Proper Fraction

A proper fraction is a fraction in which the numerator is smaller than the denominator. Thus, the value of a proper fraction is always less than 1.

${\dfrac{2}{3}}$, ${\dfrac{5}{8}}$, ${\dfrac{11}{28}}$, ${\dfrac{3}{7}}$, ${\dfrac{4}{13}}$, and ${\dfrac{15}{48}}$ are a few examples of proper fraction.

Visualizing

We can easily visualize a proper fraction using pie charts or bar models. 

For instance, if we divide a pie into two equal parts, the shaded 1 part represents the proper fraction ${\dfrac{1}{2}}$.

Similarly, dividing the pie into four equal parts, each part represents ‘quarter’ and is represented as ${\dfrac{1}{4}}$, which is also a proper fraction. 

Unlike an improper fraction, a proper fraction never completes the whole. Thus, a proper fraction cannot be converted to an improper fraction and vice versa.

On a Number Line

The value of a proper fraction always lies between 0 and 1, which can be easily represented on a number line. 

Let us represent the fraction ${\dfrac{5}{8}}$ on a number line.

We first look at the denominator to represent a proper fraction on a number line. The denominator indicates the number of divisions we need to make between 0 and 1.

Now, marking the divisions based on the numerator of the fraction,  

Here, the first interval is marked as ${\dfrac{1}{8}}$, the second is ${\dfrac{2}{8}}$, and so on, with the last mark being ${\dfrac{8}{8}}$, or 1. 

The numerator indicates where we should mark to represent the given fraction. Here, the numerator is 5. Thus, we mark the fifth interval to represent ${\dfrac{5}{8}}$.

Adding

While adding proper fractions, we consider two cases based on the denominators: whether they are the same or different.

Case 1: With the Same Denominators

Adding proper fractions with the same denominator is straightforward: keeping the denominator the same, we simply add the numerators.

For example,

${\dfrac{2}{7}+\dfrac{4}{7}}$

= ${\dfrac{2+4}{7}}$

= ${\dfrac{6}{7}}$

Case 2: With Different Denominators

When the fractions have different denominators, we find their LCM (Least Common Multiple) and then rewrite them as equivalent fractions using the LCM as a common denominator. Once the denominators are the same, we add the numerators.

For example,

${\dfrac{5}{8}+\dfrac{1}{4}}$

Here, the denominators are 8 and 4

Their LCM is 8

The denominator of the first fraction ${\dfrac{5}{8}}$ is 8 (which is the same as the LCM), while in the second fraction ${\dfrac{1}{4}}$ it is 4, which is different. 

Now, we convert ${\dfrac{1}{4}}$ to an equivalent fraction with 8 as the denominator.

${\dfrac{1\times 2}{4\times 2}}$ = ${\dfrac{2}{8}}$

Thus, the sum is 

${\dfrac{5}{8}+\dfrac{1}{4}}$ 

= ${\dfrac{5}{8}+\dfrac{2}{8}}$

= ${\dfrac{5+2}{8}}$

= ${\dfrac{7}{8}}$

Subtracting

Subtracting proper fractions follows the same principle as addition.

Case 1: With the Same Denominators

${\dfrac{4}{9}-\dfrac{2}{9}}$

= ${\dfrac{4-2}{9}}$

= ${\dfrac{2}{9}}$

Case 2: With Different Denominators

Let us subtract ${\dfrac{1}{4}}$ from ${\dfrac{5}{8}}$

Here, the denominators are 8 and 4

Their LCM is 8

The denominator of the first fraction ${\dfrac{5}{8}}$ is 8 (which is the same as the LCM), while in the second fraction ${\dfrac{1}{4}}$ it is 4, which is different. 

Now, we convert ${\dfrac{1}{4}}$ to an equivalent fraction with 8 as the denominator.

${\dfrac{1\times 2}{4\times 2}}$ = ${\dfrac{2}{8}}$

Thus, the difference is

${\dfrac{5}{8}-\dfrac{1}{4}}$ 

= ${\dfrac{5}{8}-\dfrac{2}{8}}$

= ${\dfrac{5-2}{8}}$

= ${\dfrac{3}{8}}$

Multiplying

To multiply proper fractions, we multiply the numerators and then the denominators together and, finally, simplify the result if needed.

For example,

${\dfrac{1}{2}\times \dfrac{5}{6}}$

= ${\dfrac{1\times 5}{2\times 6}}$

= ${\dfrac{5}{12}}$

Dividing

To divide a fraction by another, we multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor).

For example,

Let us divide ${\dfrac{1}{6}}$ by ${\dfrac{2}{3}}$

${\dfrac{1}{6}\div \dfrac{2}{3}}$

Here, the dividend is ${\dfrac{1}{6}}$ and the divisor is ${\dfrac{2}{3}}$

The reciprocal of ${\dfrac{2}{3}}$ is ${\dfrac{3}{2}}$

Now, multiplying ${\dfrac{1}{6}}$ by ${\dfrac{3}{2}}$, we get

${\dfrac{1}{6}\times \dfrac{3}{2}}$

= ${\dfrac{1\times 3}{6\times 2}}$

= ${\dfrac{3}{12}}$

= ${\dfrac{1}{4}}$

Proper vs. Improper Fractions

Basis	Proper Fraction	Improper Fraction
Numerators and denominators	The numerator is smaller than the denominator Numerator < Denominator	The numerator is greater than or equal to the denominator Numerator ≥ Denominator
Value	Its value is always between 0 and 1	Its value is always greater than or equal to 1
Examples	${\dfrac{3}{4}}$, ${\dfrac{5}{12}}$, and ${\dfrac{3}{7}}$	${\dfrac{4}{3}}$, ${\dfrac{12}{5}}$, and ${\dfrac{7}{3}}$

Addition with Mixed Fraction 

When adding a mixed fraction and a proper fraction, we first convert the mixed fraction to an improper fraction and then add the given fractions. After getting the sum, we convert the result to a mixed fraction again.

For example,

${1\dfrac{2}{5}+\dfrac{3}{7}}$

= ${\dfrac{7}{5}+\dfrac{3}{7}}$

= ${\dfrac{7\times 7}{5\times 7}+\dfrac{3\times 5}{7\times 5}}$

= ${\dfrac{49+15}{35}}$

= ${\dfrac{64}{35}}$

= ${1\dfrac{29}{35}}$

Solved Examples