Fundamentals Of Fractions
Comparison Of Fraction
Common Denominator Method:
Find a common denominator for the two fractions.
Convert both fractions to equivalent fractions with the common denominator.
Compare the numerators.
For example, to compare \(
\frac{1}
{4}
\) and \(
\frac{3}
{8}
\)
Find a common denominator, which is 8.
Convert \(
\frac{1}
{4}
\) to \(
\frac{2}
{8}
\)
Now, compare \(
\frac{2}
{8}
\) and \(
\frac{3}
{8}
\), \(
\frac{3}
{8}
\) and is greater.
Comparison of fraction by cross-multiplication method
If two fractions \(
\frac{a}
{b}
\) and \(
\frac{c}
{d}
\) are to be compared, we cross multiply
i) If \(
a \times d > b \times c
\), then \(
\frac{a}
{b} > \frac{c}
{d}\) ii) If \(
a \times d < b \times c
\), then \(% MathType!MTEF!2!1!+-
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\[
\frac{a}
{b} < \frac{c}
{d}
\)
iii) If \(
a \times d = b \times c
\), then \(
\frac{a}
{b} = \frac{c}
{d}
\)
Decimal Conversion:
Convert both fractions to decimals.
Compare the decimal representations.
For example, to compare \(
\frac{1}
{3}
\) to \(
\frac{2}
{5}
\)
\(
\frac{1}
{3}
\)as a decimal is approximately 0.333.
\(
\frac{2}
{5}
\)as a decimal is 0.4.
Therefore, \(
\frac{2}
{5}
\) is greater
Reducing a fraction: Reducing is what we do when we want to make a smaller version of a fraction that still has the same mathematical value as the original.
Ex: \(
\frac{4}
{8} = \frac{{4 \div 4}}
{{8 \div 4}} = \frac{1}
{2}
\)
In this numerator and denominator both are divided by 4, we get 1 over 2.
Comparison of fractions :
1) Converting fractions to decimals.
Ex:\(
\frac{3}
{8}or\frac{5}
{{12}}
\) which is greater?
Sol:\(
\frac{3}
{8} = 0.375\& \frac{5}
{{12}} = 0.4166...
\)
\(
\therefore \frac{5}
{{12}} > \frac{3}
{8} \Rightarrow \frac{5}
{{12}}
\) is greater.
2) Making fractions as like fractions.
Ex: i)\(
\frac{2}
{3}or\frac{4}
{{15}}
\) which is smaller?
Sol: Given , Here LCM of 3 & 15 is 15.
Make the denominators = LCM of denominators
\(
\frac{2}
{3} = \frac{{2 \times 5}}
{{3 \times 5}} = \frac{{10}}
{{15}}
\) and \(
\frac{4}
{{15}}
\)
\(
\therefore \frac{4}
{{15}} < \frac{{10}}
{{15}} \Rightarrow \frac{4}
{{15}} < \frac{2}
{3}
\)
\(
\therefore \frac{4}
{{15}}
\) is smaller.
ii) \(
\frac{4}
{9} < \frac{5}
{9}\left( {\because 4 < 5} \right)
\) denominator same
iii)\(
\frac{3}
{8}or\frac{5}
{{12}}
\) which is greater?
Sol: 8 x 3 = 24, 12 x 2 = 24
Make the denominator equal to LCM of denominators.
\(
\begin{gathered}
\therefore \frac{{3 \times 3}}
{{8 \times 3}} = \frac{9}
{{24}}and\frac{{5 \times 2}}
{{12 \times 2}} = \frac{{10}}
{{24}} \hfill \\
\therefore \frac{9}
{{24}} < \frac{{10}}
{{24}}\left( {\because 9 < 10} \right) \hfill \\
\end{gathered}
\)
So \(
\frac{5}
{{12}}
\) is greater.
i.e., to compare fractions we convert unlike fractions into like fractions by making denominator equal to LCM of denominators.