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Comparison Of Fractions

Comparison of Fractions having Same Denominators

  • To compare two or more fractions, we should follow the given steps:
  • Let us consider an example.

Which one is greater between \(\dfrac{2}{5}\) and \(\dfrac{4}{5}\)?

Step 1: If the denominators are same, compare the numerators.

  • The denominator \(5\) is same in both the fractions.
  • Numerator of \(\dfrac{2}{5}=2\)

Numerator of \(\dfrac{4}{5}=4\)

Step 2: The fraction with larger numerator is greater than the fraction with smaller numerator.

\(4\) is greater than \(2\).

Thus, \(\dfrac{4}{5}\) is greater than \(\dfrac{2}{5}\).

Step 3: Write the result using greater than (>), less than (<) or equal to (=) signs.

\(4>2\)

\(\therefore\;\dfrac{4}{5}>\dfrac{2}{5}\)

Illustration Questions

Which one is smaller between  \(\dfrac{3}{2}\) and \(\dfrac{1}{2}\) ?

A \(\dfrac{1}{2}\)

B \(\dfrac{3}{2}\)

C Both are equal

D

×

Given fractions: \(\dfrac{3}{2}\) and \(\dfrac{1}{2}\)

The denominator \(2\) is same in both the fractions.

Numerator of \(\dfrac{3}{2}=3\)

Numerator of \(\dfrac{1}{2}=1\)

\(1\) is smaller than \(3\)  

or  \(1<3\)

\(\therefore\;\dfrac{1}{2}\)  is smaller than  \(\dfrac{3}{2}\)

or   \(\dfrac{1}{2}<\dfrac{3}{2}\)

Hence, option (A) is correct.

Which one is smaller between  \(\dfrac{3}{2}\) and \(\dfrac{1}{2}\) ?

A

\(\dfrac{1}{2}\)

.

B

\(\dfrac{3}{2}\)

C

Both are equal

D

Option A is Correct

Comparison of Fractions having Different Denominators

  • Comparison is possible only when we convert the fractions into like fractions.
  • To compare two or more fractions having different denominators, we should follow the given steps:
  • Let us consider an example.

Compare \(\dfrac{3}{4}\) and \(\dfrac{2}{3}\).

Step 1: Find the least common multiple (L.C.M) of denominators.

L.C.M of \(4\) and \(3,\) 

Multiples of \(4=4,\;8,\;12\,...\)

Multiples of \(3=3,\;6,\;9,\;12\,...\)

L.C.M \(=12\)

Step 2: L.C.M becomes the lowest common denominator.

L.C.D \(=12\)

Step 3: Find the equivalent fractions having L.C.D as denominator.

Equivalent fraction of  \(\dfrac{3}{4}=\dfrac{3×3}{4×3}=\dfrac{9}{12}\)

Equivalent fraction of  \(\dfrac{2}{3}=\dfrac{2×4}{3×4}=\dfrac{8}{12}\)

Step 4: Compare the fractions having same denominators.

Compare \(\dfrac{9}{12}\) and \(\dfrac{8}{12}\).

\(9>8\)

\(\therefore\;\dfrac{9}{12}>\dfrac{8}{12}\)

Step 5: Rewrite the original fractions for the answer.

\(\dfrac{3}{4}>\dfrac{2}{3}\)

Illustration Questions

Which one is greater between  \(\dfrac{3}{2}\) and \(\dfrac{1}{5}\) ?

A \(\dfrac{3}{2}\)

B \(\dfrac{1}{5}\)

C Both are equal

D

×

Given: \(\dfrac{3}{2}\) and \(\dfrac{1}{5}\)

Finding the least common multiple (L.C.M) of \(2 \) and \(5\),

Multiples of \(2=2,\;4,\;6,\;8,\;10\,...\)

Multiples of \(5=5,\;10\,...\)

L.C.M \(=10\)

L.C.M becomes the lowest common denominator,

L.C.D \(=10\)

Finding the equivalent fractions having \(10\) as denominator,

Equivalent fraction of  \(\dfrac{3}{2}=\dfrac{3×5}{2×5}=\dfrac{15}{10}\)

Equivalent fraction of  \(\dfrac{1}{5}=\dfrac{1×2}{5×2}=\dfrac{2}{10}\)

Comparing the fractions having same denominators,

\(15>2\)

\(\therefore\;\dfrac{15}{10}>\dfrac{2}{10}\)

Writing the original fractions for the answer,

\(\dfrac{3}{2}>\dfrac{1}{5}\)

Hence, option (A) is correct.

