Category: Mathematics

We are all familiar with the concept of comparing two integers or two fractions and determining which is smaller or greater. Let us go a step ahead and compare Two Rational Numbers. We know fact that every positive integer is greater than 0, and a negative integer is less than 0. By knowing this fundamental rule we can infer some facts about how to compare rational numbers. They are listed below

1. Every Positive Rational Number is Greater than Zero.
2. Every Negative Rational Number is Less than Zero.
3. Comparison of Positive and Negative Rational Number is quite obvious i.e. Positive Rational Number is greater than a negative rational number.
4. Every Rational Number represented on a number line is greater than every other rational number represented present to its left.
5. Every Rational Number represented on a number line is less than every other rational number represented present to its right.

How to Compare Rational Numbers?

In order to compare any two rational numbers, you can go through the below-mentioned steps. They are as under

Step 1: Check the given rational numbers

Step 2: Write down the given rational numbers in a way that they have their denominators the same.

Step 3: Determine the Least Common Multiple of the Positive Denominators you obtained in the earlier step.

Step 4: Express rational numbers obtained in the second step using the LCM obtained as Common Denominator.

Step 5: Compare the numerators of rational numbers obtained and declare the one having a greater numerator as a greater rational number.

Solved Examples

1. Of the two rational numbers which is greater 2/3 or 5/7?

Solution: 

Given Rational Numbers are 2/3, 5/7

LCM of 3, 7 is 21

Expressing the rational numbers with the same denominator using the LCM obtained we get

Therefore, we get 2/3 = (2*7)/(3*7) = 14/21

5/7 = (5*3)/(7*3) = 15/21

See the numerators of both the rational numbers obtained i.e. 14/21, 15/21

Since 15 is greater the rational number 5/7 is greater.

Therefore, of the two rational numbers, 2/3 and 5/7,  5/7 is greater.

2. Which of the two rational numbers 2/5 and -3/4 is greater?

Solution:

Given Rational Numbers are 2/5 and -3/4

We clearly know between a positive rational number and a negative rational number positive rational number is always greater.

Therefore, 2/5 is greater than -3/4.

3. Which is greater among -1/2 and – 1/5?

Solution: 

Given rational numbers are -1/2 and -1/5

LCM of 2, 5 is 10

Expressing the rational numbers with the same denominator using the LCM obtained.

-1/2 = (-1*5)/(2*5)= -5/10

-1/5 = (-1*2)/(5*2) = -2/10

-2 > -5

Therefore, – 1/5 is greater than -1/2.

4. Which of the numbers 3/4 and -3/4 are greater?

Solution:

We know that every positive rational number is greater than a negative rational number. Therefore, 3/4 is greater than -3/4.

Let us learn about Equality of Rational Numbers using Cross Multiplication in detail here. Check out the Procedure to determine whether Rational Numbers are Equal or not using the Cross Multiplication Technique. Have a glance at the solved examples explaining the concept in detail so that you can solve related problems.

How to determine Equality of Rational Numbers using Cross Multiplication?

There are numerous methods to check the equality of rational numbers. But here we are using the Cross Multiplication Method to check whether the given rational numbers are equal or not. Follow the guidelines to check the equality of rational numbers.

Let us consider two rational numbers a/b, c/d

a/b = c/d

⇔ a × d = b × c

⇔ The Numerator of First × The Denominator of Second = The Denominator of First × The Numerator of the Second

Solved Examples

1.  Determine whether the following pair of Rational Numbers are Equal or Not?

8/4 and 6/3

Given Rational Numbers are 8/4 and 6/3

⇔ We know a × d = b × c

Multiplying Numerator of First × The Denominator of Second = The Denominator of First × The Numerator of the Second we get

8*3 = 6*4

24 = 24

Therefore, the given rational numbers 8/4 and 6/3 are equal.

2. If -8/6 = k/30, find the value of k?

Solution:

-8/6 = k/30

Cross multiplying we get

-8*30 = k*6

Performing basic math we get the value of k

(-8*30)/6 =k

k=-40

Therefore, the value of k is -40.

3. If 5/m = 40/16 determine the value of m?

Solution:

5/m = 40/16

Cross multiplying we get 5*16 = m*40

Separating m to get the value of it.

m= (5*16)/40

= 80/40

= 2

Therefore, the value of m is 2.

4. Fill in the Blank -7/10 = …/120?

Solution:

In order to express -7 as a denominator with 120, we first need to find out the number which when multiplied by 10 gives 120.

Thus, the integer is 120÷ 10 = 12

Multiplying the numerator and denominator of a given rational number with 12 we get

-7/10 = (-7*12)/(10*12)

= -84/120

Thus, the required number is -84/120.

You will learn about the Equality of Rational Numbers with Common Denominator from this article. Check out how to determine whether given Rational Numbers are equal or not using the Common Denominator. Apart from the procedure, we have even listed a few examples for better understanding.

How to determine Equality of Rational Numbers with Common Denominator?

There are many methods to check the equality of rational numbers but here in this article, we are going to discuss the method of the same denominator. We have listed the procedure on how to make denominators equal for the given rational numbers and they are in the following fashion.

Step 1: Check the given Rational Numbers.

Step 2: Multiply the numerator and denominator of the first number with the denominator of the second number.

Step 3: On the other hand, multiply the numerator and denominator of the second number with the denominator of the first number.

Step 4: Later check the numerators of the numbers obtained in steps 2, 3. If the numerators are equal then the given rational numbers are equal or else they are not equal.

Solved Examples

1. Are the Rational Numbers -7/5 and -5/3 equal?

Solution:

Given Rational Numbers are -7/5, -5/3

Multiply the second number numerator and denominator with the denominator of the first number i.e.

