Operations on Mixed Numbers

a) Addition of Mixed Numbers

To add mixed numbers, follow these steps:

1. Add the whole numbers together.

2. Add the fractions together.

3. If the sum of the fractions is an improper fraction, convert it to a mixed number and add it to the whole number sum.

Example:

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

\(2 + 1 = 3\)

\( \frac{3}{4} + \frac{1}{2} = \frac{6}{8} + \frac{4}{8} = \frac{10}{8}\)

which is \(1\frac{2}{8}\)

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

which simplifies to \(4\frac{1}{4}\).

b) Subtraction of Mixed Numbers

To subtract mixed numbers, follow these steps:

1. Subtract the whole number part of the second mixed number from the whole number part of the first mixed number.

2. Subtract the fraction part of the second mixed number from the fraction part of the first mixed number.

3. If the fraction part of the first mixed number is smaller than the fraction part of the second mixed number, borrow from the whole number part.

Example:

\(4\frac{1}{3} - 2\frac{2}{5}\)

\(4 - 2 = 2\)

\( \frac{1}{3} - \frac{2}{5} = \frac{5}{15} - \frac{6}{15} = -\frac{1}{15}\)

 As the fraction is negative, borrow from the whole number part, so \(2 - 1 = 1\)

Thus, \(4\frac{1}{3} - 2\frac{2}{5} = 1\frac{14}{15}\).

c) Multiplication of Mixed Numbers

To multiply mixed numbers, first convert them to improper fractions, then multiply the fractions, and finally convert the result back to a mixed number.

Example:

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

1. Convert both mixed numbers to improper fractions: \(2\frac{2}{3} = \frac{8}{3}\) and \(3\frac{1}{4} = \frac{13}{4}\).

2. Multiply the fractions: \( \frac{8}{3} \times \frac{13}{4} = \frac{104}{12}\).

3. Convert the improper fraction to a mixed number: \( \frac{104}{12} = 8\frac{8}{12}\), which simplifies to \(8\frac{2}{3}\).

d) Division of Mixed Numbers

To divide mixed numbers, first convert them to improper fractions, then multiply the first mixed number by the reciprocal of the second mixed number, and finally convert the result back to a mixed number.

Example:

\(5\frac{1}{2} \div 2\frac{2}{3}\)

1. Convert both mixed numbers to improper fractions: \(5\frac{1}{2} = \frac{11}{2}\) and \(2\frac{2}{3} = \frac{8}{3}\).

2. Multiply the first fraction by the reciprocal of the second fraction: \( \frac{11}{2} \times \frac{3}{8} = \frac{33}{16}\).

3. Convert the improper fraction to a mixed number: \( \frac{33}{16} = 2\frac{1}{16}\).