Fractions - Ascending or Descending order of Rational Numbers

FRACTIONS

"Arrange the following fractions... or find the smallest or greatest fraction" is a common question in exam. In this post we will learn tricks to arrange fractions in ascending or descending order within mind. There are various methods to arrange fractions or select fraction like take the LCM of denominator and compare the numerators or divide the numerator by denominator and convert the fraction in decimal numbers. But such method takes time. This new method is method of ratio to select smallest or largest fraction.

What is a fraction?

When anything is divided in certain number of parts, then each part is known as Fraction of that unit. Numbers written in the form of p/q are known as Rational Numbers or Fractions.

For example: 1/2, 3/4, 7/15 etc. We can also write fractions in the form of decimal like 0.5, 0.75, 0.46 etc. or in the form of percentage like 50%, 75%, 46% etc.

A fraction represents a part of the whole. It consist two parts: Numerator and Denominator.
Numerator represents number of equal parts we count or we have while denominator represents total number of parts the whole is divided into. Remember everything is 100% in itself.

Now the questions is: How to identify which fraction is smaller and which one is larger?
For example if someone offers us to choose profit share of 3/5 or 7/10, then how will we identify which one is greater? or if we face any question in exam to arrange fractions in ascending or descending order then how will we arrange?

One method is simply divide the numerator by denominator and convert the fraction into decimal. This is most accurate but it takes time. In this post i will explain other method to arrange fractions in ascending or descending order.

Rules to compare fractions:

 1.  For fractions increasing with constant ratio: 

• If (Increase in numerator/Increase in denominator) > first fraction, the last fraction is greatest.

• If (Increase in numerator/Increase in denominator) < first fraction, the last fraction is smallest.

• If (Increase in numerator/Increase in denominator) = first fraction, all the fractions are equal.

Example 1: Arrange the following fractions in ascending order:

a.) 3/4 ; b.) 4/5 ; c.) 5/6 

Solution: Here numerator is increasing by (4-3) = 1

      And Denominator is increasing by (5 - 4) = 1

Hence numerator and denominator are increasing by a ratio of 1 so, 4/5 > 3/4

       Similarly, 5/6 > 4/5

Hence ascending order = 3/4 < 4/5 <  5/6

In the above example we only need to see the difference between numerators and denominators and we will be able to determine which fraction is smaller or which one is larger.

Example 2: Which of the following fraction is smallest?

a.) 2/7 ; b.) 4/15 ; c.) 6/23

Solution: Here numerator is increasing by 2 and denominator is increasing by 8.

Ratio is: 2/8 which is smaller than the first fraction so smallest fraction will be: 6/23.

 2.  For fractions increasing with variable ratio:

We can use above method in this case too to find smallest or greatest fraction but to arrange multiple fractions in ascending or descending order this method needs little practice.

Example 1: Find the greatest fraction from the given fractions: 3/7, 4/5, 7/9, 1/2, 3/5
Solution: Compare first two fractions: Numerator increased by 1 but denominator decreased by 2. Clearly 4/5 is greater.
Now compare 4/5 with 7/9, Ratio = 3/4. Hence 4/5 > 7/9.
Now compare 1/2 with 4/5, Ratio = 3/3. Hence 4/5 is greater.
At last 4/5 is clearly visible to be greater than 3/5.
       ⇒ Greatest Fraction = 4/5

Here we don't need to write these lines again and again, just calculate ratio in mind and decide which fraction is greater or which fraction is smaller. It saves a lot of time as compared to taking LCM and comparing numerators or converting each fraction in percentage or decimal.

Example 2: Arrange the following in ascending order: 3/7, 4/5, 7/9, 1/2, 3/5
Solution: Start with fractions having smaller numerators as compared to others.

Compare 1/2 with 3/5, here numerator increased by 2 and denominator by 3.

Ratio = 2/3 which is greater than the first fraction. It means 3/5 > 1/2.

Now compare 3/5 with 3/7, clearly 3/5 > 3/7.

Again we have to compare 3/7 with 1/2 , Ratio will be 2/5, it means 1/2 > 3/7.

Now denominator are same so 4/5 > 3/5.

At last 7/9 will be smaller than 4/5 but larger than 3/5.

Ascending order = 3/7 < 1/2 < 3/5 < 7/9 < 4/5

It seems this method may take more time than division method depends on the number of fractions because we have to compare each fraction with one another. Also for fractions with smaller digits conventional method of changing the values into decimal will be more convenient for some.

Division method: 3/7 = 0.42 ; 4/5 = 0.8 ; 7/9 = 0.77 ; 1/2 = 0.5 ; 3/5 = 0.6
Ascending order = 3/7 < 1/2 < 3/5 < 7/9 < 4/5

Both the methods are better than taking LCM of denominators and multiplying numerator to decide which fraction is smallest or largest. Rest depends on the calculation and speed of candidates. Good Luck 👍