Informative line

Comparison And Ordering Of Rational Numbers

Comparison of Two Rational Numbers

The comparison of rational numbers can be done in the same way as fractions. We can compare the two rational numbers by following the given steps:

Case-I When both the rational numbers are positive.

Step-1: Make both the denominators same (if they are not) by multiplying both the numerator and the denominator with the required factors.

Example: \(\dfrac {2}{3}\) and \(\dfrac {1}{2}\)

Here, the denominators are not same, i.e. 3 and 2.

So, \(\dfrac {2×2}{3×2}\) and \(\dfrac {1×3}{2×3}\)

\(\Rightarrow\dfrac {4}{6}\) and \(\dfrac {3}{6}\)

Step 2: Now the rational number with the larger numerator is greater than the other.

Thus, \(\dfrac {4}{6}\) is greater than \(\dfrac {3}{6}\),

or   \(\dfrac {2}{3}\) is greater than \(\dfrac {1}{2}\).

Case-II When both the rational numbers are negative.

Step-1: Make both the denominators same (if they are not) by multiplying both the numerator and the denominator with the required factors.

Example: \(\dfrac {-1}{5}\) and \(\dfrac {-3}{4}\)

Here, the denominators are not same, i.e. 5 and 4.

So, \(\dfrac {-1×4}{5×4}\) and \(\dfrac {-3×5}{4×5}\)

\(\Rightarrow\dfrac {-4}{20}\) and \(\dfrac {-15}{20}\)

Step-2: Now the rational number having the larger absolute value of the numerator is smaller than the other.

\(|-15|>|-4|\)

\(\therefore\;\dfrac {-15}{20}\) is smaller than \(\dfrac {-4}{20}\),

or   \(\dfrac {-3}{4}\) is smaller than  \(\dfrac {-1}{5}\).

Case-III When both the rational numbers have opposite signs.

When both the rational numbers have opposite signs, then the positive rational number is always greater than the negative rational number.

Example: \(\dfrac {-1}{6}\) and \(\dfrac {5}{2}\)

Both the rational numbers have opposite signs.

\(\therefore \;\;\dfrac {-1}{6}<\dfrac {5}{2}\)

Illustration Questions

Which one of the following statements is correct?

A \(\dfrac {1}{4}\) is greater than \(\dfrac {3}{4}\)

B \(\dfrac {-2}{5}\) is less than \(\dfrac {-3}{5}\)

C \(\dfrac {8}{7}\) is greater than \(\dfrac {9}{8}\)

D \(\dfrac {1}{4}\) is less than \(\dfrac {1}{8}\)

×

Option (A):

The denominators of \(\dfrac {1}{4}\) and \(\dfrac {3}{4}\) are same, i.e. \(4\).

So, compare the numerators.

\(3>1\)

\(\therefore\) \(\dfrac {3}{4}\) is greater than \(\dfrac {1}{4}\).

Hence, option (A) is incorrect.

Option (B):

The denominators of \(\dfrac {-2}{5}\) and \(\dfrac {-3}{5}\) are same, i.e. \(5\).

\(|-2|<|-3|\)

\(\therefore\;\;\)\(\dfrac {-3}{5}\) is less than \(\dfrac {-2}{5}\).

Hence, option (B) is incorrect.

Option (C):

The denominators of \(\dfrac {8}{7}\) and \(\dfrac {9}{8}\) are not same. So, first, make them same.

\(\dfrac {8×8}{7×8}\) and \(\dfrac {9×7}{8×7}\)

\(\dfrac {64}{56}\) and \(\dfrac {63}{56}\)

Now, compare the numerators.

\(64>63\)

\(\therefore\) \(\dfrac {64}{56}\) is greater than \(\dfrac {63}{56}\),

or   \(\dfrac {8}{7}\) is greater than \(\dfrac {9}{8}\).

Hence, option (C) is correct.

Option (D):

The denominators of \(\dfrac {1}{4}\) and \(\dfrac {1}{8}\) are not same. So, first, make them same.

\(\dfrac {1×2}{4×2}\) and \(\dfrac {1×1}{8×1}\)

\(\dfrac{2}{8}\) and \(\dfrac{1}{8}\)

Now, compare the numerators. 

\(1<2\)

\(\therefore\) \(\dfrac {1}{8}\) is less than \(\dfrac {2}{8}\)

or  \(\dfrac {1}{8}\) is less than  \(\dfrac {1}{4}\).

