We will learn how to solve addition of mixed fractions or addition of mixed numbers. There are two methods to add the mixed fractions.

For example, add 2\(\frac{3}{5}\) and 1\(\frac{3}{10}\).

We can use the two methods to add the mixed numbers.

**Method 1:**

2\(\frac{3}{5}\) + 1\(\frac{3}{10}\) = (2 + 1) + \(\frac{3}{5}\) + \(\frac{3}{10}\) = 3 + \(\frac{3}{5}\) + \(\frac{3}{10}\) = 3 + \(\frac{3 × 2}{5 × 2}\) + \(\frac{3 × 1}{10 × 1}\), [L.C.M. of 5 and 10 = 10] = 3 + \(\frac{6}{10}\) + \(\frac{3}{10}\) = 3 + \(\frac{6 + 3}{10}\) = 3 + \(\frac{9}{10}\) = 3\(\frac{9}{10}\) |
Step I: We add the whole numbers, separately. Step II: To add fractions, we take L.C.M. of the denominators and change the fractions into like fractions.
Step III: We find the sum of the whole numbers and the fractions in the simplest form. |

**Method 2:**

2\(\frac{3}{5}\) + 1\(\frac{3}{10}\) = (5 × 2) + \(\frac{3}{5}\) + (10 × 1) + \(\frac{3}{10}\) = \(\frac{13}{5}\) + \(\frac{13}{10}\) = \(\frac{13 × 2}{5 × 2}\) + \(\frac{13 × 1}{10 × 1}\), [L.C.M. of 5 and 10 = 10] = \(\frac{26}{10}\) + \(\frac{13}{10}\) = \(\frac{26 + 13}{10}\) = \(\frac{39}{10}\) = 3\(\frac{9}{10}\) |
Step I: We change the mixed fractions into improper fractions. Step II: We take L.C.M. of the denominators and change the fractions into like fractions.
Step III: We add the like fractions and express the sum to its simplest form. |

**Now let us consider
some of the examples on addition of mixed numbers using Method 1.**

**1.** Add 1\(\frac{1}{6}\) , 2\(\frac{1}{8}\) and 3\(\frac{1}{4}\)

**Solution:**

1\(\frac{1}{6}\) + 2\(\frac{1}{8}\) + 3\(\frac{1}{4}\)

Let us add whole numbers and fraction parts separately.

= (1 + 2 + 3) + (\(\frac{1}{6}\) + \(\frac{1}{8}\) + \(\frac{1}{4}\))

= 6 + (\(\frac{1}{6}\) + \(\frac{1}{8}\) + \(\frac{1}{4}\))

= 6 + \(\frac{1 × 4}{6 × 4}\) + \(\frac{1 × 3}{8 × 3}\) + \(\frac{1 × 6}{4 × 6}\); [Since, the L.C.M. of 6, 8 and 4 = 24]

= 6 + \(\frac{4}{24}\) + \(\frac{3}{24}\) + \(\frac{6}{24}\)

= 6 + \(\frac{4 + 3 + 6}{24}\)

= 6 + \(\frac{13}{24}\)

= 6\(\frac{13}{24}\)

**2.** Add 5\(\frac{1}{9}\), 2\(\frac{1}{12}\) and \(\frac{3}{4}\).

**Solution:**

5\(\frac{1}{9}\) + 2\(\frac{1}{12}\) + \(\frac{3}{4}\)

Let us add whole numbers and fraction parts separately.

= (5 + 2 + 0) + (\(\frac{1}{9}\) + \(\frac{1}{12}\) + \(\frac{3}{4}\))

= 7 + \(\frac{1}{9}\) + \(\frac{1}{12}\) + \(\frac{3}{4}\)

= 7 + \(\frac{1 × 4}{9 × 4}\) + \(\frac{1 × 3}{12 × 3}\) + \(\frac{3 × 9}{4 × 9}\), [Since the L.C.M. of 9, 12 and 4 = 36]

= 7 + \(\frac{4}{36}\) + \(\frac{3}{36}\) + \(\frac{27}{36}\)

= 7 + \(\frac{4 + 3 + 27}{36}\)

= 7 + \(\frac{34}{36}\)

= 7 + \(\frac{17}{18}\),

= 7\(\frac{17}{18}\).