Which one is greater between  \(\dfrac{3}{2}\) and \(\dfrac{1}{5}\) ?

A

\(\dfrac{3}{2}\)

.

B

\(\dfrac{1}{5}\)

C

Both are equal

D

Option A is Correct

Comparison of Fractions using Cross Multiplication Method

Comparison of fractions can be done by two methods:

  1. By converting them into equivalent fractions
  2. By using cross-multiplication method

Cross-multiplication Method:

Let \(\dfrac {a}{b}\) and \(\dfrac {c}{d}\) be two fractions.

If \(ad < bc\), then \(\dfrac {a}{b}<\dfrac {c}{d}\)

  • Conversely, if \(\dfrac {a}{b}<\dfrac {c}{d}\), then \(ad<bc\)
  • This is the cross-multiplication method.
  • Here, \(ad\) and \(bc\) are the cross products.
  • This method is used to compare any two fractions.

For example:  Compare \(\dfrac {3}{2}\) and \(\dfrac {5}{6}\).

Cross multiply the fractions \(\dfrac {3}{2}\) and \(\dfrac {5}{6}\).

\(3×6=18\)  and  \(5×2=10\)  

\(18>10\)

\(\therefore\;\dfrac {3}{2}>\dfrac {5}{6}\)

\(3×6\) and \(5×2\)

\(18\) and \(10\)

\(\because\;18>10\)

\(\therefore\;\dfrac {3}{2}>\dfrac {5}{6}\)

Illustration Questions

Which one of the following options represents the correct relation between the two fractions, \(\dfrac {8}{9}\) and \(\dfrac {7}{6}\) ?

A \(\dfrac {8}{9}>\dfrac {7}{6}\)

B \(\dfrac {8}{9}=\dfrac {7}{6}\)

C \(\dfrac {8}{9}<\dfrac {7}{6}\)

D Both A and B

×

Cross-multiplication can be stated as:

Let \(\dfrac {a}{b}\) and \(\dfrac {c}{d}\) be two fractions.

If \(ad < bc\), then \(\dfrac {a}{b}<\dfrac {c}{d}\)

image

Apply cross-multiplication method:

\(8×6\\ \,\,\,\,48\)                           \(7×9\\ \,\,\,\,63\)

Here, \(48\) and  \(63\) are cross products.

image

\(48<63\)

\(\therefore\;\dfrac {8}{9}<\dfrac {7}{6}\)

Hence, option (C) is correct.

Which one of the following options represents the correct relation between the two fractions, \(\dfrac {8}{9}\) and \(\dfrac {7}{6}\) ?

A

\(\dfrac {8}{9}>\dfrac {7}{6}\)

.

B

\(\dfrac {8}{9}=\dfrac {7}{6}\)

C

\(\dfrac {8}{9}<\dfrac {7}{6}\)

D

Both A and B

Option C is Correct

Comparison of Fractions through Representation on a Number Line

  • Consider two fractions as two points on the number line.

  • On a number line, a fraction which is placed to the left is smaller than the fraction which is placed to the right side.

For example: Consider \(\dfrac{1}{4}\) and \(\dfrac{5}{4}\).

\(\dfrac{1}{4}\) is placed to the left of \(\dfrac{5}{4}\).

\(\therefore\) \(\dfrac{1}{4}<\dfrac{5}{4}\)

Note: While comparing two or more fractions, make sure that each one of them has the same denominator.

Illustration Questions

Two points namely, \(P\) and \(R\), are plotted on the given number line. Determine the values of \(P\)  and \(R\)  in the fraction form and choose their relation from the following options.

A \(P\)= \(\dfrac{9}{6}\), \(R\)= \(\dfrac{5}{6}\), \(P>R\)

B \(P\)= \(\dfrac{5}{6}\), \(R\)= \(\dfrac{9}{6}\), \(P>R\)

C \(P\)= \(\dfrac{9}{6}\), \(R\)= \(\dfrac{5}{6}\), \(R>P\)

D \(P\)= \(\dfrac{5}{6}\), \(R\)= \(\dfrac{9}{6}\), \(R>P\)

×

Given number line:

image

We can observe that each interval from 0 to 1 and 1 to 2, is subdivided into 6 equal parts.

