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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