In a Proper fraction, the numerator usually gives us an idea of how many parts of the denominator we are going to use, while the denominator is the total number of parts we have.

A good way to remember the term proper fraction is to start thinking that this is the best way to work with fractions in general. In proper fractions, the top number or the numerator is smaller than the bottom number or the denominator.

$\frac{1}{2}$ Is proper fraction compared to $\frac{2}{1}$

## Proper Fraction Definition

Proper fraction is one where the numerator or the number on top is less than the denominator or the number at the bottom and the overall value is less than 1.

Examples: $\frac{4}{9}$, $\frac{2}{5}$, $\frac{15}{16}$, $\frac{5}{6}$

In the above example, the numerator (top number) is less than the denominator (bottom number).

We can also explain proper fractions by saying that it is not proper to allow a bigger number to sit on top of an smaller number so for a proper fraction to work it is advisable to put the smaller number on top.

Let us try some of these now..

### How do I write a proper fraction?

The word fraction would tell us a part of something and it represents a type of numeral. In most of cases, the quotient of two integers with of course the top number being called the numerator which we sometimes denote as (the number of parts), and the bottom number being the denominator (how many parts the whole is divided into). These when written out, the numerator and the denominator would be separated by "/" or "_".

Usually we denote a fraction by $\frac{a}{b}$ in which both "a" and "b" are the whole numbers and "b" is not equal to zero.

Any rational number between zero and 1 can be represented by fractions and if the quotient is less than one, such as $\frac{1}{5}$ or $\frac{2}{7}$, then it is called a proper fraction.

Anything other than this representation would be considered as an improper where the quotient is greater than one or in other words if the numerator of a fraction is larger than the denominator such as $\frac{18}{8}$

Can a fraction's numerator be equal to zero?
In the context of division we have learned that it is not possible or proper to be divided by zero, which is categorized as undefined.
But we do can have zero as our numerator and any fraction with zero as numerator would result in zero so all the fractions like $\frac{0}{2}$, $\frac{0}{450}$, $\frac{0}{29}$ equal zero.

## Types of Proper Fraction

When we talk about the types of proper fractions we actually look for the denominator with like numbers and unlike numbers.

So, if a pair of fraction have the same denominators we categorize them as like fractions whereas fractions with different denominators are considered as unlike fractions.

Like fraction:

With like fractions we could add and subtract easily and all we need to do is add or subtract the numerators and put the final value over the common denominator.

$\frac{2}{13}$ + $\frac{1}{13}$ = $\frac{(2 + 1)}{13}$ = $\frac{3}{13}$

Unlike fraction:

To begin working with unlike fractions we need to find the equivalent fractions with the same denominator.
(a) Find the LCM or smallest multiple of both numbers
(b) Write the fractions in the form of equivalent fractions with the least common multiple as the denominator (now when we work with fractions the least common multiple is referred to as the least common denominator.

With unlike fraction we always need to compare the denominators.

Example: $\frac{4}{5}$ + $\frac{6}{8}$, Need to find the LCM for the denominators, for 5 and 8 the LCM is 5 × 8 = 40

$\frac{4 \times (8)}{5 \times (8)}$ + $\frac{6 \times (5)}{8 \times (5)}$ = $\frac{32}{40}$ + $\frac{30}{40}$ = $\frac{62}{40}$ = 1 $\frac{22}{40}$

Let us get the addition part of proper fractions.

When we look at the operation of addition or subtraction we need to make sure that the fractions and their measurements are performed properly.

Step 1: We have to determine first if both the fractions have the same denominator. If not then fractions must be converted to equivalent fractions with the same denominator.
Step 2: Next we have to write the equivalent fractions using the least common multiple of the two denominators as the new denominator in both fractions.
Step 3: Next we have to add only the numerators of the new fractions as provided in the problem and keeping the denominator unchanged as the equivalent fraction’s denominator.
Step 4: At the end we have to write the final answer in the simplest form or as a mixed number.

