Table Of Contents
What are Proper Fractions?
Every fraction consists of two parts, a numerator and a denominator. In this situation, the numerator represents the number of equal parts that are being counted and the denominator represents how many equal parts are there in the whole. If the numerator of a fraction is less than the denominator, the fraction is said to be a proper fraction. The value of a proper fraction is always less than 1 and greater than 0. Fractions such as \(\frac{2}{5},\frac{3}{8},and\frac{7}{9}\) are a few examples of proper fractions.
Proper fractions are categorized into two types:
- Like fractions – Fractions with the same denominators.
Example – \(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}.\)
- Unlike fractions – Fractions with different denominators.
Example – \(\frac{3}{4},\frac{2}{5},\frac{5}{6}.\)
Operations on Proper Fractions:
Just like whole numbers, arithmetic operations like addition, subtraction, multiplication and division can be performed on proper fractions.
Addition and Subtraction of Proper Fractions:
In order to add two or more proper fractions, we need to follow these steps:
Step 1: Check whether the fractions are like fractions.
Step 2: If the fractions are like fractions, directly add or subtract the numerators and keep the denominator the same to find the result of the addition or subtraction operation.
Step 3: If the fractions are unlike fractions, rewrite the fractions as equivalent fractions to make them like fractions.
Step 4: Reduce the resulting fraction to its simplest form if needed.
Multiplication of Proper Fractions:
To multiply two or more proper fractions, follow the process mentioned below.
Step 1: Multiply the numerators first and then the denominators of both the fractions.
Step 2: Reduce the resulting fraction to its simplest form if needed.
Division of Proper Fractions:
Here are the steps to follow in order to divide two or more proper fractions:
Step 1: Write the divisor in its reciprocal form.
Step 2: Now, multiply the reciprocal of the divisor with the dividend.
Step 3: Reduce the resulting fraction to its simplest form if needed.
Difference Between Proper Fractions and Improper Fractions:
Rapid Recall:
Solved Examples:
Example 1:
Find the sum of fractions: \(\frac{17}{19}+\frac{5}{19}\)
Solution:
The operation provided: \(\frac{17}{19}+\frac{5}{19}\)
Since the fractions are like fractions, add the numerators.
\(=\frac{17}{19}+\frac{5}{19}\) [Like fractions]
\(=\frac{17+5}{19}\) [Adding the numerators]
\(=\frac{22}{19}\) [Add]
Therefore, \(=\frac{17}{19}+\frac{5}{19}=\frac{22}{19}\) .
Example 2:
What will be the difference between the fractions \(\frac{37}{20}and\frac{3}{10}\) ?
Solution:
The operation mentioned in the question \(=\frac{37}{20}-\frac{3}{10}\)
Here, the fractions are unlike fractions.
So, find the equivalent fraction of \(\frac{3}{10}\) with the denominator as 20, that is, \(\frac{6}{20}\)
\(=\frac{37}{20}-\frac{3}{10}\) [Unlike fractions]
\(=\frac{37}{20}-\frac{3\times2}{10\times2}\) [Rewrite \(\frac{3}{10}\) as \(=\frac{6}{20}\) resulting in equivalent fractions]
\(=\frac{37}{20}-\frac{6}{20}\) [Like fractions]
\(=\frac{37-6}{20}\) [Subtract the numerators]
\(=\frac{31}{20}\)
Hence, the difference between the fractions \(=\frac{37}{20} and \frac{3}{10} is \frac{31}{20}\).
Example 3:
Sam has baked a chocolate cake. His sister Jane ate a portion of the cake adding up to \(\frac{1}{4}th\) of the cake. What portion of the cake is remaining?
Solution:
Let’s assume the total portion of cake = 1
Jane ate \(\frac{1}{4}\) of the cake.
To find the leftover portion of the cake, subtract the quantity of cake eaten by Jane from the whole quantity of cake.
That is, 1
\(\Rightarrow 1-\frac{1}{4}\) [Unlike fractions]
\(\Rightarrow\frac{4}{4}-\frac{1}{4}\) [Rewrite 1 as \(\frac{4}{4}\) resulting in equivalent fractions]
\(\Rightarrow \frac{4-1}{4}\) [Subtract the numerators]
\(\Rightarrow \frac{3}{4}\) [Simplify]
Thus, the portion of the cake remaining is \(\frac{3}{4}th\) of the whole cake.
An improper fraction is a fraction in which the numerator is greater than the denominator.
There are three types of fractions:
- Proper fractions
- Improper fractions
- Mixed fractions
Mixed numbers or mixed fractions are fractions that consist of two parts, a whole number and a fraction.
Fractions that have the same denominator are referred to as like fractions.