\displaystyle \textbf{DECIMAL FRACTIONS}
\displaystyle \text{Decimal fractions, when expressed in decimal form, are known as}
\displaystyle \text{decimal numbers or simply decimals.}
\displaystyle \text{Examples: }\frac{8}{10},\ \frac{64}{100},\ \frac{329}{100}\text{ are decimal fractions.}
\displaystyle \text{In decimal form, they can be written as }0.8,\ 0.64,\text{ and }3.29,
\displaystyle \text{respectively.}
\displaystyle \\

\displaystyle \textbf{PARTS OF A DECIMAL}
\displaystyle \text{A decimal has two parts, namely the whole number part and the decimal part.}
\displaystyle \text{These parts are separated by a dot }(.)\text{ called the decimal point.}
\displaystyle \text{The digits lying to the left of the decimal point form the whole number part.}
\displaystyle \text{The decimal point together with the digits lying to its right forms the}
\displaystyle \text{decimal part.}
\displaystyle \text{Example: In the decimal }57.612,\text{ the whole number part is }57\text{ and the}
\displaystyle \text{decimal part is }.612.
\displaystyle \\

\displaystyle \textbf{DECIMAL PLACES}
\displaystyle \text{The number of digits contained in the decimal part of a decimal gives the}
\displaystyle \text{number of its decimal places.}
\displaystyle \text{Examples:}
\displaystyle \text{The decimal }5.48\text{ has }2\text{ decimal places.}
\displaystyle \text{The decimal }7.067\text{ has }3\text{ decimal places.}
\displaystyle \\

\displaystyle \textbf{LIKE DECIMALS}
\displaystyle \text{Decimals having the same number of decimal places are called like decimals.}
\displaystyle \text{Examples: }0.2,\ 33.4,\ 867.6,\ 4211.5\text{ are like decimals, each having}
\displaystyle \text{one decimal place.}
\displaystyle \\

\displaystyle \textbf{UNLIKE DECIMALS}
\displaystyle \text{Decimals having different numbers of decimal places are called unlike decimals.}
\displaystyle \text{Examples: }0.421,\ 3.31,\ 36.1,\ 391\text{ are unlike decimals.}
\displaystyle \\

\displaystyle \textbf{TO CONVERT UNLIKE DECIMALS TO LIKE DECIMALS}
\displaystyle \text{Among the given unlike decimals, find the decimal having the largest number}
\displaystyle \text{of decimal places, say }n.
\displaystyle \text{Convert each of the remaining decimals into one having }n\text{ decimal places}
\displaystyle \text{by annexing the required number of zeros to the extreme right of its}
\displaystyle \text{decimal part.}
\displaystyle \\

\displaystyle \textbf{REMARK}
\displaystyle \text{Annexing any number of zeros to the extreme right of the decimal part does}
\displaystyle \text{not change the value of a decimal.}
\displaystyle \text{Thus, }4.4=4.40=4.400=\cdots
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Convert }13.42,\ 0.123,\ 9.6,\text{ and }1.97\text{ into like decimals.}
\displaystyle \textbf{Solution: }0.123\text{ has the largest number of decimal places, i.e., }3.
\displaystyle \text{Therefore, we convert all the decimals into decimals having }3\text{ places.}
\displaystyle 13.42=13.420,\qquad 0.123=0.123
\displaystyle 9.6=9.600,\qquad 1.97=1.970
\displaystyle \therefore 13.420,\ 0.123,\ 9.600,\text{ and }1.970\text{ are like decimals.}
\displaystyle \\

\displaystyle \textbf{TO WRITE A DECIMAL IN EXPANDED FORM}
\displaystyle \text{A decimal can be written in expanded form by expressing each digit}
\displaystyle \text{according to its place value.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Arrange the digits of }395.174\text{ in a place-value chart. Hence, write}
\displaystyle 395.174\text{ in expanded form.}
\displaystyle \textbf{Solution:}
\displaystyle \text{We may arrange the digits of }395.174\text{ in the place-value chart below.}
\displaystyle \begin{array}{|c|c|c|c|c|c|} \hline \text{Hundreds}&\text{Tens}&\text{Ones}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline 3&9&5&1&7&4\\ \hline \end{array}
\displaystyle \therefore 395.174=300+90+5+0.1+0.07+0.004
\displaystyle =300+90+5+\frac{1}{10}+\frac{7}{100}+\frac{4}{1000}
\displaystyle =3\times10^2+9\times10^1+5\times1+1\times\frac{1}{10}
\displaystyle \qquad +7\times\frac{1}{10^2}+4\times\frac{1}{10^3}.
\displaystyle \\