Let the two fractions be $\frac{5}{9}$ and $\frac{7}{11}$
$\frac{5}{9}$ + $\frac{7}{11}$
$\frac{5 \times (11)}{9 \times (11)}$ + $\frac{7 \times (9)}{11 \times (9)}$
$\frac{(55 + 63)}{99}$ = $\frac{118}{99}$ = 1 $\frac{19}{99}$

### Word Problem on Adding Proper Fraction:

Norman’s family ordered a king size pizza. They ate one fifth of the pizza at the pizza center and another one fifth in the car on the way home. In all, how much pizzas were eaten before they got home?

Solution:

Tips: Read and understand the problem.
Read the problem carefully and find out what are the important information you have right now.
The family ate $\frac{1}{5}$ of the pizza in the pizza center and $\frac{1}{5}$ of the pizza their way home.
What are you trying to find? How much of the pizza they ate before they got home.
So what is happening in the solution? How much they ate in pizza center and how much more they ate on their way home.

Norman’s family ate one fifth and then ate another one fifth of the pizza so we can just add both these and find out how much they ate in total before they got back home. So we have the following fractions added $\frac{1}{5}$ + $\frac{1}{5}$ = $\frac{2}{5}$ of the pizza eaten before they reached home.

### More Examples on Adding Proper Fractions:

1) Find the sum $\frac{2}{5}$ + $\frac{3}{5}$ ?

Solution:?
Step 1: Add the numerators = (2 + 3) = 5
Step 2: Retain the common denominator = 5
Step 3: Sum of the fractions = $\frac{(Sum\ of\ Numerators)}{(Common\ Denominators)}$
$\Rightarrow$ $\frac{5}{5}$
Step 4: $\frac{5}{5}$ = 1

2) Find the sum $\frac{4}{9}$ + $\frac{7}{9}$?

Solution:
Step 1: Add the numerators = (4 + 7) = 11
Step 2: Retain the common denominator = 9
Step 3: Sum of the fractions = $\frac{(Sum\ of\ Numerators)}{(Common\ Denominators)}$ $\Rightarrow$ $\frac{11}{9}$
Step 4: $\frac{11}{9}$

Steps to Add Proper Fractions with Different Denominators:
Step 1: Find the lowest or least common denominator (LCD) of the fractions.
Step 2: Express the fractions as equivalent fractions with the denominator as the LCD.
Step 3: Add the numerators of the fractions.
Step 4: Simplify the fraction if necessary.

### Examples on Adding Proper Fractions with Different Denominators :

1) Find the sum $\frac{1}{6}$ + $\frac{7}{8}$?

Solution:
Step 1: LCD of the given fractions = least common multiple (LCM) of 6 and 8 = 24
Step 2: Express the given fraction as an equivalent fraction with the common denominator 24
$\frac{1}{6}$ = $\frac{(1 \times 4)}{(6 \times 4)}$ = $\frac{4}{24}$
$\frac{7}{8}$ = $\frac{(7 \times 3)}{(8 \times 3)}$ = $\frac{21}{24}$
Step 3: Add the numerators: $\frac{4}{24}$ + $\frac{21}{24}$ $\Rightarrow$ $\frac{(4 + 21)}{24}$ $\Rightarrow$ $\frac{25}{24}$
Step 4: $\frac{25}{24}$

2) Find the sum $\frac{5}{12}$ + $\frac{11}{15}$?

Solution:
Step 1: LCD of the given fractions = least common multiple (LCM) of 12 and 15 = 60
Step 2: Express the given fractions as an equivalent fraction with the common denominator 60
$\frac{5}{12}$ = $\frac{(5 \times 5)}{(12 \times 5)}$ = $\frac{25}{60}$
$\frac{11}{15}$ = $\frac{(11 \times 4)}{(15 \times 4)}$ = $\frac{44}{60}$
Step 3: Add the numerators: $\frac{25}{60}$ + $\frac{44}{60}$ $\Rightarrow$ $\frac{(25 + 44)}{24}$ = $\frac{69}{60}$
Step 4: $\frac{69}{60}$

## Subtracting Proper Fractions

Let us understand the subtraction part of proper fractions.