\(\therefore\) Each small segment = \(\dfrac{1}{6}\)

 

 

Thus, the value of point \(R\) = \(\dfrac{5}{6}\)

and the value of point \(P\) = \(\dfrac{9}{6}\)

image

On a number line, a fraction which is placed to the left is smaller than the fraction which is placed to the right side.

\(\dfrac{5}{6}\) is placed to the left of \(\dfrac{9}{6}\).

\(\therefore\) \(\dfrac{9}{6}>\dfrac{5}{6}\)

image

Thus, \(P>R\)

Hence, option (A) is correct.

Two points namely, \(P\) and \(R\), are plotted on the given number line. Determine the values of \(P\)  and \(R\)  in the fraction form and choose their relation from the following options.

image
A

\(P\)= \(\dfrac{9}{6}\), \(R\)\(\dfrac{5}{6}\)\(P>R\)

.

B

\(P\)\(\dfrac{5}{6}\), \(R\)\(\dfrac{9}{6}\), \(P>R\)

C

\(P\)\(\dfrac{9}{6}\), \(R\)\(\dfrac{5}{6}\), \(R>P\)

D

\(P\)\(\dfrac{5}{6}\), \(R\)\(\dfrac{9}{6}\), \(R>P\)

Option A is Correct

Arranging Fractions in Ascending or Descending Order

  • Ascending order (least to greatest):

In ascending order, we arrange fractions or numbers from the smallest to the largest.

For example: \(2,\;3,\;8\)

  • Descending order (greatest to least):

In descending order, we arrange fractions or numbers from the largest to the smallest.

For example: \(5,\;4,\;1\)

Let us consider two cases:

Case I: Fractions having same denominators

 For example: Arrange the following in ascending order: \(\dfrac{5}{4},\;\dfrac{3}{4},\;\dfrac{9}{4},\;\dfrac{15}{4},\;\dfrac{1}{4}\)

Step 1: If all fractions have same denominators, compare the numerators and arrange them in increasing order.

Smallest to largest order \(=\dfrac{1}{4},\;\dfrac{3}{4},\;\dfrac{5}{4},\;\dfrac{9}{4},\;\dfrac{15}{4}\)

Case II: Fractions having different denominators

For example: Arrange the following in descending order: \(\dfrac{1}{2},\;\dfrac{3}{4},\;\dfrac{1}{3}\)

Follow the given steps:

Step 1: Find the least common multiple (L.C.M) of denominators.

For L.C.M of \(4,\;2,\;3\)

Multiples of \(4=4,\;8,\;12\,...\)

Multiples of \(2=2,\;4,\;6,\;8,\;10,\;12\,...\)

Multiples of \(3=3,\;6,\;9,\;12\,...\)

L.C.M \(=12\)

Step 2: L.C.M becomes the lowest common denominator.

L.C.D \(=12\)

Step 3: Find the equivalent fractions having L.C.D as denominator.

Equivalent fraction of \(\dfrac{3}{4}=\dfrac{3×3}{4×3}=\dfrac{9}{12}\)

Equivalent fraction of \(\dfrac{1}{2}=\dfrac{1×6}{2×6}=\dfrac{6}{12}\)

Equivalent fraction of \(\dfrac{1}{3}=\dfrac{1×4}{3×4}=\dfrac{4}{12}\)

Step 4: Compare the fractions having same denominators.

Fractions \(=\dfrac{9}{12},\;\dfrac{6}{12},\;\dfrac{4}{12}\)

They are like fractions.

So, compare the numerators of the fractions and arrange them in descending order.

Greatest to least order \(=\dfrac{9}{12},\;\dfrac{6}{12},\;\dfrac{4}{12}\)

Step 5: Write the original fractions corresponding to these fractions in the same order.

