DECIMAL FRACTION :- Decimal fraction is a fraction in which the denominator is power of 10.
For eg. $\mathbf{\frac{1}{10}, \frac{3}{100}, \frac{7}{1000}}$, etc.
To read a deciaml number, the digits right side to the point are read digit by digit not together.
For eg. 0.34 is read as ( zero point three,four. ), not read as zero point thirty four.
CONVERTING A DECIMAL INTO A VULGAR FRACTION :-
RULE :- Write down the numerator without the decial point, and for the denominator put 1 under the decimal point and put as many zeros as the digits after the decimal point( in the numerator ).
For eg. Convert 0.51 into vulgar fraction.
$\mathbf{0.51=\frac{051}{100}=\frac{51}{100}}$.
NOTE :-
1. Adding zeros to the right of a decimal fraction does not change its value.
Thus, 0.7 = 0.70 = 0.700 etc.
2. If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
Thus, $\boldsymbol{\frac{1.71}{2.51}=\frac{171}{251}}$.
OPERATIONS ON DECIMAL FRACTIONS :-
1. ADDITION OF DECIMAL :- The numbers are so placed upon each other that the decimal point lie in one column. The numbers so arranged can now be added in the usual way.
For eg. Add: 5.02, 1.13 and 0.9.
$\begin{align*}5.02\\+\ \: 1.13\\0.9\! \; \;\\\hline7.05\\\hline\end{align*}$
2. SUBTRACTION OF DECIAML :- The numbers are so placed upon each other that the decimal point lie in one column. The numbers so arranged can now be subtracted in the usual way.
For eg. Subtract 6.001 from 21.1
$\begin{align*}21.100\\-\, \: 6.001\\ \hline15.099\\ \hline\end{align*}$
3. MULTIPLICATION OF DECIMAL FRACTION BY 10, 100, 1000, etc. (power of 10) :- Shift the decimal point as many places to the right, as many as there are zeros in the multiplier.
For eg. Multiply 35.02 by 1000.
sol. 35.02 $\times $ 1000
= 35020.
4. MULTIPLICATION OF DECIMAL FRACTION :- Multiply the given number as integers and in the product mark as many decimal places as there are in the case of the multiplier and the
multiplicand together, and prefixing zeros if necessary.
For eg. Multiply 0.7 by 0.15
sol. 7 $\times $ 15 = 105
Sum of decimal places ( 1 + 2) = 3
$ \therefore$ 0.7 $\times $ 0.15 = .105
5. DIVISION OF DECIMAL FRACTION BY A COUNTING NUMBER :- Divide the given number as usual way ( i.e without considering the decimal point ) by the counting number, and put the decimal point as many places given in the divident to the quotient so obtained.
For eg. Divide .0042 by 6.
sol. $\frac{42}{6}= 7$
$\Rightarrow$ $\frac{.0042}{6}=.0007$ ( four places of decimal )
6. DIVISION OF DECIMAL FRACTION BY DECIMAL :- Make the divisior as a whole number by multiplying the given dividend and divisior by a suitable multiple of 10, and solve it by the method "division of decimal fraction by a counting number".
For eg. Divide 0.00072 by 0.09.
sol. $\frac{0.00072\times 100}{0.09\times 100}=\frac{0.072}{9}=.008$
7. COMPARISON OF FRACTIONS :- First convert all fractions into decimal form and then arranged them ascending or descending order.
For eg. Arrange the fraction $\frac{3}{7},\frac{5}{9}\, and\, \frac{4}{5}$ in descending order.
sol. convert all fractions into decimal form,
$\frac{3}{7}=0.428,\frac{5}{9}=0.555,\frac{4}{5}=0.8$
now, 0.8 > 0.555 > 0.428
so, $\frac{4}{5}>\frac{5}{9}>\frac{3}{7}$
For eg. $\mathbf{\frac{1}{10}, \frac{3}{100}, \frac{7}{1000}}$, etc.
To read a deciaml number, the digits right side to the point are read digit by digit not together.
For eg. 0.34 is read as ( zero point three,four. ), not read as zero point thirty four.
CONVERTING A DECIMAL INTO A VULGAR FRACTION :-
RULE :- Write down the numerator without the decial point, and for the denominator put 1 under the decimal point and put as many zeros as the digits after the decimal point( in the numerator ).
For eg. Convert 0.51 into vulgar fraction.
$\mathbf{0.51=\frac{051}{100}=\frac{51}{100}}$.
NOTE :-
1. Adding zeros to the right of a decimal fraction does not change its value.
Thus, 0.7 = 0.70 = 0.700 etc.
2. If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
Thus, $\boldsymbol{\frac{1.71}{2.51}=\frac{171}{251}}$.
OPERATIONS ON DECIMAL FRACTIONS :-
1. ADDITION OF DECIMAL :- The numbers are so placed upon each other that the decimal point lie in one column. The numbers so arranged can now be added in the usual way.
For eg. Add: 5.02, 1.13 and 0.9.
$\begin{align*}5.02\\+\ \: 1.13\\0.9\! \; \;\\\hline7.05\\\hline\end{align*}$
2. SUBTRACTION OF DECIAML :- The numbers are so placed upon each other that the decimal point lie in one column. The numbers so arranged can now be subtracted in the usual way.
For eg. Subtract 6.001 from 21.1
$\begin{align*}21.100\\-\, \: 6.001\\ \hline15.099\\ \hline\end{align*}$
3. MULTIPLICATION OF DECIMAL FRACTION BY 10, 100, 1000, etc. (power of 10) :- Shift the decimal point as many places to the right, as many as there are zeros in the multiplier.
For eg. Multiply 35.02 by 1000.
sol. 35.02 $\times $ 1000
= 35020.
4. MULTIPLICATION OF DECIMAL FRACTION :- Multiply the given number as integers and in the product mark as many decimal places as there are in the case of the multiplier and the
multiplicand together, and prefixing zeros if necessary.
For eg. Multiply 0.7 by 0.15
sol. 7 $\times $ 15 = 105
Sum of decimal places ( 1 + 2) = 3
$ \therefore$ 0.7 $\times $ 0.15 = .105
5. DIVISION OF DECIMAL FRACTION BY A COUNTING NUMBER :- Divide the given number as usual way ( i.e without considering the decimal point ) by the counting number, and put the decimal point as many places given in the divident to the quotient so obtained.
For eg. Divide .0042 by 6.
sol. $\frac{42}{6}= 7$
$\Rightarrow$ $\frac{.0042}{6}=.0007$ ( four places of decimal )
6. DIVISION OF DECIMAL FRACTION BY DECIMAL :- Make the divisior as a whole number by multiplying the given dividend and divisior by a suitable multiple of 10, and solve it by the method "division of decimal fraction by a counting number".
For eg. Divide 0.00072 by 0.09.
sol. $\frac{0.00072\times 100}{0.09\times 100}=\frac{0.072}{9}=.008$
7. COMPARISON OF FRACTIONS :- First convert all fractions into decimal form and then arranged them ascending or descending order.
For eg. Arrange the fraction $\frac{3}{7},\frac{5}{9}\, and\, \frac{4}{5}$ in descending order.
sol. convert all fractions into decimal form,
$\frac{3}{7}=0.428,\frac{5}{9}=0.555,\frac{4}{5}=0.8$
now, 0.8 > 0.555 > 0.428
so, $\frac{4}{5}>\frac{5}{9}>\frac{3}{7}$
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