We have to repeat the same methods of finding the LCM for fractions to be subtracted and follow it up with the steps given below.

Step 1: First determine if both the fractions have the same denominator. If not then the fractions must be converted to equivalent fractions with the same denominator.
Step 2: Next write the equivalent fractions using the least common multiple of the two denominators as the new denominator in both fractions.
Step 3: Next subtract only the numerators of the new fractions as provided in the problem keeping the denominator unchanged as the equivalent fraction’s denominator.
Step 4: At the end write the final answer in the simplest form or as mixed number.

Let’s take two fractions for subtractions
as $\frac{5}{11}$ and $\frac{3}{10}$ so we have $\frac{5}{11}$ - $\frac{3}{10}$
Find the LCM of the denominators.
$\frac{5 \times (10)}{11 \times (10)}$ - $\frac{3 \times (11)}{10 \times (11)}$
$\frac{50}{110}$ - $\frac{33}{110}$ = $\frac{17}{110}$

### Examples on Subtracting Proper Fractions

1) Find $\frac{3}{7}$ - $\frac{2}{7}$?

Solution:
Step 1: The difference between the numerators = 3 - 2 = 1
Step 2: Retain the common denominator = 7
Step 3: The difference between proper fractions = $\frac{(Difference\ between\ the\ Numerators)}{Common\ Denominator}$ = $\frac{8}{23}$
Step 4: $\frac{1}{7}$

2) Find $\frac{11}{23}$ - $\frac{3}{23}$?

Solution:
Step 1: The difference between the numerators = 11 - 3 = 8
Step 2: Retain the common denominator = 23
Step 3: The difference between the proper fractions = $\frac{(Difference\ between\ the\ Numerators)}{Common\ Denominator}$ = $\frac{8}{23}$
Step 4: $\frac{8}{23}$

Steps to Subtract Proper Fractions with Different Denominators:
Step 1: Find the lowest or least common denominator (LCD) of the fractions.
Step 2: Express the fractions as equivalent fractions with the denominator as the LCD.
Step 3: Find the difference between the numerators.
Step 4: Simplify the fraction if necessary.

Examples on Subtracting Proper Fractions with Different Denominators:

1) Find $\frac{5}{9}$ - $\frac{3}{8}$?

Solution:
Step 1: LCD of the given fractions = least common multiple (LCM) of 9 and 8 = 72
Step 2: Express the fractions as equivalent fractions with the common denominator 72
$\Rightarrow$ $\frac{5}{9}$ = $\frac{(5 \times 8)}{(9 \times 8)}$ = $\frac{40}{72}$
$\Rightarrow$ $\frac{3}{8}$ = $\frac{(3 \times 9)}{(8 \times 9)}$ = $\frac{27}{72}$
Step 3: Subtract the numerators: $\frac{40}{72}$ - $\frac{27}{72}$
$\Rightarrow$ $\frac{(40 - 27)}{72}$ = $\frac{13}{72}$
Step 4: $\frac{13}{72}$

2) Find $\frac{7}{12}$ - $\frac{5}{24}$?

Solution:
Step 1: LCD of the given fractions = least common multiple (LCM) of 12 and 24 = 24
Step 2: Express the fractions as equivalent fractions with the common denominator 24
$\Rightarrow$ $\frac{7}{12}$ = $\frac{(7 \times 2)}{(12 \times 2)}$ = $\frac{14}{24}$
$\Rightarrow$ $\frac{5}{24}$ = $\frac{(5 \times 1)}{(24 \times 1)}$ = $\frac{5}{24}$
Step 3: Subtract the numerators: $\frac{14}{24}$ - $\frac{5}{24}$
$\Rightarrow$ $\frac{(14 - 5)}{24}$ = $\frac{9}{24}$
Step 4: Divide the numerator and denominator by 3
$\Rightarrow$ $\frac{(9 \div 3)}{(24 \div 3)}$ = $\frac{3}{8}$

## Multiplying Proper Fractions

Multiplying a fraction is found to be much easier than adding or even subtracting them. It is easier because in multiplication we do not need to find a common denominator instead we need to change the mixed numbers if any into improper fractions and then put it back to mixed form after the calculations.