Hence, option (D) is incorrect.

Which one of the following statements is correct?

A

\(\dfrac {1}{4}\) is greater than \(\dfrac {3}{4}\)

.

B

\(\dfrac {-2}{5}\) is less than \(\dfrac {-3}{5}\)

C

\(\dfrac {8}{7}\) is greater than \(\dfrac {9}{8}\)

D

\(\dfrac {1}{4}\) is less than \(\dfrac {1}{8}\)

Option C is Correct

Ordering of Rational Numbers

The arrangement of rational numbers either in increasing or decreasing order is called ordering of rational numbers.

Increasing order:

The arrangement of rational numbers from least to greatest is called increasing order.

Less than (<) sign is used to show an increasing order.

Example: \(\dfrac {-3}{5}<\dfrac {-2}{5}<\dfrac {1}{3}<\dfrac {1}{2}\)

Decreasing order:

The arrangement of rational numbers from greatest to least is called decreasing order.

Greater than (>) sign is used to show a decreasing order.

Example: \(\dfrac {1}{2}>\dfrac {1}{3}>\dfrac {-2}{5}>\dfrac {-3}{5}\)

To arrange the rational numbers, first, make the denominators same.

Example: \(\dfrac {3}{5}\),  \(\dfrac {5}{4}\),  \(\dfrac {7}{2}\)

Making the denominators same.

\(\Rightarrow\dfrac {3×4}{5×4},\;\;\dfrac {5×5}{4×5},\;\;\dfrac {7×10}{2×10}\)

\(\Rightarrow\dfrac {12}{20},\;\;\dfrac {25}{20},\;\;\dfrac {70}{20}\)

Now, compare all the numerators. The rational number that has the largest numerator is the greatest rational number.

\(70\) is greater than \(25\) and \(25\) is greater than \(12\)

\(\therefore\;\dfrac {70}{20}\)  is greater than \(\dfrac {25}{20}\) and \(\dfrac {25}{20}\) is greater than \(\dfrac {12}{20}\),

or \(\dfrac {7}{2}\) is greater than \(\dfrac {5}{4}\) and  \(\dfrac {5}{4}\)  is greater than \(\dfrac {3}{5}\).

Now, arrange all the given numbers from least to greatest for increasing order and greatest to least for decreasing order.

Increasing order: \(\dfrac {3}{5}<\dfrac {5}{4}<\dfrac {7}{2}\)

Decreasing order: \(\dfrac {7}{2}>\dfrac {5}{4}>\dfrac {3}{5}\)

Illustration Questions

Find the correct increasing order of the following rational numbers: \(\dfrac {1}{2},\;\dfrac {-3}{4}\) and \(0.4\)

A \(\dfrac {1}{2}<\dfrac {-3}{4}<0.4\)

B \(\dfrac {-3}{4}<0.4<\dfrac {1}{2}\)

C \(\dfrac {1}{2}>0.4>\dfrac {-3}{4}\)

D \(\dfrac {1}{2}>\dfrac {-3}{4}>0.4\)

×

Given Rational numbers: \(\dfrac {1}{2},\;\dfrac {-3}{4}\) and \(0.4\) or \(\dfrac {4}{10}\)

First, make the denominators of all the numbers same.

\(\Rightarrow\dfrac {1×10}{2×10},\;\;\dfrac {-3×5}{4×5},\;\;\dfrac {4×2}{10×2}\)

\(\Rightarrow\dfrac {10}{20},\;\;\dfrac {-15}{20},\;\;\dfrac {8}{20}\)

Now, compare all the numerators. The rational number that has the smallest numerator is the least rational number.

\(-15\)  is smaller than \(8\) and \(8\) is smaller than \(10\)

\(\therefore\;\dfrac {-15}{20}\) is smaller than \(\dfrac {8}{20}\) and \(\dfrac {8}{20}\) is smaller than \(\dfrac {10}{20}\),

or \(\dfrac {-3}{4}\) is smaller than \(0.4\) and \(0.4\)  is smaller than \(\dfrac {1}{2}\).

Now, arrange all the given numbers from least to greatest for increasing order.

Thus, the increasing order of \(\dfrac {1}{2},\;\dfrac {-3}{4}\) and \(0.4\) is 

\(\dfrac {-3}{4}<0.4<\dfrac {1}{2}\)

Hence, option (B) is correct.

