# Addition of Mixed Fractions

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.

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