Steps for multiplying proper fractions:
Step 1: Multiply the numerators together and the denominators together.
Step 2: Reduce the answer if necessary.

Let us do a simple multiplication with proper fractions.
Let the fractions be $\frac{2}{5}$ and $\frac{3}{7}$
Solving:
Step 1: multiply the denominators 5 and 7
Step 2: multiply the numerators 2 and 3
Step 3: write the values in fractional form

$\frac{2}{5}$ $\times$ $\frac{3}{7}$ = $\frac{(2 \times 3)}{(5 \times 7)}$ = $\frac{6}{35}$

Word problem on proper fraction multiplication:

Question: Amy ate $\frac{4}{5}$ of a $\frac{1}{2}$ pound cake. How much cake did she eat?

Step 1: Multiply the denominators 5 and 2
Step 2: Multiply the numerators 4 and 1
Step 3: Write the values in their fractional form

Amy ate $\frac{4}{5}$ $\times$ $\frac{1}{2}$ = $\frac{4 \times 1}{5 \times 2}$ = $\frac{4}{10}$ or $\frac{2}{5}$ of the cake.

Examples on Multiplying Proper Fractions:

1) Find the product of $\frac{2}{7}$ and $\frac{3}{8}$?

Solution:
Step 1: Multiply the numerators = 2 x 3 = 6
Step 2: Multiply the denominators = 7 x 8 = 56
Step 3: The product of the proper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{6}{56}$
Step 4: Divide numerator and denominator by 2
$\Rightarrow$ $\frac{(6 \div 2)}{(56 \div 2)}$
$\Rightarrow$ $\frac{3}{28}$

2) Find the product of $\frac{3}{5}$ and $\frac{2}{9}$?

Solution:
Step 1: Multiply the numerators = 3 x 2 = 6
Step 2: Multiply the denominators = 5 x 9 = 45
Step 3: The product of the proper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{6}{45}$
Step 4: Divide numerator and denominator by 3 $\Rightarrow$ $\frac{(6 \div 3)}{(45 \div 3)}$ $\Rightarrow$ $\frac{2}{15}$

3) Find the product of $\frac{8}{15}$ and $\frac{2}{15}$?

Solution:
Step 1: Multiply the numerators = 8 x 2 = 16
Step 2: Multiply the denominators = 15 x 15 = 225
Step 3: The product of the proper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{16}{225}$
Step 4: $\frac{16}{225}$

4) Find the product of $\frac{3}{11}$ and $\frac{2}{11}$?

Solution:
Step 1: Multiply the numerators = 3 x 2 = 6
Step 2: Multiply the denominators = 11 x 11 = 121
Step 3: The product of the proper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{6}{121}$
Step 4: $\frac{6}{121}$

## Dividing Proper Fractions

Dividing a proper fraction involves inverting the divisor.
The number that is being divided is the dividend and the number that divides into the dividend is the divisor and in a fractional division we need to invert the divisor of the fraction.

Steps for dividing proper fraction:
Step 1: Write each number in a fraction form.
Step 2: Invert the divisor and change $\div$ into $\times$ sign.
Step 3: Follow the rules for multiplying.

Let us work out one such division for proper fraction.
Let us divide 7 by $\frac{3}{5}$
Step 1: Write 7 as an improper fraction with a denominator of one (1)
Step 2: Invert into and change the $\div$ sign into $\times$
Step 3: Follow the rules for multiplication.

