Comparison between Two Rational Numbers

As we know that rational numbers are numbers which are represented in the form of \(\frac{p}{q}\) where ‘p’ and ‘q’ are the integers with both negative and positive signs and ‘q’ is not equal to zero. In this topic of rational number we’ll compare the two rational numbers. Comparison is done between two numbers so as to find the greatest of two numbers. Comparison in this case will be somewhat similar to that of comparison we used to do between two whole numbers. But, there will be some differences from whole numbers’ case depending upon the type of rational numbers we are comparing.

We are aware that rational numbers are fractions. So, they can be classified into following types:

I. Proper rational number (fraction): Proper rational numbers are those which are less than 1. In this type of rational number denominator is greater than numerator, i.e., ‘p’ is less than ‘q’ in \(\frac{p}{q}\) form.

For example: \(\frac{2}{3}\), \(\frac{4}{5}\), \(\frac{7}{9}\), etc. are all examples of proper fractions.

II. Improper rational numbers (fraction): Improper rational numbers are those which are greater than 1. In such type of rational numbers numerator is greater than denominator, i.e., ‘p’ is greater than q’ in \(\frac{p}{q}\) form.

For example: \(\frac{4}{3}\), \(\frac{9}{8}\), \(\frac{34}{12}\), etc. are all examples of improper rational numbers.

III. Positive rational number: In this type of rational number, both of the numerator and denominator are either positive or both of them are negative. These are always greater than zero.

For example: \(\frac{2}{3}\), \(\frac{-4}{-5}\), etc. are all examples of positive rational numbers.

IV. Negative rational number: In this type of rational number, either numerator is negative or denominator is negative. These are always less than zero.

For example: \(\frac{-2}{5}\), \(\frac{3}{-8}\), etc. are all examples of negative rational numbers.

Comparison between the numbers:

1. Before going to the comparison of rational numbers always remember following points:

(i) Every positive number is greater than zero.

(ii) Every negative number is less than zero.

(iii) Every positive number is greater than negative number.

(iv) Every number on the right of number line is greater than number on its left on the number line.

2. For comparison between two rational numbers we need to follow the below mentioned steps:

Step I: Firstly make sure that the denominators of the given rational numbers are positive. If not so multiply both numerator and denominator of the rational number by -1 to convert the negative denominator into positive. This will result into negative numerator and positive denominator.

Step II: Secondly, check for the rational numbers for like rational numbers (which have same denominator) and unlike rational numbers (which have different denominators).

Step III: If the rational numbers are like fractions, then we just need to compare the numerators and the one having higher denominator will be greater of the two. Don’t forget to check for negative and positive rational numbers.

Step IV: If the rational numbers are unlike fractions then convert them into like fractions by taking L.C.M. of the denominators and then compare them as given in step 1.

In short:

Let \(\frac{a}{b}\) and \(\frac{c}{d}\) be two rational numbers.

If one is positive and the other is negative, the positive number is greater than the negative number.

If both are positive (or negative), change both the numbers into fractions with common (positive) denominator. Next, compare the numerators. The fraction having the greater numerator is larger.

Solved examples on Comparison between Two Rational Numbers

1. Compare 2 and -4.

Solution:

We know that every positive number is greater than every negative number. Hence, 2 is greater than -4, i.e., 2 > (-4).

2. Compare \(\frac{1}{3}\) and \(\frac{5}{3}\).

Solution:

The given problem is of like fraction where denominators of the rational fraction are same and we just need to compare the numerators and the one having greater numerator will be the largest of the two. In this case 5 is greater than 1 and denominators of both are same, hence \(\frac{1}{3}\) is less than \(\frac{5}{3}\), i.e., \(\frac{1}{3}\) < \(\frac{5}{3}\).

3. Compare \(\frac{1}{3}\) and \(\frac{5}{6}\).

Solution: 

The given problem is of unlike fraction where denominator of the rational fractions are different and for comparing them we need to take L.C.M. of the denominators and solve as shown below:

The L.C.M. of the denominators is 6.

Now, numbers will become

 \(\frac{1 × 2}{6}\) and \(\frac{5}{6}\), i.e., numbers will be \(\frac{2}{6}\) and \(\frac{5}{6}\). Now the example becomes of the like fraction type and since their denominators have become same, we only need to compare the numerators. Since, 2 is less than 5, so \(\frac{2}{6}\) will be less than \(\frac{5}{6}\). Hence, \(\frac{1}{3}\) is less than \(\frac{5}{6}\), i.e., \(\frac{1}{3}\) < \(\frac{5}{6}\).

4. Compare \(\frac{-2}{3}\) and \(\frac{9}{-4}\)

Solution:

Since, the denominator \(\frac{9}{-4}\)is negative, we need to make it positive by multiplying both numerator and denominator by (-1). After multiplication we get \(\frac{-9}{4}\).

Now, we have to make comparison between \(\frac{-2}{3}\) and 

\(\frac{-9}{4}\). Now the example becomes of type comparison between unlike rational fractions.

Now, L.C.M. of the denominators is equal to 12.

Further the problem is solved by comparing the following two:

\(\frac{(-2) × 4}{12}\) and \(\frac{(-9) × 3}{12}\) 

Now the comparison is of like rational fractions.

\(\frac{-8}{12}\)and \(\frac{-27}{12}\)

Since, denominator is same we only need to compare only denominators. The one having more numerator will be greater of the two rational fractions. Since, both of the numerators are negative in nature so the one to the right in number line will be more than the left one. Since, (-8) is at the right side and (-27) is on the left one. Hence, (-8) is greater than (-27). So, \(\frac{-8}{12}\) is greater than \(\frac{-27}{12}\).

Hence, \(\frac{-2}{3}\) is greater than \(\frac{9}{-4}\).

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