Mixed Fraction

A mixed fraction is a combination of a whole number and a proper fraction. Take a look at this mixed fraction, \(3 \dfrac{3}{4}\) (or three and three-fourths), which is a numerical expression. Mixed fractions represent numbers greater than a whole number but less than the next whole number.  A mixed fraction can be denoted as:    

\( \text{Whole number} + \dfrac{\text{numerator}}{\text{denominator}} \)

Any number that can be expressed as a fraction \(\dfrac{a}{b}\) is a rational number, where b is not zero and both a and b are whole numbers. Now, any mixed number can be converted to a fraction. Therefore, all mixed numbers are rational numbers. For example, take the mixed fraction \(5 \dfrac{3}{4}\):

We can then convert the mixed fraction into an improper fraction. \( \dfrac{(5 \times 4) + 3}{4} \) = \( \dfrac{20 + 3}{4} \) = \(\dfrac{23}{4}\)

This is a rational number, in which the numerator and the denominator are whole numbers. When we divide,

23 ÷ 4 = 5.75 

The result is a terminating decimal, which means it stops after a certain number of digits. Therefore, \(\dfrac{23}{4}\) is a rational number.

In an improper fraction, the numerator is always greater than the denominator. \(\dfrac{14}{10}\), \(\dfrac{3}{1}\), and \(\dfrac{16}{13}\) are a few examples of improper fractions. Here, the top number is greater than the bottom number. To convert an improper fraction into a mixed fraction, follow the given steps:  
 

Step 1: Divide the numerator by the denominator.

Step 2: Identify the quotient and remainder. 

Step 3: Write them in the given form as \( Q\dfrac{R}{D} \).

Here, Q is the quotient, R is the remainder, and D is the denominator of the improper fraction. 

Let us take an example to better understand how to convert improper fractions into mixed fractions. Convert the improper fraction \(\dfrac{14}{5}\) into a mixed fraction. 

Step 1: We can divide the numerator by the denominator. 

14 ÷ 5

Step 2: Quotient = 2

Remainder = 4

Step 3: \(2 \dfrac{4}{5}\) 

\(\dfrac{14}{5}\) = \(2 \dfrac{4}{5}\) 

Any arithmetic operations such as addition, subtraction, multiplication, and division are applicable on mixed fractions. 

We must follow certain steps while adding mixed fractions. 

Step 1: In the first step, we should convert mixed numbers to improper fractions. 

Step 2: Verify whether the denominators are equal. 

Step 3: If the denominators are the same, add the numerators and write down the answer. 

Step 4: If the denominators are different, find the LCD and convert them to like fractions. 

Step 5: Add all the numerators and find the answer. 

For example, add \(1 \dfrac{3}{5}\) and \(2 \dfrac{4}{5}\)

Step 1: ​​\(1 \dfrac{3}{5}\) =  \(\dfrac{(1 \times 5 + 3)}{5}\) = \(\dfrac{8}{5}\)

\(2 \dfrac{4}{5}\) =  \(\dfrac{(2 \times 5 + 4)}{5}\) = \( \dfrac{14}{5}\) 

Step 2: The denominators of both the fractions are 5. 

Step 3: \( \frac{8}{5} + \frac{14}{5} = \frac{8 + 14}{5} = \frac{22}{5} \)

Step 4: 22 ÷ 5 → Quotient = 4, Remainder = 2;

Mixed fraction = \(4 \dfrac{2}{5} \)

The given steps should be followed while subtracting mixed fractions: 

Step 1: Convert mixed fractions to improper fractions. 

Step 2: Check if the denominators are equal. 

Step 3: If the denominators are equal, then simply subtract the numerators. 

Step 4: If the denominators are different, find the LCD and adjust the fractions. 

Step 5: Subtract only the numerators and retain the denominator. 

For example, subtract \(3 \dfrac{2}{4}\) and \(4 \dfrac{3}{4}\)

Step 1: \( 3 \frac{2}{4} = \frac{(3 \times 4) + 2}{4} = \frac{14}{4} \)
\(4 \frac{3}{4} = \frac{(4 \times 4) + 3}{4} = \frac{19}{4}\)

Step 2: The denominators of both the fractions are 4.