7 $\div$ $\frac{3}{5}$ into $\frac{7}{1}$ $\times$ $\frac{5}{3}$

$\frac{7}{1}$ $\div$ $\frac{3}{5}$ = $\frac{7}{1}$ $\times$ $\frac{5}{3}$ = $\frac{35}{3}$ = 11 $\frac{2}{3}$

Let’s work out another one
Divide $\frac{5}{11}$ $\div$ $\frac{7}{13}$
Steps of division:
Step 1: Invert the divisor $\frac{7}{13}$ into $\frac{13}{7}$ and change the $\div$ into $\times$ sign
Step 2: Follow the rules for multiplication.

$\frac{5}{11}$ $\div$ $\frac{7}{13}$ into $\frac{5}{11}$ $\times$ $\frac{13}{7}$
$\frac{5}{11}$ $\div$ $\frac{7}{13}$ = $\frac{5}{11}$ $\times$ $\frac{13}{7}$ = $\frac{65}{77}$

Examples on Dividing Proper Fractions:

1) Divide $\frac{3}{7}$ with $\frac{2}{3}$?

Solution:
Step 1: Reciprocal of the second fraction = $\frac{2}{3}$
$\Rightarrow$ $\frac{3}{2}$
Step 2: Multiply the numerators and denominators of both the fractions
$\Rightarrow$ $\frac{3}{7}$ $\times$ $\frac{3}{2}$
$\Rightarrow$ $\frac{(3 \times 3)}{(7 \times 2)}$
$\Rightarrow$ $\frac{9}{14}$
Step 3: $\frac{9}{14}$

2) Divide $\frac{3}{13}$ with $\frac{2}{9}$?

Solution:
Step 1: Reciprocal of the second fraction $\frac{2}{9}$ is $\frac{9}{2}$
Step 2: Multiply the numerator and denominator of both the fractions
$\Rightarrow$ $\frac{3}{13}$ $\times$ $\frac{9}{2}$
$\Rightarrow$ $\frac{(3 \times 9)}{(13 \times 2)}$
$\Rightarrow$ $\frac{27}{26}$
Step 3: $\frac{27}{26}$

3) Divide $\frac{2}{15}$ with $\frac{7}{15}$?

Solution:
Step 1: Reciprocal of the second fraction $\frac{7}{15}$ is $\frac{15}{7}$
Step 2: Multiply the numerator and denominator of both the fractions
$\Rightarrow$ $\frac{2}{15}$ $\times$ $\frac{15}{7}$
$\Rightarrow$ $\frac{(2 \times 15)}{(15 \times 7)}$
$\Rightarrow$ $\frac{30}{105}$
Step 3: Divide numerator and denominator by 15, we get
$\Rightarrow$ $\frac{(30 \div 15)}{(105 \div 15)}$
$\Rightarrow$ $\frac{2}{7}$

4) Divide $\frac{4}{11}$ with $\frac{5}{11}$?

Solution:
Step 1: Reciprocal of the second fraction $\frac{5}{11}$ is $\frac{11}{5}$
Step 2: Multiply the numerator and denominator of both the fractions
$\Rightarrow$ $\frac{4}{11}$ $\times$ $\frac{11}{5}$
$\Rightarrow$ $\frac{(4 \times 11)}{(11 \times 5)}$
$\Rightarrow$ $\frac{44}{55}$
Step 3: Divide numerator and denominator by 11
$\Rightarrow$ $\frac{(44 \div 11)}{(55 \div 11)}$
$\Rightarrow$ $\frac{4}{5}$

## Comparing Proper Fraction

On some occasion we need to compare two fractions to find out the bigger of the two. There are three ways we can do this and that can either be done by multiplication, by using same denominator or by comparing the decimals.

How to compare fractions:

Method 1:
To compare fractions we will use multiplication.
Step 1: Write the fractions leaving enough space above and between them.
Step 2: Multiply each denominator by the other numerator making a criss-cross pattern or X.
Step 3: Compare the fraction.

