending order.

(a) 15 , 58 , 38

Ascending Order

15
58
38

Descending Order

15
58
38

(b) 15 , 115 , 45 , 35 , 75

Ascending Order

15
115
45
35
75

Descending Order

15
115
45
35
75

(c)17 , 37 , 137 , 117 , 77

Ascending Order

17
37
137
117
77

Descending Order

17
37
137
117
77

Two fractions are unlike if they have denominators.

For example: 13 and 15 are unlike fractions. So are 23 and 45

Unlike fractions with the same numerator.

Consider a pair of unlike fractions 13 and 15 , in which the numerator is the same.

Which is greater 13 or 15 ?

Answer :

In 13, we divide the whole into equal parts and take .

In 15 we divide the whole into equal parts and take .

Note that in 13 , the whole is divided into a number of parts than in 15.

The equal part that we get in 13 is, therefore, larger than the equal part we get in 15.

Since in both cases we take the same number of parts (i.e. one), the portion of the whole showing 13 is larger than the portion showing 15 , and therfore 13 15.

In the same way we can say 23 > 25. In this case, the situation is the same as in the case above, expect that the common numerator is 2, not 1.

The whole is divided into a number of equal parts for 25 than for 23.

Therefore, each equal part of the whole in case of 23 is than that in case of 25.

Therefore, the portion of the whole showing 23 is larger than the portion showing 25 and hence, 23 25.

We can see from the above example that if the numerator is the same in two fractions, the fraction with the denominator is greater of the two.

Thus, 18 110 , 35 37 , 49 411 and so on.

Let us arrange 21,213,29,25,27 in increasing order.

25
21
29
27
213

All these fractions are unlike, but their numerator is the same. Hence, in such case, the larger the denominator, the smaller is the fraction.

The smallest is as it has the largest denominator.

The next three fractions in order are 29 27 25. The greatest fraction is 21 (It is with the smallest denominator).

1. Arrange the following in ascending and descending order :

(a) 112, 123,15,17,150,19,17

Ascending order

1/12
1/23
1/7
1/9
1/5
1/17
1/50

Descending order

1/12
1/9
1/23
1/17
1/50
1/7
1/5

(b) 37 , 311 , 35 , 32 , 313 , 34 , 317.

Ascending order

37
311
35
32
313
34
317

Descending order

37
311
35
32
313
34
317

(c)Arrange them in ascending and descending order.

(i) 213 , 225 , 26 , 28 , 210 , 217

Ascending Order

213
225
26
28
210
217

Descending Order

213
225
26
28
210
217

(ii) 56 , 517 , 514 , 57 , 512 , 58 , 511

Ascending Order

56
517
514
57
512
58
511

Descending Order

56
517
514
57
512
58
511

(iii) 411 , 413 , 45 , 47 , 419 , 417 , 415

Ascending Order

411
413
45
47
419
417
415

Descending Order

411
413
45
47
419
417
415

Compare 45 and 56.

Instruction

The fractions are fractions. Their numerators are too. Let us write their equivalent fractions.
45 = 810 = 1215 = 1620 = 2025 = 2430 = = .....
and 56 = 1012 = 1518 = 2024 = 2530 = = .....
The equivalent fractions with the same denominator are : 45 = and 56 =
Since, 2530 2430, 56 45.
Note that the common denominator of the equivalent fractions is 30 which is 5 × 6. It is a common multiple of both 5 and 6.
So, when we compare two unlike fractions, we first get their equivalent fractions with a denominator which is a common multiple of the denominators of both the fractions.

Compare 56 and 1315.

Instruction

The fractions are .
We should first get their equivalent fractions with a denominator which is a common multiple of 6 and 15.
Now, 5×56×5 = , 13×215×2 = .
Since 2630 2530, we have 1315 56.

Take the product of 6 and 15 is which is 90.

Obviously 90 is also a common multiple of 6 and 15. We may use either 90 or (LCM of 6 and 15) and get the correct answer.

But we are also aware that it is easier and more convenient to work with smaller numbers. Thus, the common multiple that we take is as small as possible. This is why the LCM of the denominators of the fractions is preferred as the common denominator.