Greatest to least order \(=\dfrac{3}{4},\;\dfrac{1}{2},\;\dfrac{1}{3}\)

Illustration Questions

Arrange the following fractions in ascending order: \(\dfrac{1}{2},\;\dfrac{1}{6},\;\dfrac{4}{3}\)

A \(\dfrac{1}{2},\;\dfrac{1}{6},\;\dfrac{4}{3}\)

B \(\dfrac{1}{2},\;\dfrac{4}{3},\;\dfrac{1}{6}\)

C \(\dfrac{1}{6},\;\dfrac{1}{2},\;\dfrac{4}{3}\)

D \(\dfrac{4}{3},\;\dfrac{1}{2},\;\dfrac{1}{6}\)

×

Given fractions: \(\dfrac{1}{2},\;\dfrac{1}{6},\;\dfrac{4}{3}\)

Finding the least common multiple of denominators (L.C.M),

Multiples of \(3=3,\;6\,...\)

Multiples of \(6=6\,...\)

Multiples of \(2=2,\;4,\;6\,...\)

L.C.M of \(2,\;3,\;6=6\)

L.C.M becomes the lowest common denominator, 

L.C.D \(=6\)

Finding the equivalent fractions having L.C.D as denominator,

Equivalent fraction of \(\dfrac{1}{6}=\dfrac{1×1}{6×1}=\dfrac{1}{6}\)

Equivalent fraction of  \(\dfrac{1}{2}=\dfrac{1×3}{2×3}=\dfrac{3}{6}\)

Equivalent fraction of  \(\dfrac{4}{3}=\dfrac{4×2}{3×2}=\dfrac{8}{6}\)

The fractions have same denominators, so comparing only the numerators and arranging the fractions accordingly.

Ascending order (least to greatest)  \(\dfrac{1}{6},\;\dfrac{3}{6},\;\dfrac{8}{6}\).

Writing the original fractions in ascending order, 

\(\dfrac{1}{6},\;\dfrac{1}{2},\;\dfrac{4}{3}\)

Hence, option (C) is correct.

Arrange the following fractions in ascending order: \(\dfrac{1}{2},\;\dfrac{1}{6},\;\dfrac{4}{3}\)

A

\(\dfrac{1}{2},\;\dfrac{1}{6},\;\dfrac{4}{3}\)

.

B

\(\dfrac{1}{2},\;\dfrac{4}{3},\;\dfrac{1}{6}\)

C

\(\dfrac{1}{6},\;\dfrac{1}{2},\;\dfrac{4}{3}\)

D

\(\dfrac{4}{3},\;\dfrac{1}{2},\;\dfrac{1}{6}\)

Option C is Correct

Comparison of Fractions (Using Figures)

  • Comparison of fractions can be easily understood using figures.

For example:

  • Two blocks are given, each having some red colored parts.
  • Which one of the two blocks represents the larger fraction of red colored parts?

Total number of equal parts is same in both the blocks.

Thus, consider the number of red colored parts for comparison.

Number of red colored parts in block \(I=4\)

Number of red colored parts in block \(II=2\)

\(4>2\)

\(\therefore\dfrac {4}{5}>\dfrac {2}{5}\)

Thus, block-\(I\) represents the larger fraction of the red colored parts than block-\(II\).

\(\because\) Total number of equal parts is same in both the blocks.

\(\therefore\) Consider the number of shaded parts to compare their fractions.

Number of shaded parts in block \(I=4\)

Number of shaded parts in block \(II=2\)

\(\because\;4>2\)

\(\therefore\dfrac {4}{5}>\dfrac {2}{5}\)

Thus, block-\(II\) represents a larger fraction of the shaded parts.

Illustration Questions

Which option correctly represents the largest fraction of red colored parts?

A

B

C

D

×

In option (A),

Total number of equal parts = \(8\)

Number of red colored parts = \(7\)

image

In option (B),

Total number of equal parts = \(8\)

Number of red colored parts = \(3\)

image

In option (C),

Total number of equal parts = \(8\)

Number of red colored parts = \(4\)

image

In option (D),

Total number of equal parts = \(8\)

Number of red colored parts = \(5\)

image

We can see that the total number of equal parts are same in all the bars.

Thus, the bar having the maximum number of red colored parts represents the largest fraction of red colored part.

Hence, option (A) is correct.

Which option correctly represents the largest fraction of red colored parts?

A image
B image
C image
D image

Option A is Correct

Ordering of Fractions and Decimals

  • Ordering refers to the arrangement of numbers. This arrangement can be in the order of least to greatest or greatest to least.
  • For arranging fractions and decimals in ascending or descending order, we first need to compare them.

