Comparison of rational number

In this chapter we will learn the method to compare two or more rational number with solved examples.

Comparison of two or more rational number means we will try to find which number is greater and which one is smaller.

Comparing rational number

We know that rational numbers are represented in the form of \mathtt{\frac{P}{Q}} .

Where P & Q are integers.

P is called the numerator.

Q is called the denominator.

Now in order to compare the numbers we have to make denominators same and then compare the numerator.

For this we have to use LCM concept.

To do the comparison, follow the below steps;

(a) write all the rational number in fraction form.

(b) Find LCM of denominators

(c) Now multiply the numerator & denominator of each rational number so that denominator becomes equal to LCM

(d) Finally compare the numerator and arrange the number from greatest to smallest.

I hope you understood the above steps. Let us solve some problems for more clarity.

Example 01
Which is greater \mathtt{\frac{2}{3} \ or\ \frac{3}{5}}

Solution

(a) Find LCM of denominator

LCM (3, 5) = 15

(b) Multiply each rational numbers to make denominator 15.

(i) Rational number 2/3

Multiply numerator and denominator by 5. This operation will make denominator equal to LCM.

\mathtt{\Longrightarrow \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 5}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{15}}

(ii) Rational number 3 / 5

Multiply numerator and denominator by 3.

\mathtt{\Longrightarrow \frac{3}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 3}{5\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{15}}

Now we have got rational number with same denominators.

(c) Compare the numerator

Since both rational numbers have same denominator, simply compare the numerator and find the greatest number.

\mathtt{\Longrightarrow \frac{10}{15} >\ \frac{9}{15}}

Getting the original number.

\mathtt{\Longrightarrow \frac{2}{3} >\ \frac{3}{5}}

Example 02
Compare the rational number \mathtt{\frac{6}{9} \ \&\ \frac{\ 12}{13}}

Solution
Do the following steps;

(a) Find LCM of denominators

LCM (9, 13) = 117

(b) Multiply numerator and denominator of rational number to make denominator equal to 117

(i) Rational number 6 / 9

Multiply numerator and denominator by 13

\mathtt{\Longrightarrow \frac{6}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\times 13}{9\times 13}}\\\ \\ \mathtt{\Longrightarrow \ \frac{78}{117}}

(ii) Rational number 12 / 13

Multiply numerator and denominator by 9

\mathtt{\Longrightarrow \frac{12}{13}}\\\ \\ \mathtt{\Longrightarrow \ \frac{12\times 9}{13\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{108}{117}}

Now we got the fraction with same denominator.

(c) Compare the numerator of calculated numbers.

\mathtt{\Longrightarrow \frac{78}{117} < \ \frac{108}{117}}

Putting the original values, we get;

\mathtt{\Longrightarrow \frac{6}{9} < \ \frac{12}{13}}

Example 03
Compare the rational number \mathtt{\frac{8}{6} \ \&\ \frac{9}{10}}

Solution

(a) Find LCM of denominators

LCM (6, 10) = 60

(b) Multiply the rational numbers to make denominator equal to LCM.

(i) Fraction 8 / 6

Multiply numerator and denominator by 10.

\mathtt{\Longrightarrow \frac{8}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8\times 10}{6\times 10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{80}{60}}

(ii) Fraction 9 / 10

Multiply numerator & denominator by 6

\mathtt{\Longrightarrow \frac{9}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9\times 6}{10\times 6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{54}{60}}

So we got the numbers with same denominator.

(c) Now compare the numerators.

\mathtt{\Longrightarrow \frac{80}{60} \ >\ \frac{54}{60}}

Putting the original values.

\mathtt{\Longrightarrow \frac{8}{6} \ >\ \frac{9}{10}}

Example 04
Compare the rational number \mathtt{\frac{-52}{3} \ \&\ \frac{23}{2}}

Solution
Among the given numbers one number is negative and other is positive.

There is no need to do any calculation as negative number is always less than positive number.

\mathtt{\frac{-52}{3} \ < \ \frac{23}{2}}

Example 05
Compare the rational numbers \mathtt{\frac{-11}{4} \&\ \frac{-15}{7}} Rational number -11/4

Solution
(a) Find LCM of denominator

LCM (4, 7) = 28

(b) Multiply the rational number to make denominator equal to 28

(i) Rational number -11/4

Multiply numerator & denominator by 7

\mathtt{\Longrightarrow \frac{-11}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-11\times 7}{4\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-77}{28}}

(ii) Rational number -15 / 7

Multiply numerator and denominator by 4

\mathtt{\Longrightarrow \frac{-15}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-15\times 4}{7\times 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-60}{28}}

Now we got the rational numbers with same denominator.

(c) Compare the numerators

\mathtt{\Longrightarrow \frac{-77}{28} < \ \frac{-60}{28}}

Putting the original values;

\mathtt{\Longrightarrow \frac{-11}{4} < \ \frac{-15}{7}}

Next chapter : Questions on finding rational number between two numbers