**3.** Add \(\frac{5}{6}\), 2\(\frac{1}{2}\) and 3\(\frac{1}{4}\)

**Solution:**

\(\frac{5}{6}\) + 2\(\frac{1}{2}\) + 3\(\frac{1}{4}\)

Let us add whole numbers and fraction parts separately.

= (0 + 2 + 3) + \(\frac{5}{6}\) + \(\frac{1}{2}\) + \(\frac{1}{4}\)

= 5 + \(\frac{5}{6}\) + \(\frac{1}{2}\) + \(\frac{1}{4}\)

= 5 + \(\frac{5 × 2}{6 × 2}\) + \(\frac{1 × 6}{2 × 6}\) + \(\frac{1 × 3}{4 × 3}\), [Since, the L.C.M. of 6, 2 and 4 = 12]

= 5 + \(\frac{10}{12}\) + \(\frac{6}{12}\) + \(\frac{3}{12}\)

= 5 + \(\frac{10 + 6 + 3}{12}\)

= 5 + \(\frac{19}{12}\); [Here, fraction \(\frac{19}{12}\) can write as mixed number.]

= 5 + 1\(\frac{7}{12}\)

= 5 + 1 + \(\frac{7}{12}\)

= 6\(\frac{7}{12}\)

**4.** Add 3\(\frac{5}{8}\) and 2\(\frac{2}{3}\).

**Solution:**

Let us add whole numbers and fraction parts separately.

3\(\frac{5}{8}\) + 2\(\frac{2}{3}\)

= (3 + 2) + (\(\frac{5}{8}\) + \(\frac{2}{3}\))

= 5 + (\(\frac{5}{8}\) + \(\frac{2}{3}\))

L.C.M. of denominator 8 and 3 = 24.

= 5 + \(\frac{5 × 3}{8 × 3}\) + \(\frac{2 × 8}{3 × 8}\), (Since, L.C.M. of 8 and 3 = 24)

= 5 + \(\frac{15}{24}\) + \(\frac{16}{24}\)

= 5 + \(\frac{15 + 16}{24}\)

= 5 + \(\frac{31}{24}\)

= 5 + 1\(\frac{7}{24}\).

= 6\(\frac{7}{24}\).

**Now let us consider some of the examples on addition of mixed numbers using Method 2.**

**1. **Add 2\(\frac{3}{9}\), 1\(\frac{1}{6}\) and 2\(\frac{2}{3}\)

**Solution:**

2\(\frac{3}{9}\) + 1\(\frac{1}{6}\) + 2\(\frac{2}{3}\)

= \(\frac{(9 × 2) + 3}{9}\) + \(\frac{(6 × 1) + 1}{6}\) + \(\frac{(3 × 2) + 2}{3}\)

= \(\frac{21}{9}\) + \(\frac{7}{6}\) + \(\frac{8}{3}\), (L.C.M. of 9, 6 and 3 = 18)

= \(\frac{21 × 2}{9 × 2}\) + \(\frac{7 × 3}{6 × 3}\) + \(\frac{8 × 6}{3 × 6}\)

= \(\frac{42}{18}\) + \(\frac{21}{18}\) + \(\frac{48}{18}\)

= \(\frac{42 + 21 + 48}{18}\)

= \(\frac{111}{18}\)

= \(\frac{37}{6}\)

= 6\(\frac{1}{6}\)

**2.** Add2\(\frac{1}{2}\), 3\(\frac{1}{3}\) and 4\(\frac{1}{4}\).

**Solution:**

2\(\frac{1}{2}\) + 3\(\frac{1}{3}\) + 4\(\frac{1}{4}\)

= \(\frac{(2 × 2) + 1}{2}\) + \(\frac{(3 × 3) + 1}{3}\) + \(\frac{(4 × 4) + 1}{3}\)

= \(\frac{5}{2}\) + \(\frac{10}{3}\) + \(\frac{17}{4}\), (L.C.M. of 2, 3 and 4 = 12)

= \(\frac{5 × 6}{2 × 6}\) + \(\frac{10 × 4}{3 × 4}\) + \(\frac{17 × 3}{4 × 3}\), (Since, L.C.M. of 2, 3 and 4 = 12)

= \(\frac{30}{12}\) + \(\frac{40}{12}\) + \(\frac{51}{12}\)

= \(\frac{30 + 40 + 51}{12}\)

= \(\frac{121}{12}\)

= 10\(\frac{1}{12}\)

**3.** Add 3\(\frac{5}{8}\) and 2\(\frac{2}{3}\).

**Solution:**

3\(\frac{5}{8}\) + 2\(\frac{2}{3}\)

Let us convert the mixed fractions into improper fractions.

