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Examples

May 29, 2020
Category: Examples

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(1) 0.000008569 when expressed in scientific notation, is :

Soln. 0.000008569 = 8.569 $\times$ $10^{-6}$

( Decimal–point changes its place 6 digits right and so we have $10^{-6}$ )

(2) 47896 when expressed in scientific notation, is :

Soln. 47896 = 4.7896 $\times$ $10^{4}$

( Decimal–point changes its place 4 digits left and so we have $10^{4}$ )

When the numerator and the denominator of the proper fractions increase by a constant value, the last fraction is the biggest.

In general, in the group of fractions $\frac{x}{y},\frac{x+a}{y+b},\frac{x+2a}{y+2b},\frac{x+3a}{y+3b},...,\frac{x+na}{y+nb};$

$\frac{x+na}{y+nb}$ has the highest value.

(3) Which one of the following is the greatest ?

(i) $\frac{3}{4},\frac{5}{6},\frac{7}{8}$ (ii) $\frac{1}{8},\frac{2}{11},\frac{3}{14}$ (iii) $\frac{4}{7},\frac{22}{25},\frac{13}{16}$ (iv) $\frac{8}{19},\frac{2}{5},\frac{5}{12}$

Solution:

(i) $\frac{3}{4},\frac{5}{6},\frac{7}{8}$

We see that the numerators as well as denominators of the above fractions increase by 2, so the last fraction i.e. $\frac{7}{8}$ is the greatest fraction.

(ii) $\frac{4}{7},\frac{22}{25},\frac{13}{16}=\frac{4}{7},\frac{13}{16},\frac{22}{25}$

We see that the numerators as well as denominators of the above fractions increase by 9, so the last fraction i.e. $\frac{22}{25}$ is the greatest fraction.

(iii) $\frac{1}{8},\frac{2}{11},\frac{3}{14}$

We see that the numerators as well as denominators of the above fractions increase by constant values ( numerators by 1 and denominators by 9 ) , so the last fraction i.e. $\frac{3}{14}$ is the greatest fraction.

(iv) $\frac{8}{19},\frac{2}{5},\frac{5}{12}$ = $\frac{2}{5},\frac{5}{12},\frac{8}{19}$

We see that the numerators as well as denominators of the above fractions increase by constant values ( numerators by 3 and denominators by 7 ) , so the last fraction i.e. $\frac{8}{19}$ is the greatest fraction.