Comparison of numbers can be understood in the following steps:

1. First, convert all the decimals and mixed fractions into simple fractions.

2. If all the fractions have same denominators, look at their numerators and compare them by taking two at a time.

3. If the fractions have different denominators, use the least common denominator (LCD) to obtain equivalent fractions having the same denominator. Now the fractions have same denominators, so look at their numerators and compare them by taking two at a time.

4. After comparison, the numbers can be arranged in the order asked.

For example:

Arrange the following numbers in the order of least to greatest : \(0.5,\,\dfrac{5}{6},\,0.60\) and \( 2\dfrac{1}{4}\).

  • First, convert all the decimals into simple fractions, 

\(0.5\) = \(\dfrac{5}{10}=\dfrac{1}{2}\) and

 \(0.60\) = \(\dfrac{60}{100}=\dfrac{3}{5}\)

  • Also convert the mixed fraction into a simple fraction.

  •  \(2\dfrac{1}{4}\) is converted to \(\dfrac{9}{4}\).

  • After the conversion, it is observed that all the fractions have different denominators. So, use least common denominator (LCD) to obtain the equivalent fractions having same denominators.

For \(4,2,6\) and \(5\), the LCD is \(60\).

\(\dfrac{9\times15}{4\times 15}=\dfrac{135}{60}\)

\(\dfrac{1\times30}{2\times 30}=\dfrac{30}{60}\)

\(\dfrac{5\times10}{6\times10}=\dfrac{50}{60}\)

\(\dfrac{3\times12}{5\times12}=\dfrac{36}{60}\)

On comparing the numerators of the equivalent fractions, we obtain the following order:

\(30<36<50<135\)

So, the order of the given corresponding numbers from least to greatest is as follows:  \(0.5,\,0.60,\,\dfrac{5}{6},\,2\dfrac{1}{4}\)

Illustration Questions

Arrange the following numbers in the order of greatest to least values: \(\dfrac{3}{7},\,0.9,\,\dfrac{7}{9}\) and \(0.7\)

A \(\dfrac{3}{7},\,\dfrac{7}{9},\,0.9,\,0.7\)

B \(0.9,\,\dfrac{7}{9},\,0.7,\,\dfrac{3}{7}\)

C \(0.7,\,\dfrac{7}{9},\,\dfrac{3}{7},\,0.9\)

D \(\dfrac{7}{9},\,0.7,\,0.9,\,\dfrac{3}{7}\)

×

Given: \(\dfrac{3}{7},\,0.9,\,\dfrac{7}{9},\,0.7\)

Convert all the decimals into simple fractions, 

\(0.9\) is converted to \(\dfrac{9}{10}\) and 

\(0.7\) is converted to \(\dfrac{7}{10}\).

The resulting fractions are  \(\dfrac{3}{7},\dfrac{9}{10},\dfrac{7}{9},\dfrac{7}{10}\).

The resulting fractions have different denominators. So, use least common denominator for \(7,10,9\;and\;10\), which is \(630\), to obtain the equivalent fractions with same denominators. 

Thus,

\(\dfrac{3\times90}{7\times90}=\dfrac{270}{630}\)

\(\dfrac{9\times63}{10\times63}=\dfrac{567}{630}\)

\(\dfrac{7\times70}{9\times70}=\dfrac{490}{630}\)

\(\dfrac{7\times63}{10\times63}=\dfrac{441}{630}\)

On comparing the numerators of the equivalent fractions, we obtain the following order:

\(567>490>441>270\)

So, the order of the corresponding given numbers from greatest to least values is:

\(0.9,\,\dfrac{7}{9},\,0.7,\,\dfrac{3}{7}\)

Hence, option (B) is correct.

Arrange the following numbers in the order of greatest to least values: \(\dfrac{3}{7},\,0.9,\,\dfrac{7}{9}\) and \(0.7\)

A

\(\dfrac{3}{7},\,\dfrac{7}{9},\,0.9,\,0.7\)

.

B

\(0.9,\,\dfrac{7}{9},\,0.7,\,\dfrac{3}{7}\)

C

\(0.7,\,\dfrac{7}{9},\,\dfrac{3}{7},\,0.9\)

D

\(\dfrac{7}{9},\,0.7,\,0.9,\,\dfrac{3}{7}\)

Option B is Correct

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