$\frac{1}{5}$ or $\frac{3}{7}$ ?

7 $\times$ 1 = 7 and 5 $\times$ 3 = 15 and in this case 15 greater than 7
So, $\frac{1}{5}$ < $\frac{3}{7}$

Method 2:
Compare by using the same denominator for the fractions
Step 1: Write the LCM of the denominators of the fractions used and make the fractions equivalent.
Step 2: Use the factor of the LCM to multiply both the numerator and the denominator.
Step 3: Compare the fractions.

$\frac{2}{3}$ or $\frac{3}{5}$ ?
$\frac{(2 \times 5)}{(3 \times 5)}$ and $\frac{3 \times 3}{5 \times 3}$
$\frac{10}{15}$ and $\frac{9}{15}$
And so we have 10 > 9 therefore, $\frac{2}{3}$ > $\frac{3}{5}$

### Examples on Comparing Proper Fractions:

1) Compare the fractions $\frac{7}{8}$ and $\frac{2}{8}$?

Solution:
Step 1: Compare the numerators
$\Rightarrow$ 7 is greater than 2
Step 2: So, $\frac{7}{8}$ > $\frac{2}{8}$

2) Which of the fractions $\frac{14}{17}$ and $\frac{16}{17}$ is greater?

Solution:
Step 1: Compare the numerators, 14 is less than 16
Step 2: So, $\frac{14}{17}$ < $\frac{16}{17}$

3) Compare the fractions $\frac{5}{8}$ and $\frac{3}{11}$?

Solution:
Step 1: Least common multiple of the denominators (LCD), 8 and 11 = 88
Step 2: Write the fractions in equivalent fraction form with the denominator as the LCD.
$\Rightarrow$ $\frac{5}{8}$ = $\frac{(5 \times 11)}{(8 \times 11)}$ = $\frac{55}{88}$
$\Rightarrow$ $\frac{3}{11}$ = $\frac{(3 \times 8)}{(11 \times 8)}$ = $\frac{24}{88}$
Step3: Compare the numerators
$\Rightarrow$ 55 is greater than 24
So, $\frac{5}{8}$ > $\frac{3}{11}$

4) Compare the fractions $\frac{6}{7}$ and $\frac{5}{8}$?

Solution:
Step 1: Least common multiple of the denominators (LCD), 7 and 8 = 56
Step 2: Write the fractions in an equivalent fraction form with the denominator as the LCD.
$\Rightarrow$ $\frac{6}{7}$ = $\frac{(6 \times 8)}{(7 \times 8)}$ = $\frac{48}{56}$
$\Rightarrow$ $\frac{5}{8}$ = $\frac{(5 \times 7)}{(8 \times 7)}$ = $\frac{35}{56}$
Step3: Compare the numerators, 48 is greater than 35
So, $\frac{6}{7}$ > $\frac{5}{8}$

### Practice Problems on Comparing Proper Fractions:

Problem 1: Compare the fractions $\frac{3}{7}$ and $\frac{2}{9}$?
Problem 2: Compare the fractions $\frac{5}{12}$ and $\frac{13}{17}$?

Tips for better result:
The more you practice the better you will get.
If you get stuck, move to the next problem.
Solve the ones you know first and you will feel more confident.
Go back later and the try the problems you skipped.

## Ordering of Proper Fractions

Ordering of proper fractions is nothing but an arrangement of these fractions from the least to the greatest or from the greatest to the least.

This can be done by comparing it first and then set the order for each participating fractions.

