# Compare Fraction || Arrange Fraction in Ascending or Descending order

In this post we will learn ” how to compare two or more fraction”.

We we learn to analyze the given fractions and try to find “which fraction is greater among all”.

## Methods to compare fraction

There are two methods to compare the fractions:

(a) By converting fraction into decimal
(b) By making all denominators same

### By converting fraction into decimal

For fraction comparison, do the following steps:

(a) Convert all the fraction into decimal form.
We know that fractions can be represented in form of division. Do the simple division and find the decimal.

(b) Compare the decimal numbers
Comparison of decimal numbers are straight forward. Just compare all the numbers and select the one which is greatest.

Example 01
Which of the fraction is greater \mathtt{\frac{2}{7} \ \ or\ \frac{2}{5}}

(a) Find the decimal values of all number

\mathtt{\frac{2}{7} \ =\ 2\ \div \ 7\ =\ 0.286} \\\ \\

\mathtt{\frac{2}{5} \ =\ 2\ \div \ 5\ =\ 0.4}

(b) Compare the decimal values

We know that 0.286 < 0.4.

Hence, \mathtt{\ \frac{2}{7} \ < \ \ \frac{2}{5} \ }

Example 02
Which of the fraction is greater \mathtt{\frac{8}{6} \ or\ \frac{9}{7\ }}

(a) Find the decimal value of both fraction

\mathtt{\frac{8}{6} \ =\ 8\ \div \ 6\ =\ 1.33} \\\ \\

\mathtt{\frac{9}{7} \ =\ 9\ \div \ 7\ =\ 1.29} \\\ \\

(b) Compare the decimal values

we know that 1.33 > 1.29

Hence, \mathtt{\ \frac{8}{6} \ > \ \ \frac{9}{7} \ }

Example 03
Compare the fractions using decimal method
\mathtt{\frac{4}{5} \ or\ \frac{3}{2\ } \ or\ \ \frac{7}{4} \ }

(a) Find the decimal value of all fractions

\mathtt{\frac{4}{5} \ =\ 4\ \div \ 5\ =\ 0.8} \\\ \\

\mathtt{\frac{3}{2} \ =\ 3\ \div \ 2\ =\ 1.5} \\\ \\

\mathtt{\frac{7}{4} \ =\ 7\ \div \ 4\ =\ 1.75} \\\ \\

(b) Compare the decimal values

We know that; 1.75 > 1.5 > 0.8

Hence, \mathtt{\frac{7}{4} \ >\ \frac{3}{2} \ >\ \frac{4}{5}}

### Comparing Fractions by Making all denominators same

In this method we make all denominator same so that the fraction can be compared using the numerator only.

How to make all denominator same?

This can be done by two methods:
(A) L.C.M method
(B) Common Denominator methods

#### (A) L.C.M Method

Do the following steps:

(i) Find LCM of the denominator of given fractions

(ii) Multiply/Divide numbers to ensure that the denominator is same as L.C.M

(iii) Now compare the fractions

Let us understand the process with help of examples:

Example 01
Compare the fractions \mathtt{\frac{2}{3} \ or\ \frac{3}{5\ }}

(i) Find LCM of denominators

LCM (3, 5) = 15

We have to make denominator 15 on both the fractions.

(ii) Multiply and divide the fractions

For Fraction (2/3)
Multiply 5 on both numerator and denominator, we get;

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

For Fraction (3/5)
Multiply 3 on both numerator and denominator

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

(iii) Compare the fractions
Since the denominator is same, we have to compare only numerators

we know that; \mathtt{\frac{10}{15} \ >\ \ \frac{9}{15} \ }

Hence, fraction (2/3) > (3/5)

Example 02
Compare the fractions \mathtt{\frac{7}{8} \ or\ \frac{6}{12\ }}

(i) Find the L.C.M of denominators

LCM (8, 12) = 24

Now we have to make denominator 24 on both the fraction

(ii) Multiply or Divide the fraction

For Fraction (7/8);
Multiply 3 on both numerator and denominator

\mathtt{\frac{7\times 3}{8\times 3} \ =\ \frac{21}{24}}

For Fraction (6/12);
Multiply 2 on numerator and denominator

\mathtt{\frac{6\ \times 2}{12\times 2} \ =\ \frac{12}{24}}

(iii) Compare the fractions
Since we have same denominator on both the fraction, we have to only compare numerator

It’s clear that; \mathtt{\frac{21}{24} \ >\ \frac{12}{24}}

Hence, (7/8) > (6/12)

Example 03
Compare the fractions \mathtt{\frac{2}{7} ,\ \frac{1}{6} \ \&\ \frac{3}{5}}

(i) Find the L.C.M of denominators

LCM (7, 6, 5) = 210

Hence, we have to make denominator 210 on all fractions

(ii) Multiply or divide the fraction

For fraction (2/7)
Multiply 30 on both numerator and denominator
\mathtt{\frac{2\ \times 30}{7\times 30} \ =\ \frac{60}{210}}

For Fraction (1/6)
Multiply numerator and denominator by 35
\mathtt{\frac{1\ \times 35}{6\ \times 35} \ =\ \frac{35}{210}}

For Fraction (3/5)
Multiply numerator and denominator by 42
\mathtt{\frac{3\ \times 42}{5\ \times 42} \ =\ \frac{126}{210}}

(iii) Compare the fractions
Since denominator is same in all fraction we have to just compare numerator

It’s clear that;
\mathtt{\frac{126}{210} >\ \frac{60}{210} \ >\ \frac{35}{210}}

Hence, (3/5) > (2/7) > (1/6)

#### (B)Common Denominator Method

In this method we multiply each fraction by denominator of other fraction.

