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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