COMPARING FRACTIONS - ARRANGING IN ORDER, FRACTION(S) BETWEEN TWO FRACTIONS

Please study
Fractions before Comparing Fractions
if you have not already done so.

There we studied about half, quarter, three fourth with examples and exercises.

Also study Fractions Made Easy.

There we studied about concept of fraction in general with examples and exercises.

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Here, we will see which fractions are bigger(smaller) and arranging them in ascending/descending order, Fraction(s) Lying between two given Fractions, with number of solved examples and problems for practice with answers.

Rules for deciding which fractions are bigger(smaller)

Rule 1 :

Between two fractions with same denominator, the one with greater numerator is greater of the two.

Examples :
(i) (2⁄3) > (1⁄3) ; (ii) (7⁄5) > (6⁄5) ; (iii) (4⁄7) > (2⁄7) ; (iv) (3⁄4) > (1⁄4)

Rule 2 :

Between two fractions with same numerator, the one with smaller denominator is greater of the two.

Examples :
(i) (1⁄3) > (1⁄4) ; (ii) (7⁄4) > (7⁄5) ; (iii) (3⁄2) > (3⁄4) ; (iv) (5⁄7) > (5⁄8)

Comparing fractions with different numerators and denominators

Method 1: Cross Products Method

Let a⁄b and c⁄d be two fractions.

Find cross products ad and bc.

If ad > bc, then a⁄b > c⁄d.

If ad < bc, then a⁄b < c⁄d.

If ad = bc, then a⁄b = c⁄d.

Solved Example 1 of Comparing Fractions

Compare 5⁄6 with 11⁄13

Solution:
Find the cross products 5 x 13 and 6 x 11.

We know 5 x 13 = 65; 6 x 11 = 66.
65 < 66 ⇒ 5 x 13 < 6 x 11 ⇒ 5⁄6 < 11⁄13. Ans.

Cross products method is useful for comparing two fractions.

For comparing a number of fractions (ordering of the fractions), Method 2 is more convenient.

Method 2 : L.C.M. method

Change each one of the given fractions into an   equivalent fraction   with denominator equal to the  L.C.M.  of the denominators of the given fractions. Now the new fractions are like fractions which can be compared by Rule 1.

Solved Example 2 of Comparing Fractions

Arrange the following fractions in ascending order :

17⁄32, 7⁄12, 19⁄48, 13⁄24, 9⁄16.

Solution:

Find the L.C.M. of the denominators by
common division method.

        4|    32     12     48     24     16
       -----------------------------------------
        2|     8      3     12      6      4
       -----------------------------------------
        2|     4      3      6      3      2
       -----------------------------------------
        3|     2      3      3      3      1
       -----------------------------------------
               2      1      1      1      1

Let us convert each of the given fractions into an
equivalent fraction with denominator equal to 96.

17⁄32 = (17 x 3)⁄(32 x 3) = 51⁄96
7⁄12 = (7 x 8)⁄(12 x 8) = 56⁄96
19⁄48 = (19 x 2)⁄(48 x 2) = 38⁄96
13⁄24 = (13 x 4)⁄(24 x 4) = 52⁄96
9⁄16 = (9 x 6)⁄(16 x 6) = 54⁄96

Now these fractions are like fractions (denominators same).
By Rule 1, the fraction with bigger numerator is bigger.

So, the ascending order of these fractions is
38⁄96, 51⁄96, 52⁄96, 54⁄96, 56⁄96
i.e. 19⁄48, 17⁄32, 13⁄24, 9⁄16, 7⁄12 Ans.

Exercise 1 on Comparing Fractions

Solve the following problems on Comparing Fractions

1. Compare 8⁄9 with 15⁄17
2. Arrange the following fractions in descending order :
   17⁄27, 11⁄12, 3⁄10, 13⁄15, 7⁄18.

Fraction Lying between two given Fractions

If a⁄b and c⁄d are two fractions,
then the fraction (a + c)⁄(b + d) lies between a⁄b and c⁄d.
Thus, a⁄b < (a + c)⁄(b + d) < c⁄d.

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Solved Example 3 of Comparing Fractions

Between 1⁄2 and 3⁄4

1. Insert one fraction
2. Insert two fractions
3. Insert three fractions

Thus,Two fractions lying between 1⁄2 and 3⁄4 are
3⁄5 and 2⁄3. Ans.

(iii) Solution :
We have, 1⁄2 < 3⁄5 < 2⁄3 < 3⁄4
To get one more fraction in between 1⁄2 and 3⁄4,
let us get a fraction in between 2⁄3 and 3⁄4.

Applying the above formula, again,
the fraction lying between
2⁄3 and 3⁄4 is
(2 + 3)⁄(3 + 4) = 5⁄7. Ans.
Thus, 1⁄2 < 3⁄5 < 2⁄3 < 5⁄7 < 3⁄4.

Thus,Three fractions lying between 1⁄2 and 3⁄4 are
3⁄5, 2⁄3 and 5⁄7. Ans.

Exercise 2 on Comparing Fractions

Between 1⁄3 and 8⁄9

1. Insert one fraction
2. Insert two fractions
3. Insert three fractions

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Answers to Exercise 1 on Comparing Fractions

1. 8⁄9 > 15⁄17
2. 11⁄12, 13⁄15, 17⁄27, 7⁄18, 3⁄10

Answers to Exercise 2 on Comparing Fractions

1. 3⁄4
2. 4⁄7, 3⁄4
3. 4⁄7, 3⁄4, 11⁄13