Addition and Subtraction of Unlike Fractions

In addition and subtraction of unlike fractions, we first convert them into corresponding equivalent like fractions and then they are added or subtracted.

Following steps are used to do the same.

Step I:

Obtain the fractions and their denominators.

Step II:

Find the LCM (least common multiple) of the denominators.

Step III:

Convert each fraction into an equivalent fraction having its denominator equal to the LCM (least common multiple) obtained in Step II.

Step IV:

Add or subtract like fractions obtained in Step III.

For Example:

1. Add ²/₃ and ³/₇.

Solution:

The LCM (least common multiple) of the denominators 3 and 7 is 21.

So, we convert the given fractions into equivalent fractions with denominator 21.

We have,

²/₃ + ³/₇

= ^{(2 × 7)}/_{(3 × 7)} + ^{(3 × 3)}/_{(7 × 3)}

[since 21 ÷ 3 = 7 and 21 ÷ 7 = 3]

= ¹⁴/₂₁ + ⁹/₂₁

= ^{(14 + 9)}/₂₁

= ²³/₂₁

2. ¹/₆ + ³/₈

Solution:

The LCM (least common multiple) of the denominators 6 and 8 is 24.

So, we convert the given fractions into equivalent fractions with denominator 24.

We have,

= ¹/₆ = ^{(1 × 4)}/_{(6 × 4)}= ⁴/₂₄ [since 24 ÷ 6 = 4]

and, ³/₈ = ^{(3 × 3)}/_{(8 × 3)} = ⁹/₂₄ [since 24 ÷ 8 = 3]

Thus, ¹/₆ + ³/₈ = ⁴/₂₄ + ⁹/₂₄

= ^{(4 + 9)}/₂₄

= ¹³/₂₄

3. Add 2⁴/₅ and 3⁵/₆.

Solution:

We have,

2⁴/₅ = ^{(2 × 5 + 4)}/₅ = ^{(10 + 4)}/₅ = ¹⁴/₅

and, 3⁵/₆ = ^{(3 × 6 + 5)}/₆ = ²³/₆

Now, we will compute ¹⁴/₅ + ²³/₆

The LCM (least common multiple) of the denominators 5 and 6 is 30.

So, we convert the given fractions into equivalent fractions with denominator 30.

We have,

= ¹⁴/₅ = ^{(14 × 6)}/_{(5 × 6)} = ⁸⁴/₃₀ [since 30 ÷ 5 = 6]

and, ²³/₆ = ^{(23 × 5)}/_{(6 × 5)} = ¹¹⁵/₃₀ [since 30 ÷ 6 = 5]

Thus, ¹⁴/₅ + ²³/₆ = ⁸⁴/₃₀ + ¹¹⁵/₃₀

= ^{(84 + 115)}/₃₀

= ¹⁹⁹/₃₀

= 6¹⁹/₃₀

4. Find the difference of ¹⁷/₂₄ and ¹⁵/₁₆.

Solution:

The LCM (least common multiple) of the denominators 24 and 16 is 48.

[Therefore, LCM = 2 × 2 × 2 × 2 × 3 = 48]

So, we convert the given fractions into equivalent fractions with denominator 48.

We have,

= ¹⁷/₂₄ = ^{(17 × 2)}/_{(24 × 2)} = ³⁴/₄₈ [since 48 ÷ 24 = 2]

and, ¹⁵/₁₆ = ^{(15 × 3)}/_{(16 × 3)} = ⁴⁵/₄₈ [since 48 ÷ 16 = 3]

Clearly, ⁴⁵/₄₈ > ³⁴/₄₈

Therefore, ¹⁵/₁₆ > ¹⁷/₂₄

Hence, difference = ¹⁵/₁₆ – ¹⁷/₂₄

= ⁴⁵/₄₈ – ³⁴/₄₈

= ^{(45 – 34)}/₄₈

= ¹¹/₄₈.

5. Simplify: 4²/₃ – 3¹/₄ + 2 1/6

Solution:

We have,

4²/₃ – 3¹/₄ + 2¹/₆

= ^{(4 × 3 + 2)}/₃ – ^{(3 × 4 + 1)}/₄ + ^{(2 × 6 +1)}/₆

= ^{(12 + 2)}/₃ – ^{(12 +1)}/₄ + ⁽¹²⁺¹⁾/₆

= ¹⁴/₃ – ¹³/₄ + ¹³/₆

The LCM (least common multiple) of the denominators 3, 4 and 6 is 12.

[Therefore, LCM = 2 × 2 × 3 = 12]

So, we convert the given fractions into equivalent fractions with denominator 12.

We have,

= ^{(14 × 4)}/_{(3 × 4)} – ^{(13 × 3)}/_{(4 × 3)} + ^{(13 × 2)}/_{(6 × 2)}

= ⁵⁶/₁₂ – ³⁹/₁₂ + ²⁶/₁₂

= ^{(56 – 39 + 26)}/₁₂

= ^{(82 – 39)}/₁₂

= ⁴³/₁₂