Comparison of Fractions

☼ Comparison of Fractions : Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M. as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

e.g. Find the greater fraction from $\frac{2}{3}$ and $\frac{3}{4}$ .

Soln. [ 3, 4 ] = 12.

$\frac{2}{3}$ = $\frac{2\times 4}{3\times4}$ = $\frac{8}{12}$ & $\frac{3}{4}$ = $\frac{3\times 3}{4 \times 3}$ = $\frac{9}{12}$

Hence, $\frac{3}{4}$ is greater one.

Denominators of two fractions are same, then the fraction with greater numerator is greater.

e.g. $\frac{15}{7}$ > $\frac{15}{8}$

Numerators of two fractions are same, then the fraction with smaller denominator is greater.

e.g. $\frac{15}{7}$ > $\frac{15}{8}$

e.g. Arrange the fractions $\frac{1}{3},\frac{3}{5},\frac{2}{9},\frac{5}{2}$ in descending order.

Method 1:

Take L.C.M. of all denominators.

$\therefore$ [ 3, 5, 9, 8 ] = 90

$\therefore$ $\frac{1}{3}=\frac{30}{90}$ , $\frac{3}{5}=0.6$ , $\frac{1}{3}=0.33$ , $\frac{5}{2}=\frac{255}{90}$

$\therefore$ Descending order is : $\frac{255}{90}$ , $\frac{54}{90}$ , $\frac{30}{90}$ , $\frac{20}{90}$ or $\frac{5}{2}$ > $\frac{3}{5}$ > $\frac{1}{3}$ > $\frac{2}{9}$

Method 2: (Shortcut Method)

Convert each fraction into decimals.

$\frac{5}{2}=2.5$ , $\frac{3}{5}=0.6$ , $\frac{1}{3}=0.33$ , $\frac{2}{9}=0.22$

Clearly, 2.5 > 0.6 > 0.33 > 0.22

$\therefore$ $\frac{5}{2}$ > $\frac{3}{5}$ > $\frac{1}{3}$ > $\frac{2}{9}$

Important Rule : For fraction less than 1, if the difference between numerator and denominator of each of the given fractions remains the same, then the one with greater numerator is greater.

e.g. Arrange the following fractions in descending order.

$\frac{17}{19}$ , $\frac{25}{27}$ , $\frac{41}{43}$ , $\frac{13}{15}$ , $\frac{5}{7}$

Soln. In the given example difference between the numerator and denominator of all these fractions is same and equal to 2. The one with larger numerator is larger. Proceeding in similar way.

Hence, we have

$\frac{41}{43}$ > $\frac{25}{27}$ > $\frac{17}{19}$ > $\frac{13}{15}$ > $\frac{5}{7}$

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