Comparing Unlike Fractions

In comparing unlike fractions, we first convert them into like fractions by using the following steps and then compare them.

Step I:

Obtain the denominators of the fractions and find their LCM (least common multiple).

Step II:

Each fractions are converted to its equivalent fraction with denominator equal to the LCM (least common multiple) obtained in Step I.

Step III:

Compare the numerators of the equivalent fractions whose denominators are same.

For example:

1. Which is larger ³/₄ or ⁵/₁₂ ?

Solution:

Let us first find the LCM (least common multiple) of the denominators 4 and 12.

We have,

$Comparing Unlike Fractions$

Therefore, LCM (least common multiple) of 4 and 12 is 2 × 2 × 3 = 12.

Now we convert the given fractions to equivalent fractions with denominator 12

We have,

3/4 = (3 × 3)/(4 × 3) = 9/12

5/12 = (5 × 1)/(12 × 1) = 5/12

Now we will observe the numerator, that is 9 > 5.

So, ⁹/₁₂ > ⁵/₁₂

Therefore, ³/₄ > ⁵/₁₂.

2. Compare ⁷/₈ and ⁵/₆.

First we find the LCM of denominators .

We have,

$Comparing Unlike Fractions$

Therefore, LCM (least common multiple) = 2 × 2 × 2 × 3 = 24.

Now, we convert each fraction into equivalent fraction with 24 as its denominator.

We have,

7/8 = (7 × 3)/(8 × 3) = 21/24 [since, 24 ÷ 8 = 3]

5/6 = (5 × 4)/(6 × 4) = 20/24 [since 24 ÷ 6 = 4]

Now we will observe the numerator, that is 20 < 21

So, 20/24 < 21/24

Therefore, ⁵/₆ < ⁷/₈.

3. Arrange the fractions ⁵/₈, ⁵/₆, ⁷/₄, ³/₅ in ascending order.

Solution:

Let us first find the LCM (least common multiple) of the denominators:

We have,

$Comparing Unlike Fractions$

Therefore, LCM (least common multiple) = 2 × 2 × 2 × 3 × 5 = 120. Now, we convert each fraction into equivalent fraction with 120 as its denominator.

We have,

5/8 = (5 × 15)/(8 × 15) = 75/120, [since 120 ÷ 8 = 15].

5/6 = (5 × 20)/(6 × 20) = 100/120, [since 120 ÷ 6 = 20].

7/4 = (7 × 30)/(4 × 30) = 210/120, [since 120 ÷ 4 = 30].

3/5 = (3 × 24)x/(5 × 24) = 72/120, [since 120 ÷ 5 = 24].

Now we will observe the numerator, that is 72 < 75 < 100 < 210.

So, 72/120 < 75/120 < 100/120 < 210/120.

Therefore, ³/₅, ⁵/₈, ⁵/₆, ⁷/₄.

4. Arrange the following fractions in descending order ³/₈, ⁵/₆, ²/₄, ¹/₃, ⁶/₈.

Solution:

We observe that the given fractions neither have common denominator nor common numerator.

So, first we convert them into like fractions i.e. fractions having common denominator. For this, we first find the LCM (least common multiple) of the denominators of the given fractions.

Denominators are 8, 6, 4, 3, 8.

We have,

$Comparing Unlike Fractions$

Therefore, LCM (least common multiple) = 2 × 2 × 2 × 3 = 24.

Now, we convert each fraction into equivalent fraction with 24 as its denominator.

Thus,

3/8 = (3 × 3)/( 8 × 3) = 9/24 [since 24 ÷ 8 = 3]

5/6 = (5 × 4)/(6 × 4) = 20/24 [since 24 ÷ 6 = 4]

2/4 = (2 × 6)/(4 × 6) = 12/24 [since 24 ÷ 4 = 6]

1/3 = (1 × 8)/(3 × 8) = 8/24 [since 24 ÷ 3 = 8]

6/8 = (6 × 3)/(8 × 3) = 18/24 [since 24 ÷ 8 = 3]

Now we will observe the numerator, that is 20 > 18 > 12 > 9 > 8.

So, 20/24 > 18/24 > 12/24 > 9/24 > 8/24.

Therefore, 5/6 > 6/8 > 2/4 > 3/8 > 1/3