Steps of Ordering:
Step 1: Find the LCM of the denominators
Step 2: multiplying numerator and denominator of each fraction by a number to make the LCM
Step 3: Ordering the fractions with like denominators

Let us do one comparing and ordering of fractions.
$\frac{2}{5}$, $\frac{4}{7}$, $\frac{5}{8}$
The LCM for the denominators is 280
Multiply the numerator and the denominator of each fraction by the fractor to make LCM
$\frac{2 \times 56}{5 \times 56}$ = $\frac{112}{280}$ ; $\frac{4 \times 40}{7 \times 40}$ = $\frac{160}{280}$ ; $\frac{5 \times 35}{8 \times 35}$ = $\frac{175}{280}$
From the above values the ordering in least to greatest would be
$\frac{2}{5}$ < $\frac{4}{7}$ < $\frac{5}{8}$

### Examples on Ordering Proper Fractions with the Same Denominators:

1) Write the order of the given fractions: $\frac{5}{8}$, $\frac{7}{8}$, $\frac{1}{8}$, $\frac{3}{8}$?

Solution:

Step 1: Compare the numerators, 1 < 3 < 5 < 7
Step 2: So, $\frac{1}{8}$ < $\frac{3}{8}$ < $\frac{5}{8}$ <$\frac{7}{8}$

2) Write the order of the given fractions: $\frac{15}{11}$, $\frac{17}{11}$, $\frac{12}{11}$, $\frac{13}{11}$?

Solution:
Step 1: Compare the numerators $\Rightarrow$ 12 < 13 < 15 < 17
Step 2: So, $\frac{12}{11}$ < $\frac{13}{11}$ < $\frac{15}{11}$ < $\frac{17}{11}$

### Examples on Ordering Proper Fractions with Different Denominators:

1) Write the order of the given fractions: $\frac{2}{7}$, $\frac{4}{9}$, $\frac{1}{6}$, $\frac{3}{14}$?

Solution:
Step 1: Least common multiple of the denominators (LCD) 7, 9, 6 and 14 = 126
Step 2: Make the fractions as equivalent fractions with the denominator as the LCD.
$\frac{2}{7}$ = $\frac{(2 \times 18)}{(7 \times 18)}$ = $\frac{36}{126}$
$\frac{4}{9}$ = $\frac{(4 \times 14)}{(9 \times 14)}$ = $\frac{56}{126}$
$\frac{1}{6}$ = $\frac{(1 \times 21)}{(6 \times 21)}$ = $\frac{21}{126}$
$\frac{3}{14}$ = $\frac{(3 \times 9)}{(14 \times 9)}$ = $\frac{27}{126}$

Step 3: Compare the numerators $\Rightarrow$ 21< 27 < 36 < 56
Step 4: So, $\frac{1}{6}$ < $\frac{3}{14}$ < $\frac{2}{7}$ < $\frac{4}{9}$

2) Write the order of the given fractions: $\frac{1}{5}$, $\frac{3}{25}$, $\frac{2}{9}$, $\frac{4}{3}$?

Solution:
Step 1: Least common multiple of the denominators (LCD) 5, 25, 9 and 3 = 225
Step 2: Make the fractions as equivalent fractions with the denominator as the LCD.
$\frac{1}{5}$ = $\frac{(1 \times 18)}{(5 \times 45)}$ = $\frac{18}{225}$
$\frac{3}{25}$ = $\frac{(3 \times 9)}{(25 \times 9)}$ = $\frac{27}{225}$
$\frac{2}{9}$ = $\frac{(2 \times 25)}{(9 \times 25)}$ = $\frac{50}{225}$
$\frac{4}{3}$ = $\frac{(4 \times 75)}{(3 \times 75)}$ = $\frac{300}{225}$

Step 3: Compare the numerators $\rightarrow$ 18 < 27 < 50 < 300
Step 4: So, $\frac{1}{5}$ < $\frac{3}{25}$ < $\frac{2}{9}$ < $\frac{4}{3}$

### Practice Problems on Ordering Proper Fractions:

Problem 1: Write the order of the given fractions: $\frac{2}{7}$, $\frac{1}{5}$, $\frac{3}{14}$, $\frac{4}{21}$?
Problem 2: Write the order of the given fractions: $\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{6}$, $\frac{7}{8}$?