Suppose \mathtt{\frac{a}{b} \ \ \&\ \ \frac{c}{d} \ } are the two fractions.

In this method, we follow below steps:

(i) Multiply fraction a/b with denominator d (both numerator & denominator)

\mathtt{\Longrightarrow \frac{a\ \times \ d}{b\ \times \ d}}

(ii) Multiply fraction c/d with denominator b

\mathtt{\Longrightarrow \frac{c\ \times \ b}{d\ \times \ b}}

(iii) After the above two steps we will have fractions with common denominator. Now just compare the numerators and find the answer.

Example 01
Compare the fraction
\mathtt{\frac{4}{3} \ \&\ \frac{2}{7}}

(i) Multiply fraction (4/3) with 7

\mathtt{\Longrightarrow \frac{4\ \times \ 7}{3\ \times \ 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28}{21}}

(ii) Multiply fraction (2/7) with 3

\mathtt{\Longrightarrow \frac{2\ \times \ 3}{7\ \times \ 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6}{21}}

(iii) Now both the fraction 28/21 and 6/21 have common denominator. So comparing the numerator; 28 > 6.

Hence, (4/3) > (2/7)

Example 02
Compare the fraction
\mathtt{\frac{12}{13} \ \&\ \frac{19}{17}}

(i) Multiply fraction 12/13 with 17

\mathtt{\Longrightarrow \frac{12\ \times \ 17}{13\ \times \ 17}}\\\ \\ \mathtt{\Longrightarrow \ \frac{204}{221}}

(ii) Multiply 19/17 with 13

\mathtt{\Longrightarrow \frac{19\ \times \ 13}{17\ \times \ 13}}\\\ \\ \mathtt{\Longrightarrow \ \frac{247}{221}}

(iii) Now we have fraction we same denominator.
Comparing the numerator 204 < 247

Hence, 19/17 > 12/13

## Questions on fraction comparison

(01) Find which of the fraction is greater using decimal method?
\mathtt{\Longrightarrow \ \frac{15}{21} \ \ \&\ \ \frac{36}{17} \ }

(a) 15/21 > 36/17
(b) 15/21 < 36/17

Option (b) is correct

Calculating decimal values of both fraction
\mathtt{\Longrightarrow \ \frac{15}{21} \ =\ 15\div \ 21\ =\ 0.714} \\\ \\

\mathtt{\Longrightarrow \ \frac{36}{17} \ =36\ \div \ 17\ =2.117} \\\ \\

we know that; 2.117 > 0.714

Hence, 36/17 > 15/21

(02) Find which of the fraction is greatest using LCM method.
\mathtt{\Longrightarrow \ \frac{25}{50} \ \ \&\ \ \frac{19}{20} \ }\\\ \\

(a) 19/20 > 25/20
(b) 19/20 < 25/20

Option (a) is correct

Explanation

Find L.C.M of denominators
LCM (50, 20) = 100

We have to make denominator 100 on both fraction

For fraction 25/50
Multiply by 2 on both numerator & denominator

\mathtt{\Longrightarrow \frac{25\ \times \ 2}{50\ \times \ 2} \ =\ \frac{50}{100}}

For Fraction 19/20
Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \frac{19\ \times \ 5}{20\ \times \ 5} \ =\ \frac{95}{100}}

Now in both the fractions, denominator is same.
So comparing the numerator; 95 > 50

Hence, 19/20 > 25/50

(03) Compare the below fraction using LCM method.
\mathtt{\Longrightarrow \ \frac{1}{10} \ \ ,\ \ \frac{2}{11} \ \ and\ \frac{3}{13}}

(a) 1/10 > 2/11 > 3/13
(b) 1/10 < 2/11 < 3/13
(c) 1/10 > 2/11 < 3/13

Option (b) is correct

Explanation

Find LCM of denominators
L.C.M (10, 11, 13) = 1430

Make denominator 1430 in all fractions

Fraction 1/10
Multiply 143 in both numerator & denominator

\mathtt{\Longrightarrow \frac{1\ \times \ 143}{10\ \times \ 143} \ =\ \frac{143}{1430}}

Fraction 2/11
Multiply 130 in both numerator & denominator

\mathtt{\Longrightarrow \frac{2\ \times \ 130}{11\ \times \ 130} \ =\ \frac{260}{1430}}

Fraction 3/13
Multiply 120 on both numerator & denominator

\mathtt{\Longrightarrow \frac{3\ \times \ 120}{13\ \times \ 120} \ =\ \frac{360}{1560}}

Now denominator is same in all fractions.
Compare the numerators; 143 < 260 < 360

Hence, 1/10 < 2/11 < 3/13

(04) Compare the below fractions using direct multiplication method
\mathtt{\Longrightarrow \ \ \frac{4}{7} \ \ and\ \frac{9}{5}}

(a) 4/7 > 9/5
(b) 4/7 < 9/5

Option (b) is correct

Explanation

Multiply fraction 4/7 with 5
\mathtt{\Longrightarrow \frac{4\ \times \ 5}{7\ \times \ 5} \ =\ \frac{20}{35}}

Multiply fraction 9/5 with 7
\mathtt{\Longrightarrow \frac{9\ \times \ 7}{5\ \times \ 7} \ =\ \frac{63}{35}}

Both the fraction have same denominator.
Comparing the numerator; 20 < 63.

Hence, 9/5 > 4/7