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