Find the correct increasing order of the following rational numbers: \(\dfrac {1}{2},\;\dfrac {-3}{4}\) and \(0.4\)

A

\(\dfrac {1}{2}<\dfrac {-3}{4}<0.4\)

.

B

\(\dfrac {-3}{4}<0.4<\dfrac {1}{2}\)

C

\(\dfrac {1}{2}>0.4>\dfrac {-3}{4}\)

D

\(\dfrac {1}{2}>\dfrac {-3}{4}>0.4\)

Option B is Correct

Comparison of Rational Numbers on Number line

We can compare the rational numbers on a number line by following the given steps:

Step 1: Draw a number line.

Step 2: Plot the given numbers on it.

Step 3: The number that is placed to the left of any number on the number line is smaller than that number while the number that is placed to the right of any number on the number line is greater than that number.

Consider the two rational numbers, \(\dfrac {-5}{3 }\)  and \(\dfrac {-1}{2}\)  for comparison.

First, draw a number line.

Now, plot both the numbers, i.e. \(\dfrac {-5}{3}\) and \(\dfrac {-1}{2}\) on it.

Here, \(\dfrac {-5}{3} = \dfrac {-10}{6}\; and\; \dfrac {-1}{2} = \dfrac {-3}{6}\)

 

Since\(\dfrac {-1}{2}\) is plotted on the right side of \(\dfrac {-5}{3}\),

 therefore, \(\dfrac {-1}{2}\) is greater than\(\dfrac {-5}{3}\).

Note: Positive rational numbers are always greater than the negative rational numbers as they are plotted on the right side of the negative rational numbers.

Illustration Questions

Compare the rational numbers that are plotted on the number line as shown.

A \(\dfrac {-1}{4}\) is less than \(\dfrac {-3}{2}\)

B \(\dfrac {-1}{4}\) is greater than \(\dfrac {-3}{2}\)

C Both are equal

D

×

Given :

image

The number that is placed to the right of any number on the number line is greater than that number.

\(\dfrac {-1}{4}\) is plotted on the right side of \(\dfrac {-3}{2}\).

\(\therefore\) \(\dfrac {-1}{4}\) is greater than \(\dfrac {-3}{2}\).

Hence, option (B) is correct.

Compare the rational numbers that are plotted on the number line as shown.

image
A

\(\dfrac {-1}{4}\) is less than \(\dfrac {-3}{2}\)

.

B

\(\dfrac {-1}{4}\) is greater than \(\dfrac {-3}{2}\)

C

Both are equal

D

Option B is Correct

Opposite of Rational Numbers

  • Rational numbers are fractions, integers, decimals and their opposites.

For example:

Opposite of \(-2\) is \(2\) and vice versa.

Opposite of \(\dfrac {1}{5}\) is \(\dfrac {-1}{5}\).

Opposite of \(2.4\) is \(-2.4\)

  • If two rational numbers are placed at the same distance from zero on both sides of the number line, then they are opposites of each other.
  • Thus, \(\dfrac {-5}{2}\) and \(\dfrac {5}{2}\) are opposites of each other.
  • The opposite of a fraction is a negative rational number.

Example:

\(\dfrac {-5}{2}\) is the opposite of the fraction \(\dfrac {5}{2}\).

Here, \(\dfrac {-5}{2}\) is a negative rational number.

The opposite of a negative rational number is a positive rational number.

Example: The opposite of \(\dfrac {-1}{2} = -\left( \dfrac{-1}{2} \right)\) or \(\dfrac {1}{2}\)

Here, \(\dfrac {1}{2}\) is a positive rational number.

Note:

The opposite of the opposite of a rational number is the number itself.

For example:

The opposite of \(\dfrac {2}{6}\) is \(\dfrac {-2}{6}\).

Now, the opposite of the opposite of \(\dfrac {2}{6}\)

\(=-\left (\dfrac {-2}{6}\right)\)

\(=\dfrac {2}{6}\)

Illustration Questions

Which one of the following number lines represents the location of \(\dfrac {-9}{4}\) and its opposite?

A

B

C

D

×

Rational numbers are fractions, integers, decimals and their opposites.

So, the opposite of \(\dfrac {-9}{4}\) is \(\dfrac {9}{4}\).

On a number line, the opposite of a rational number has the same distance from zero as the rational number on the other side. So,

image

Hence, option (A) is correct.

Which one of the following number lines represents the location of \(\dfrac {-9}{4}\) and its opposite?

A image
B image
C image
D image

Option A is Correct

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