Which one is greater between \(\dfrac{2}{5}\) and \(\dfrac{4}{5}\)?
Step 1: If the denominators are same, compare the numerators.
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.
\(4>2\)
\(\therefore\;\dfrac{4}{5}>\dfrac{2}{5}\)
A \(\dfrac{1}{2}\)
B \(\dfrac{3}{2}\)
C Both are equal
D
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.
\(\dfrac{3}{4}>\dfrac{2}{3}\)
A \(\dfrac{3}{2}\)
B \(\dfrac{1}{5}\)
C Both are equal
D
Comparison of fractions can be done by two methods:
Cross-multiplication Method:
Let \(\dfrac {a}{b}\) and \(\dfrac {c}{d}\) be two fractions.
If \(ad < bc\), then \(\dfrac {a}{b}<\dfrac {c}{d}\)
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 \(10\)
\(\because\;18>10\)
\(\therefore\;\dfrac {3}{2}>\dfrac {5}{6}\)
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
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.
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 \(P\)= \(\dfrac{5}{6}\), \(R\)= \(\dfrac{9}{6}\), \(R>P\)
In ascending order, we arrange fractions or numbers from the smallest to the largest.
For example: \(2,\;3,\;8\)
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 \(=\dfrac{3}{4},\;\dfrac{1}{2},\;\dfrac{1}{3}\)
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 \(\dfrac{4}{3},\;\dfrac{1}{2},\;\dfrac{1}{6}\)
For example:
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.
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 \( 2\dfrac{1}{4}\).
\(0.5\) = \(\dfrac{5}{10}=\dfrac{1}{2}\) 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: \(0.5,\,0.60,\,\dfrac{5}{6},\,2\dfrac{1}{4}\)
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 \(\dfrac{7}{9},\,0.7,\,0.9,\,\dfrac{3}{7}\)