= \(\frac{(8 × 3) + 5}{8}\) + \(\frac{(3 × 2) + 2}{3}\)

= \(\frac{29}{8}\) + \(\frac{8}{3}\),

L.C.M. of denominator 8 and 3 = 24.

= \(\frac{29 × 3}{8 × 3}\) + \(\frac{8 × 8}{3 × 8}\), (Since, L.C.M. of 8 and 3 = 24)

= \(\frac{87}{24}\) + \(\frac{64}{24}\)

= \(\frac{87 + 64}{24}\)

= \(\frac{151}{24}\)

= 6\(\frac{7}{24}\).

Word Problem on Addition of Mixed Fraction:

The doctor advises every child to drink 3\(\frac{1}{2}\) litres of water in morning, 4\(\frac{1}{4}\) litres in the after noon and \(\frac{1}{2}\) litre before going to bed. How much water should a child drink every day?

**Solution:**

**3\(\frac{1}{2}\) + **4\(\frac{1}{4}\) + \(\frac{1}{2}\)

Let us add whole numbers and fraction parts separately.

= (3 + 4 + 0) + (\(\frac{1}{2}\) + \(\frac{1}{4}\) + \(\frac{1}{2}\))

= 7 + (\(\frac{1}{2}\) + \(\frac{1}{4}\) + \(\frac{1}{2}\))

L.C.M. of denominators 2, 4 and 2 = 4.

= 7 + \(\frac{1 × 2}{2 × 2}\) + \(\frac{1 × 1}{4 × 1}\) + \(\frac{1 × 2}{2 × 2}\), [Since, the L.C.M. of 2, 4 and 2 = 4.]

= 7 + \(\frac{2}{4}\) + \(\frac{1}{4}\) + \(\frac{2}{4}\)

= 7 + \(\frac{2 + 1 + 2}{4}\)

= 7 + \(\frac{5}{4}\)

[Here, the fraction \(\frac{5}{4}\) can write as mixed number.]

= 7 + 1\(\frac{1}{4}\)

= 8\(\frac{1}{4}\)

Therefore, 8\(\frac{1}{4}\) litres of water should a child drink every day.

● **Related Concepts**

**Fraction of a Whole Numbers****Representation of a Fraction****Equivalent Fractions****Properties of Equivalent Fractions****Finding Equivalent Fractions****Reducing the Equivalent Fractions****Verification of Equivalent Fractions****Finding a Fraction of a Whole Number****Like and Unlike Fractions****Comparison of Like Fractions****Comparison of Fractions having the same Numerator****Comparison of Unlike Fractions****Fractions in Ascending Order****Fractions in Descending Order****Types of Fractions****Changing Fractions****Conversion of Fractions into Fractions having Same Denominator****Conversion of a Fraction into its Smallest and Simplest Form****Addition of Fractions having the Same Denominator****Addition of Unlike Fractions****Addition of Mixed Fractions****Word Problems on Addition of Mixed Fractions****Worksheet on Word Problems on Addition of Mixed Fractions****Subtraction of Fractions having the Same Denominator****Subtraction of Unlike Fractions****Subtraction of Mixed Fractions****Word Problems on Subtraction of Mixed Fractions****Worksheet on Word Problems on subtraction of Mixed Fractions****Addition and Subtraction of Fractions on the Fraction Number Line****Word Problems on Multiplication of Mixed Fractions****Worksheet on Word Problems on Multiplication of Mixed Fractions****Multiplying Fractions****Dividing Fractions****Word Problems on Division of Mixed Fractions****Worksheet on Word Problems on Division of Mixed Fractions**

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