Introduction to Ordering Fractions: Examples and Applications :

In the field of mathematics, ordering fractions is how we arrange the fractions in two ways i.e. smallest to greatest and greatest to smallest. For this purpose, we need to compare the fractions. The LCM (least common multiple) of both denominators is used to determine whether the denominator is the same when comparing fractions. We can quickly determine the order of fractions after equating the denominators. It is assumed that the fraction with the larger numerator is bigger than the other. In this article, the process of ordering fractions, ways to determine the order of fractions, examples, and applications in everyday life will be discussed.

 

What is meant by the ordering of fractions ?

Comparing two or more fractions to discover whether one is bigger or smaller than the others is
known as ordering fractions.
Usually, this process is done to find the following orders of fractions

• Descending order (To arrange fractions from greater to smaller)

• Ascending order (To arrange fractions from smaller to greater)

  
  

  How to Order Fractions?

  
  Here are some steps to follow while ordering fractions:
  1) Find a common denominator for all fractions you want to order. For this, we have to
  take L.C.M of all denominators.
  2) Create a similar fraction for each fraction using the same denominator.
  3) Order the fraction by comparing their numerators.
  4) Simplify the ordered fractions if possible.

  
  Methods of ordering
  There are two common methods used for ordering fractions
  1. Equating the denominators of all fractions
  2. Converting fractions into decimal

  
  

  Examples
  (By the method of equating denominators)
  Example 1:

   Arrange the following in ascending order 2/3, 5/7, 5/3, 1/5, 3/10
  Solution:
  Step 1: Taking the L.C.M of all denominators of the above fractions, we have
  2 3,7,3,5,10
  3 3,7,3,5,5
  5 1,7,1,5,5
  7 1,7,1,1,1
  1,1,1,1,1
  
  L.C.M = 2×3×5×11×7 =210
  Step 2: To equate all the denominators i.e. 210
  (2/3) × 210/210 = 140/210
  (5/7) × 210/210 = 150/210
  (5/3) × 210/210 = 350/210
  (1/5) × 210/210 = 42/210
  
  (3/10) × 210/210 = 63/210
  Step 3: Arrange all fractions according to numerators like smallest to biggest, we have
  140 /210, 150 /210, 350 /210, 42 /210, 63 /210
  Hence the ascending order is 5/3, 5/7, 2/3, 3/10, 1/5

  
  

  Example 2: Arrange the following in descending order 1/3, 4/7, 2/9, 4/5, 7/10
  Solution:
  Step 1: Taking the L.C.M of all denominators of the above fractions, we have
  3 3,7,9,5,10
  3 1,7,3,5,10
  5 1,7,1,5,10
  2 1,7,1,1,2
  7 1,7,1,1,1
  1,1,1,1,1
  
  L.C.M = 3×3×5×2×7 = 630
  Step 2: To equate all the denominators i.e. 630
  (1/3) × 630/630 = 210/630
  (4/7) × 630/630 = 360/630
  (2/9) × 630/630 = 140/630
  (4/5) × 630/630 = 504/630
  (7/10) × 630/630 = 441/630
  Step 3: Arrange all fractions according to numerators like biggest to lowest, we have
  504 /630, 441 /630, 360 /630, 210 /630, 140 /630
  Hence the descending order is 4/5, 7/10, 4/7, 1/3, 2/9
  A least to greatest calculator can be used to verify the results of solved problems.

  
  

  Example 3: (By the method of converting the fractions into decimals)
  Arrange 2/3, 1/5, and 5/6 in ascending order.
  Solution:
  Step 1:
  Convert the fractions to decimals
  Divide the numerator by the denominator using long division or a calculator to change fractions
  to decimals. Once all of the fractions have been converted to decimals, you may compare them
  to discover which one is the smallest or largest.
  For example, let's say we want to order the fractions 2/3, 1/5, and 5/6 in decimal form:
  2/3 = 0.666666...
  1/5 = 0.20
  5/6 = 0.833333...
  Now that we have to change the fractions to decimals, we can see that the order from least to
  greatest is:
  
  1/5 = 0.20
  < 2/3 = 0.666666...
  < 5/6 = 0.833333...
  Now,
  1/5 < 2/3< 5/6
  Example 4: Check the order in the following fractions by the method of converting decimals.
  1/5, 3/10, 2/3, 5/7, 8/11
  Solution:
  Converting the given fractions into decimals, we have
  1/5 = 0.20
  3/10 = 0.30
  2/3 = 0.66
  5/7 = 0.71
  8/11 = 0.82
  Clearly, we can say that the given order of fractions is ascending as it starts from the lowest one
  and ends at the greatest one.
  1/5 < 3/10 < 2/3 < 5/7 < 8/11

  
  Applications

  
  Fractional thinking comes in quite handy in everyday life. Fractions frequently lend a helping
  hand in solving our issues. Here are some instances when the ordering-of-fractions approach is
  frequently employed:

  In Shopping: 

  Whenever we go outside shopping, we often see discounts on different
  articles of different brands. For example, if we have to buy a shoe on which 40% is off,
  we have to use the concept of fractions to find out the payment we have to pay

  > In Cooking: 

  While cooking, we have to keep check and balance in the ingredients
  required for a specific recipe. Without keeping the order or sequence, our recipe will
  not be tasteful. After ordering the recipe’s required material, we can serve a tasty
  recipe in front of guests including you as well. Pizza is the best example to modify the
  
  concept of fraction, as on average a large pizza has 8 slices. If you are 4 friends, all of
  you will get 8/2 slices equally.

  In Sports: In all games, we have to follow the rules that are set by game officials.
  Without keeping balance in them, we can’t qualify for it. For this purpose, fractions are
  used. For example, in a T20 Cricket match, a single bowler can bowl only 20/4 overs per
  inning. No more than 20/4 overs are allowed for all other bowlers as well.

  > In the Medical field: For medical purposes, the dosage of suitable medicine prescribed by a doctor varies from age to age. For example for an adult person, a specific medicine is prescribed 4/5 tablespoon as per dosage by a doctor and only 2/5 is prescribed for a child of age less than 10.
   

  Summary
  In the discussion above, we learned about the idea of ranking fractions as well as many
  approaches and instances. Both approaches may be utilized to locate a precise answer at any
  place. The use of fractions in daily life acknowledges both the value of fractions and their ability
  to be ordered in many ways.