Adding Mixed Fractions

Numbers that consist of a whole and a proper fraction are known as mixed fractions. For example: In the mixed number 3 1/4, 3 is the whole number and 1/4 is the proper fraction.

We add mixed fractions to calculate the total amounts in situations like cooking. For example, if you need to double a quantity like 2½ cups of sugar, you would add 2½ + 2½ to calculate the total.

To add mixed numbers, we use the following steps:

• Identify the whole number part and the fractional part.

• Add the whole number part and the fractional part separately.

• Convert improper fractions into mixed fractions.

We add mixed numbers with the same denominators like how we add like fractions. For example: 1225 and 1425 are two mixed numbers with like denominators. However, we need to understand the key facts about mixed numbers:

• The common way to express any mixed number x y/z is x + (y/z).

• To convert a mixed fraction to an improper fraction:
  Begin by multiplying the whole number by the denominator, and then add the numerator. 
  
  For example: 2⅗ → (2 × 5) + 3 = 13/5

• To convert an improper fraction to a mixed fraction:
  
  We divide the numerator of the improper fraction by its denominator. Now, the quotient becomes the whole number part, the remainder becomes the numerator, and the denominator is retained.
  
  Let’s take an example: Convert 24/5 to a mixed number
  
  Division of 24 by 5 gives a quotient of 4 and a remainder of 4. So we can convert 24/5 to a mixed number 4 4/5.

In the case of mixed numbers with different denominators, we must first make their denominators equal before adding them.

For example: Add 314 and 612

Method 1: Convert to Improper Fractions

Let’s first convert both mixed numbers into improper fractions.

• 314 = 13/4
   
• 612 = 13/2

Next, find a common denominator using the LCD method.

• The LCD of 2 and 4 is 4.

Make the denominators the same.

• 13/2 becomes 26/4

Then, add the fractions.

• 13/4 + 26/4 = 39/4

Now, we convert 39/4 to a mixed number.

• 39 ÷ 4 = 9 remainder 3 → 934

Method 2: Add Whole and Fraction Parts Separately

As the first step, we separate the whole number and fraction parts:

• (3 + 6) + (1/4 + 1/2)

Since the denominators are different, we convert fractions to like denominators:

• To add 1/4 + 1/2, we need to find the LCD of 4 and 2
  
  The LCD of 4 and 2 is 4.
  
  The denominator of 1/2 can be changed to 4 by multiplying both the numerator and the denominator by 2. So 1/2 = 1 × 2/2 × 2 = 2/4
  
  Adding the fractions, 1/4 + 2/4 = 3/4

Now, we add both parts:

• 9 + 3/4 = 934

The addition of mixed numbers and proper fractions follows similar steps, with a few exceptions; while adding mixed numbers, we add whole numbers and fractions separately. On the other hand, the addition of proper fractions involves only the fractional parts. Let’s learn this through examples.

Fractions with the same denominator:

Example: Add 4 3/7 + 2/7

The first step is to express the mixed number as a sum:
 4 3/7 = 4 + 3/7

Now, add the fractions:
 4 + 3/7 + 2/7 = 4 + 5/7

⇒ 4 5/7

Fractions with different denominators:

Example: Add 5 1/3 + 3/4

Let’s first convert the mixed number to an improper fraction:

5 1/3 = (5 × 3 + 1)/3 = 16/3

Find the LCM of 3 and 4, which is 12

Convert both fractions to have the same denominator:

16/3 = 64/12

3/4 = 9/12

Add the fractions:

64/12 + 9/12 = 73/12

Convert the answer to a mixed number:

73 ÷ 12 = 6 remainder 1 → 6 1/12

Mixed fractions have many applications. They appear in various real-life situations. Let’s look at a few examples:

• Cooking and Baking: Mixed fractions are added to calculate the total amount of ingredients required in a recipe. For e.g., adding 212 cups of water to 314 cups of milk.

• Tracking Study Hours: Students can find their total study time by adding different durations expressed in mixed fractions.

• Travel Distance: The total distance of a journey can be calculated by adding the distances expressed in mixed fractions.