-5*5/3*5

= -25/15

Multiply the first number numerator and denominator with the denominator of the second number i.e.

-7*3/5*3

= -21/15

Check the numerators obtained in the earlier steps and see whether they are equal or not.

Since 21, 25 aren’t equal both the given rational numbers aren’t equal.

Therefore, -7/5 and -5/3 are not equal.

2. Show that the Rational Numbers -6/8 and -9/12  are equal?

Solution:

Given Rational Numbers are -6/8, -9/12

Multiply both numerator and denominator of the second number with the denominator of the first number.

= (-9/12)*8

= -72/96

Multiply both numerator and denominator of the first number with the denominator of the second number.

= (-6/8)*12

= -72/96

Therefore, Rational Numbers -6/8 and -9/12 are equal.

In this article of ours, we tried covering everything about the concept of Equality of Rational Numbers using Standard Form. You will find all about how to determine whether two rational numbers are equal or not. However, there are various methods to know whether given rational numbers are equal or not but we will employ the standard form method here.

How to determine the Equality of Rational Numbers using Standard Form?

To check whether the given rational numbers are equal or not you need to find out the standard form of both of them individually. If the standard form of both the rational numbers is equal then the rational numbers are equal or else not equal.

Solved Examples

1. Determine whether the Rational Numbers 4/-9 and -16/36 equal or not using a standard form?

Solution:

Given Rational Numbers are 4/-9, -16/36

Check for the denominators in both the rational numbers if they aren’t positive change them to positive.

4/-9 = 4*(-1)/-9*(-1)

=-4/9

GCD(4,9) = 1, thus -4/9 is the standard form.

-16/36 since it has a positive denominator it remains unchanged

Find the GCD of absolute values of the numerator and denominator for the rational numbers.

GCD(16, 36) = 4

To reduce the rational number to standard form divide both numerator and denominator with GCD obtained.

-16/36 = (-16÷4)/(36÷4)

= -4/9

-4/9 is the standard form of -16/36

Since the standard forms of rational numbers are equal both the given rational numbers 4/-9 and -16/36 are equal.

2. Determine whether the Rational Numbers 2/3 and 5/7 equal or not using a Standard Form?

Solution:

Given Rational Numbers are 2/3 and 5/7

Since both the denominators are positive you need not multiply or divide to make them positive.

Find the GCD of absolute values of numerator and denominator in the given rational numbers.

GCD(2, 3) =1

GCD(5, 7) = 1

Since both the rational numbers have GCD 1 and the numbers are relatively prime. Both the Rational Numbers are in Standard Form.

2/3 is not equal to 5/7

Therefore, Rational Numbers 2/3 and 5/7 are not equal.

Standard Form of a Rational Number: A rational number is said to be in standard form if the common factor between numerator and denominator is only 1 and the denominator is always positive. Furthermore, the numerator can have a positive sign. Such Numbers are called Rational Numbers in Standard Form. Check out a few examples that illustrate the procedure of expressing Rational Number in Standard Form to be familiar with the concept even better.

What is the Standard Form of a Rational Number?

Usually, a rational number a/b is said to be in standard form if it has no common factors other than 1 between the numerator and denominator alongside the denominator b should be positive.

How to Convert a Rational Number into Standard Form?

Go through the below-listed guidelines to express a Rational Number into Standard Form. The Detailed Procedure is explained for better understanding and they are along the lines

Step 1: Have a look at the given rational number.

Step 2: Firstly, find whether the denominator is positive or not. If it is not positive multiply or divides numerator and denominator with -1 so that the denominator no longer remains negative.

Step 3: Determine the GCD of the absolute values of both numerator and Denominator.

Step 4: Divide the numerator and denominator with the GCD obtained in the earlier step. Thereafter, the rational number obtained is the standard form of the given rational number.

Solved Examples

1.  Determine whether the following Rational Numbers are in Standard Form or Not?

(i) -8/23 (ii) -13/-39

Solution:

-8/23 is said to be in Standard Form since both the numerator and denominator doesn’t have any common factors other than 1. In fact, the denominator is also positive. Thus, the given rational number -8/23 is said to be in its Standard Form.

-13/-39 is not in standard form since it has common factor 13 along with 1. Moreover, the denominator is not positive. Thus we can say the given rational number is not in standard form.

2. Express the Rational Number 18/45 in Standard Form?

Solution:

Given Rational Number 18/45

Check for the denominator in the given rational number. Since it is positive you need not do anything.

Later find the GCD of the absolute values of numerator 18, denominator 45

GCD(18, 45) = 9

Thus, to convert the given rational number 18/45 to standard form simply divide both the numerator and denominator by 9

18/45 = (18÷9)/(45÷9)

= 2/5

Therefore, 18/45 expressed in standard form is 2/5.

3. Find the Standard Form of 12/-18?

Solution:

Given Rational Number is 12/-18

Check for the denominator in the given Rational Number

Since the denominator, -18 has a negative sign multiply both numerator and denominator with -1 to make it positive.

12/-18 = 12*(-1)/-18*(-1)

= -12/18

Find the GCD of absolute values of both numerator and denominator

GCD(12, 18) = 6

To convert a given rational number to its standard form multiply and divide both numerator and denominator by 6.

-12/18 = ((-12)÷6)/(18÷6)

= -2/3

Thus, the standard form of Rational Number 12/-18 is -2/3.

4. Reduce 3/15 to Standard Form?

Solution:

Given Rational Numbers is 3/15

Since the denominator is positive you need not do anything to change it to positive.

Find the GCD of absolute values of numerator and denominator of the given rational number.

GCD(3, 15) = 3

Divide numerator and denominator with GCD obtained.

3/15 = (3÷3)/(15÷3)

= 1/5

Therefore, 3/15 Reduced to Standard Form is 1/5.