Addition and subtraction of mixed fractions

There are two ways to add mixed numbers. You can choose the method that works best for you.

Method 1: Add Whole Numbers and Fractions Separately

Steps:

1. Add the whole numbers together
2. Add the fractions together (remember to use a common denominator)
3. Combine the two results

Example

$2\frac{1}{4} + 1\frac{3}{5} = ?$

Solution

Step 1: Add the whole numbers: $2 + 1 = 3$

Step 2: Add the fractions $\frac{1}{4} + \frac{3}{5}$

• Find the LCM of 4 and 5, which is 20
• $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$
• $\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$
• $\frac{5}{20} + \frac{12}{20} = \frac{17}{20}$

Step 3: Combine: $3 + \frac{17}{20} = 3\frac{17}{20}$

Therefore, $2\frac{1}{4} + 1\frac{3}{5} = 3\frac{17}{20}$

Method 2: Convert to Improper Fractions First

Steps:

1. Convert each mixed number to an improper fraction
2. Find a common denominator
3. Add the fractions
4. Change back to a mixed number if needed

Example

$2\frac{1}{4} + 1\frac{3}{5} = ?$

Solution

Convert to improper fractions:

• $2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}$
• $1\frac{3}{5} = \frac{(1 \times 5) + 3}{5} = \frac{8}{5}$

Find the LCM of 4 and 5, which is 20:

• $\frac{9}{4} = \frac{9 \times 5}{4 \times 5} = \frac{45}{20}$
• $\frac{8}{5} = \frac{8 \times 4}{5 \times 4} = \frac{32}{20}$

Add: $\frac{45}{20} + \frac{32}{20} = \frac{77}{20}$

Convert back to mixed number: $\frac{77}{20} = 3\frac{17}{20}$

Simple Case: Fractions Have the Same Denominator

When the fractions have the same denominator, subtraction is straightforward.

Example

$3\frac{2}{5} - 1\frac{1}{5} = ?$

Solution

Step 1: Subtract whole numbers: $3 - 1 = 2$

Step 2: Subtract fractions: $\frac{2}{5} - \frac{1}{5} = \frac{1}{5}$

Step 3: Combine: $2 + \frac{1}{5} = 2\frac{1}{5}$

Therefore, $3\frac{2}{5} - 1\frac{1}{5} = 2\frac{1}{5}$

Harder Case: Fractions Have Different Denominators

When the fractions have different denominators, we need to find a common denominator first.

Example

$6\frac{1}{6} - 2\frac{1}{2} = ?$

Solution

Step 1: Convert to improper fractions:

• $6\frac{1}{6} = \frac{(6 \times 6) + 1}{6} = \frac{37}{6}$
• $2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$

Step 2: Find the LCM of 6 and 2, which is 6

• $\frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6}$

Step 3: Subtract: $\frac{37}{6} - \frac{15}{6} = \frac{22}{6}$

Step 4: Simplify: $\frac{22}{6} = \frac{11}{3} = 3\frac{2}{3}$

Therefore, $6\frac{1}{6} - 2\frac{1}{2} = 3\frac{2}{3}$

Important: When Subtracting Requires Borrowing

If the fraction you are subtracting is larger than the fraction part of the mixed number, you need to borrow 1 from the whole number.

Example

$5\frac{1}{4} - 2\frac{3}{4} = ?$

Here, $\frac{3}{4}$ is larger than $\frac{1}{4}$, so we need to borrow.

Solution

• Borrow 1 from 5, which becomes $\frac{4}{4}$. Add to $\frac{1}{4}$: $\frac{1}{4} + \frac{4}{4} = \frac{5}{4}$
• Now subtract: $5\frac{1}{4}$ becomes $4\frac{5}{4}$
• $4\frac{5}{4} - 2\frac{3}{4} = (4-2) + (\frac{5}{4} - \frac{3}{4}) = 2 + \frac{2}{4} = 2\frac{1}{2}$

Therefore, $5\frac{1}{4} - 2\frac{3}{4} = 2\frac{1}{2}$

• Always check if the fractions have the same denominator before adding or subtracting
• The LCM (Least Common Multiple) helps you find a common denominator
• Always simplify your final answer if possible
• When subtracting mixed numbers, you may need to borrow from the whole number

In Tanzania, mixed numbers are very useful in everyday situations. For example, when a farmer at the market sells $2\frac{1}{2}$ kg of rice to one customer and $1\frac{3}{4}$ kg to another customer, they need to add mixed numbers to know the total rice sold. Similarly, if a mama lishe cooks with $3\frac{1}{4}$ litres of water but only uses $1\frac{1}{2}$ litres, she subtracts mixed numbers to find out how much water is left. Understanding how to add and subtract mixed numbers helps in shopping, cooking, farming, and managing money.