Proper Fraction

A fraction contains two parts: a numerator and a denominator, separated by a bar. There are various types of fractions based on the values of the numerator and denominator. A proper fraction is a type of fraction where the numerator is less than the denominator.

For example, \(\frac{15}{24} \) and \(\frac{23}{40} \) are some examples of proper fractions.

To find whether a fraction is a proper fraction, we follow these steps:

Step 1: Identify whether the numerator is smaller than the denominator.

Step 2: Check the denominator and identify if it is greater than the numerator.

Step 3: Ensure both numerator and denominator are either positive or negative (not mixed signs).

If the numerator < denominator, it is a proper fraction. 

If the numerator ≥ denominator, it is an improper fraction.

Here are some of the main differences between improper and proper fractions:

By converting an improper fraction into a mixed number, we will get a whole number and a proper fraction. Let us now convert \(\frac{17}{7} \) to a mixed fraction using the following steps:

Step 1: We first divide the numerator by the denominator. We divide \(\frac{17}{7} \), we get the quotient which is 2, and the remainder is 3.

Step 2: The quotient is 2, which will be the whole number in the mixed fraction.

Step 3: The numerator will be the remainder, in this example the numerator is 3.

Step 4: The denominator will remain the same. So the final answer will be \(2 \frac{3}{7} \). 

To add a mixed fraction and a proper fraction, we first need to convert the mixed fraction into an improper fraction. Once the mixed number is converted, we will then add both fractions in the usual way of the addition of fractions. Let us take the earlier mixed number \(2 \frac{3}{7} \) and add it to \( \frac{4}{7} \).

Step 1: Convert 2 3/7 into an improper fraction that is \(\frac{17}{7} \)
 

Step 2: Add \(\frac{17}{7}\) + \(\frac{4}{7}\) using the usual method. If the fractions have similar denominators, add the numerators directly and keep the same denominator. If the denominators are different, find the least common multiple of the denominators and adjust the fractions before adding. 
 

Step 3: \(\frac{17}{7}\)+ \(\frac{4}{7}\) = \(\frac{17+4}{7} \) = \(\frac{21}{7}\) = \(3\)

Similar to most fractions, proper fractions can be added, subtracted, multiplied, or divided with others. To add two fractions a/b and c/d, the formulas of each operation are as follows:

The concept of proper fractions is simple and to master it, use the tips and tricks given below. 

• Visualize fractions with everyday objects like pizza, chocolate bars, so on to demonstrate the fractional portions. 
   
• Mark fractions like \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), etc., on a number line, so it can be easily understood and compared. 
   
• Simplify fractions by dividing both the numerator and denominator by the greatest common divisor(GCD). For example, \(\frac{6}{8}\) will be simplified to \(\frac{3}{4}\). Hence, it can be easier to understand and compare. 
   
• To convert an improper fraction to a mixed fraction, divide the numerator by the denominator. The quotient becomes the whole number and the remainder can be taken as the numerator of the fractional part, while the original denominator remain the same. For example, \(\frac{7}{4}\) can be written as \(1\frac{3}{4}\) after converting to a mixed number. 
   
• Regular practice is a key to master fractions. Solve various fraction problems daily to build confidence and proficiency. 
  
   

Proper fractions are commonly used across different fields and industries. Here are a few real-world applications:

• Cooking: Proper fractions are used in recipes to measure ingredients like flour, milk, and sugar.
   
• Construction: During construction work, measurements like 5/8 inches or feet are used to cut materials efficiently and accurately.
   
• Finance and budgeting: Proper fractions are used in dividing money when allocating a budget or trying to divide your salary for rent and groceries.
   
• Time management: Fractions are used to divide hours or minutes. For instance, \(\frac{3}{4}\) of an hour for studying or \(\frac{1}{2}\) hour for exercise is often said in many occasions. 
   
• Sharing and distribution: Proper fractions are applied when sharing items among people. For instance, a cake is divided into 8 equal parts, and taking 3 pieces represents \(\frac{3}{8}\) of the cake.