Mixed Numbers

A mixed number consists of a whole number and a proper fraction combined. It represents a value greater than a whole, but it is not a whole number by itself.

The parts of a mixed number include the whole number, the numerator, and the denominator.

The numerator represents how many parts are taken, while the denominator shows the total number of equal parts in one whole.

 The properties of mixed numbers are mentioned below:
 

• Mixed numbers are a combination of whole numbers and fractions.
   
• They can be converted to an improper fraction by getting a new numerator. We can get it by multiplying the denominator and the whole number, and then adding it with the numerator.
   
• It can be converted to a decimal by performing long division.
   
• It can be used in basic arithmetic operations.
   
• It obeys the commutative property for addition and multiplication.
   
• It does not follow the commutative property for subtraction and division.
   
• It follows the associative property for addition and multiplication.
   
• It follows the identity property.

Steps to add mixed numbers:

Step 1: For easier addition, you can convert mixed numbers into improper fractions.

Step 2: If the fractions have different denominators, find the Least Common Denominator (LCD) and convert both fractions.

Step 3: Add whole numbers and fractions separately and then write them together.

Step 4: Simplify (if necessary)

If the fraction part is improper, convert it to a mixed number and adjust the final answer. Simplify the fraction at the end.

To subtract mixed numbers, the steps are mentioned below:

Step 1: Convert mixed numbers to improper fractions (optional, but useful for complex problems). If the mixed numbers have different denominators or borrowing is needed. Converting them into improper fractions makes subtraction easier. However, if the whole numbers and fractions can be subtracted directly, you may skip this step.

Step 2: If the fractions have different denominators, find the Least Common Denominator (LCD) and convert both fractions.

Step 3: As we saw above, the process of subtraction between the whole numbers and the fractions must be done separately.

Step 4: If the fraction in the first number is smaller than that in the second, borrow 1 from the whole number and convert it into an equivalent fraction.

For converting improper fractions to mixed numbers, the following steps are used:

Step 1: Divide the numerator by the denominator by perform long division.

Step 2: The quotient from Step 1 becomes the whole number of the mixed number.

Step 3: The remainder from Step 1 becomes the numerator of the fraction. The denominator remains the same.

Step 4: Write the mixed number by combining the whole number and fraction to form the mixed number.

To convert mixed numbers to improper fractions, follow the steps mentioned below:

Step 1: A mixed number consists of a whole number and a fraction \(\frac{\text{numerator}}{\text{denominator}} \)

Step 2: Multiply the whole number by the denominator of the fraction. Whole number × denominator

Step 3: Take the result from Step 2 and add the numerator of the fraction.

Step 4: Write the result from Step 3 as the numerator, keeping the denominator the same.

To convert mixed numbers to decimals, follow the steps mentioned below:

Step 1:  mixed number consists of a whole number and a fraction. Therefore, we should identify the whole number and fraction

Example: Convert \(3 \dfrac{1}{2} \) to a decimal.

Whole number = 3

Fraction = \(\frac{1}{2} \)

Step 2: To convert the fraction to a decimal, divide the numerator by the denominator. ​

\(\dfrac{1}{2} = 1 \div 2 = 0.5 \)

So, \(\dfrac{1}{2} = 0.5 \)

Step 3: Now, add the decimal from Step 2 to the whole number.

\(3 + 0.5 = 3.5\)

Therefore, \(3 \dfrac{1}{2} = 3.5 \) in decimal form.

Here are some tips and tricks to understand mixed numbers deeply and to master it.
 

• Always convert mixed fractions to improper fractions before performing multiplication or division. You can do the same for addition and subtraction if the denominators are the same.
   
• Always simplify fractions to their lowest terms after calculations. 
   
• Use some real-life examples, like calculating time or adding money, to help visualize numbers in daily life.
   
• Try to use fraction bars, circles, or number lines to represent the fraction and the whole number. This visual representation makes it easier to add, subtract, or compare mixed numbers.
   
• For addition and subtraction, always make the denominators the same before performing operations. 

The mixed numbers have numerous applications across various fields. Let us explore how mixed numbers are used in different areas:
 

• Cooking and baking: Mixed numbers are essential in cooking and baking, where ingredients are measured precisely, such as \(2 \dfrac{1}{2} \) cups of flour or \(3 \dfrac{3}{4} \) teaspoons of sugar. These measurements ensure that flavors and textures are balanced and that recipes turn out as expected.
   
• Time management and scheduling: Mixed numbers are commonly used when planning and managing time. Appointments, travel durations, and schedules often involve expressions like \(1 \dfrac{1}{2} \) hours or \(2 \dfrac{3}{4} \) hours. For example, if a movie starts at 6:30 PM and lasts \(1 \dfrac{3}{4} \) hours, knowing how to work with mixed numbers helps determine when it will end. It will end at 8:15 PM 
   
• Construction and carpentry: Mixed numbers are used in measuring materials for construction. For measuring the dimensions like \(4 \dfrac{3}{8} \) inches or \(6 \tfrac{1}{2} \) feet for door frame, such mixed numbers are used for accuracy. These precise measurements ensure that parts fit together correctly, preventing structural issues.
   
• Sports and fitness: Mixed numbers are capable of tracking distances, scores or timings precisely. They help us in planning and exercise routines. 
   
• Travel: Mixed numbers can be used for distance and fuel calculations. They make budgeting and planning easier. For example, if we need \(1 \dfrac{1}{2} \) liters of fuel for a short trip and \(2 \dfrac{3}{4} \) liters of fuel for a long trip, we can calculate the total liters of fuel required by adding these two mixed fractions, that is, \(4 \dfrac{1}{4} \) liters.