Step 3: \( \frac{14}{4} - \frac{19}{4} = \frac{14 - 19}{4} = -\frac{5}{4} \)
\( -\frac{5}{4} = -1 \frac{1}{4} \) (Remainder positive: 1)

The following steps are used to multiply mixed fractions:

Step 1: Convert the mixed fractions to improper fractions. 

Step 2: Multiply the numerators together and the denominators together. 

Step 3: The fraction can be simplified into its lowest form, or convert the product into a mixed fraction or an improper form. 

For example, multiply \(2 \dfrac{1}{4}\) and \(3 \dfrac{2}{4}\)

Step 1: \( 2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4} \)

\( 3 \frac{2}{4} = \frac{(3 \times 4) + 2}{4} = \frac{12 + 2}{4} = \frac{14}{4} \)

Step 2: \( \frac{9}{4} \times \frac{14}{4} = \frac{9 \times 14}{4 \times 4} = \frac{126}{16} \)

Step 3: Simplify 126 and 16 by finding the greatest common divisor (GCD) of 126 and 16. 

To find the GCD, we must identify the prime factorization of 126. 
126 = 2 × 3² × 7 

The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126.

Prime factorization of 16 = 2⁴

The factors of 16 include 1, 2, 4, 8, and 16.

The greatest common divisor of 126 and 16 is 2. 

Now, we can divide the numerator and denominator by 2. 

\( \frac{126 \div 2}{16 \div 2} = \frac{63}{8} \)

Step 4: Next, we can convert the fraction into a mixed fraction. 

63 ÷ 8

Quotient = 7

Remainder = 7

\( \frac{63}{8} = 7 \frac{7}{8} \)

The given steps are followed to divide mixed fractions: 

Step 1: Convert the given mixed fractions to improper fractions. 

Step 2: Multiply the reciprocal of the second fraction by the first fraction. 

Step 3: The result can be simplified to its lowest form if possible, or convert the result to a mixed or improper fraction. 

For example, divide \(1 \dfrac{1}{3}\) and \(2 \dfrac{2}{3}\)

Step 1:  \( 1 \frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{4}{3} \)

\( 2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3} \)

Step 2: Now we can multiply the reciprocal of the second fraction by the first fraction.

The reciprocal of \(\dfrac{8}{3}\) is \(\dfrac{3}{8}\)

\( \frac{4}{3} \times \frac{3}{8} = \frac{12}{24} \)

Step 3: \(\dfrac{12}{24}\) can be simplified to \(\dfrac{1}{2}\)

Therefore, \( 1 \frac{1}{3} \div 2 \frac{2}{3} = \frac{1}{2} \)

Let's take a look at some tips and tricks in mixed fractions which would help us in mastering the subject. It allows us in doing such calculations very easily.
 

• Understand that mixed fractions are in the form of a whole number linked with a proper fraction. They can be converted to improper fraction and vice versa.
   
• While adding mixed fractions, always convert them to improper fractions, add them and then convert back to mixed fraction.
   
• While performing operations, try to round the fractions to its nearest whole to check the answer quickly.
   
• Try to simplify them early. Reduce the fractions before operating to save time. 
   
• Memorize the common mixed fraction conversions to calculate easily. 

A mixed fraction is a number that consists of two parts: a whole and a proper fraction. In our daily lives, from cooking to construction and engineering, we use mixed fractions to represent various measurements or calculations. Here are some real-life applications of mixed fractions:
 

• When we cook, we usually measure ingredients in mixed fractions. This helps us to easily understand the needed measurements and quantities without getting too many ingredients. For example, \(2 \dfrac{1}{2}\) tablespoons of coconut oil are added to \(4 \dfrac{1}{2}\) cups of flour.
   
• In construction and engineering, accurate measurements are vital to work and complete projects effectively. For example, engineers use mixed fractions, such as \(3 \dfrac{1}{2}\) to cut a piece of iron rod for a room.
   
• We use mixed fractions for time management, which helps to estimate the time required for a task or project. It is especially useful while reading clocks.
   
• Mixed fractions are used to denote precise numbers. Therefore, they are widely used in finance, economics, and budgeting, where accurate numbers are displayed.
   
• In sports and fitness, we use mixed fractions for performance measurements. They allow precise tracking.