Do you want to figure out which rational number is greater and smaller among a given set of rational numbers? Then this article is quite handy as it gives you a complete idea of the comparison between two rational numbers. We will discuss all about the classification of rational numbers, common facts about comparing rational numbers, how to compare two rational numbers with enough examples. Step by Step Solutions provided makes it easy for you to understand the concept as well as to improve your problem-solving skills.
Classification of Rational Numbers
Rational Numbers are a type of fractions. They are classified into the following types
Proper Rational Numbers: Proper Rational Numbers are numbers that are less than 1. In these kinds of rational numbers, the denominator is greater than the numerator. Example: 2/4, 5/6, 7/8 are all proper fractions.
Improper Rational Numbers: Improper Rational Numbers are numbers that are greater than 1. In these rational numbers, the numerator is greater than the denominator. Example 4/3, 7/6, 9/10 are all improper rational numbers.
Positive Rational Numbers: In the Case of Positive Rational Numbers both the numerator and denominator are either positive or negative. These are always greater than zero. Example: 3/4, -5/-6, -7/-8 are all Positive Rational Numbers.
Negative Rational Numbers: In this type of rational numbers, either numerator or denominator anyone is negative. These are always less than zero. Example: -3/4, 7/-8, 1/-2 are all Negative Rational Numbers.
Facts about Rational Numbers Comparison
Comparison of Rational Numbers is the same as Comparison of Integers and Fractions. Here is a list of common facts you need to know about Comparing Rational Numbers. They are along the lines
- Every positive rational number is greater than zero.
- Every rational number < 0 is a negative rational number.
- A Positive rational number is greater than a negative rational number.
- Every Rational number denoted by a point on the right of a number line is greater than all rational numbers represented by points on the number line’s left.
Also, See:
How do you Compare Two Rational Numbers?
Follow the simple steps listed below to compare two rational numbers. They are as under
- Firstly, identify the rational numbers from the given data.
- Later convert the denominators of the mentioned rational numbers to positive if they aren’t by simply multiplying both numerator and denominator with -1.
- Next, check if the given rational numbers are like rational numbers or unlike rational numbers.
- If the rational numbers are like fractions just check the numerators of the fractions and the one that is higher is the larger rational number. Remember to check if they are positive or negative rational numbers.
- If the rational numbers are unlike fractions calculate the LCM of the Denominators and express the fractions in terms of the common denominator and then compare them.
Let us understand the process better by considering an example.
Example:
Which is greater of the rational numbers -4/8, 5/-12?
Solution:
Given Rational Numbers are -4/8, 5/-12
Since the given rational numbers 5/-12 is having a negative denominator we need to multiply both numerator and denominator with -1 to get the positive denominator.
5/1-12 = 5*(-1)/-12*(-1) = -5/12
Now since the given rational numbers are unlike rational numbers let us calculate the LCM of denominators.
LCM(8, 12) = 24
Expressing the given rational numbers in terms of the common denominator we can rewrite them as
-4/8 = (-4*3)/(8*3) = -12/24
-5/12 = (-5*2)/(12*2) = -10/24
Now let us compare the numerators of both the rational numbers i.e. -12, -10
-10 is greater thus 5/-12 is the greater rational number.
Comparing Rational Numbers Examples
Example 1.
Compare the rational numbers -7/8, -9/10?
Solution:
Given Rational Numbers are -7/8, -9/10
Since the given rational numbers are unlike fractions let us find the LCM of Denominators.
LCM(8,10) = 40
Expressing the given rational numbers in terms of common denominators we have
-7/8 = -(7*5)/(8*5) = -35/40
-9/10 = (-9*4)/(10*4) = -36/40
Now compare the numerators of both the fractions i.e. -35, -36
Thus the fraction with numerator -35 is greater i.e. -7/8
Example 2.
Compare the Rational Numbers 1/3 and -4/3?
Solution:
Given Rational Numbers are 1/3 and -4/3
Here both the fractions are like fractions so we will check the numerators of the fractions and decide which one is greater.
1>-4
Therefore, rational number 1/3 is greater than -4/3
Example 3.
Compare rational numbers 5 and -7?
Solution:
Given Rational Numbers are 5/1 and -7/1
Here both the fractions are like fractions let us check the numerators of the fractions and decide which one os greater.
5>-7
Therefore, rational number 5 is greater than -7.