\displaystyle \textbf{COMPARISON OF TWO DECIMALS}
\displaystyle \text{1. Convert the given decimals into like decimals, if necessary.}
\displaystyle \text{2. First compare their whole number parts. The decimal having the greater}
\displaystyle \text{whole number part is greater.}
\displaystyle \text{3. If the whole number parts are equal, compare the digits in the tenths}
\displaystyle \text{place. The decimal having the greater tenths digit is greater.}
\displaystyle \text{4. If the tenths digits are also equal, compare the hundredths digits and}
\displaystyle \text{continue in the same way until unequal digits are found.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Compare }17.63\text{ and }13.9.
\displaystyle \textbf{Solution:}
\displaystyle \text{First, convert the given decimals into like decimals.}
\displaystyle 17.63=17.63\qquad\text{and}\qquad13.9=13.90.
\displaystyle \text{Now compare their whole number parts. Since }17>13,
\displaystyle \therefore 17.63>13.90.
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Write }9.03,\ 4.85,\ 0.974,\ 7.5,\ 4.92\text{ and }0.7\text{ in descending order.}
\displaystyle \textbf{Solution:}
\displaystyle \text{Converting the given decimals into like decimals, we get:}
\displaystyle 9.030,\quad4.850,\quad0.974,\quad7.500,\quad4.920,\quad0.700.
\displaystyle \text{Clearly,}
\displaystyle 9.030>7.500>4.920>4.850>0.974>0.700.
\displaystyle \therefore \text{The given decimals in descending order are:}
\displaystyle 9.03,\quad7.5,\quad4.92,\quad4.85,\quad0.974,\quad0.7.
\displaystyle \\

\displaystyle \textbf{OPERATIONS ON DECIMALS}

\displaystyle \textbf{ADDITION OF DECIMALS}
\displaystyle \text{1. Convert the given decimals into like decimals, if necessary.}
\displaystyle \text{2. Write the addends one below the other, keeping their decimal points in}
\displaystyle \text{the same vertical column.}
\displaystyle \text{3. Add as in whole numbers. In the sum, place the decimal point directly}
\displaystyle \text{below the decimal points in the given numbers.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Add: }24.6,\ 8.57,\ 0.9,\ 136.2358\text{ and }3.07.
\displaystyle \textbf{Solution:}
\displaystyle \text{Converting the given decimals into like decimals, we get:}
\displaystyle 24.6000,\quad8.5700,\quad0.9000,\quad136.2358,\quad3.0700.
\displaystyle \text{Adding them column-wise, we get:}
\displaystyle \begin{array}{r} \phantom{1}24.6000\\ \phantom{1}\phantom{2}8.5700\\ \phantom{1}\phantom{2}0.9000\\ 136.2358\\ \phantom{1}\phantom{2}3.0700\\ \hline 173.3758\\ \hline \end{array}
\displaystyle \therefore 24.6+8.57+0.9+136.2358+3.07=173.3758.
\displaystyle \\

\displaystyle \textbf{SUBTRACTION OF DECIMALS}
\displaystyle \text{1. Convert the given decimals into like decimals, if necessary.}
\displaystyle \text{2. Write the subtrahend below the minuend, keeping their decimal points}
\displaystyle \text{in the same vertical column.}
\displaystyle \text{3. Subtract as in whole numbers. In the difference, place the decimal point}
\displaystyle \text{directly below the decimal points in the given numbers.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Subtract }19.56\text{ from }25.2.
\displaystyle \textbf{Solution:}
\displaystyle \text{Converting the given decimals into like decimals, we get }25.20\text{ and }19.56.
\displaystyle \text{Subtracting column-wise, we get:}
\displaystyle \begin{array}{r} 25.20\\ -19.56\\ \hline 5.64\\ \hline \end{array}
\displaystyle \therefore 25.20-19.56=5.64.
\displaystyle \\

\displaystyle \textbf{MULTIPLICATION OF DECIMALS}

\displaystyle \textbf{MULTIPLICATION OF A DECIMAL BY }10,\ 100,\ 1000,\textbf{ ETC.}
\displaystyle \text{On multiplying a decimal by }10,\ 100,\ 1000,\text{ etc., move the decimal}
\displaystyle \text{point to the right by as many places as there are zeros in the multiplier.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Multiply:}
\displaystyle \text{(i) }9.3\text{ by }10\qquad\text{(ii) }9.32\text{ by }100
\displaystyle \text{(iii) }58.4\text{ by }1000\qquad\text{(iv) }976.40\text{ by }100
\displaystyle \textbf{Solution:}
\displaystyle \text{(i) }9.3\times10=93
\displaystyle \text{The decimal point is moved }1\text{ place to the right.}

\displaystyle \text{(ii) }9.32\times100=932
\displaystyle \text{The decimal point is moved }2\text{ places to the right.}

\displaystyle \text{(iii) }58.4\times1000=58400
\displaystyle \text{The decimal point is moved }3\text{ places to the right.}

\displaystyle \text{(iv) }976.40\times100=97640
\displaystyle \text{The decimal point is moved }2\text{ places to the right.}
\displaystyle \\

\displaystyle \textbf{MULTIPLICATION OF A DECIMAL BY A WHOLE NUMBER}
\displaystyle \text{1. Ignoring the decimal point, multiply the decimal by the whole number}
\displaystyle \text{in the same way as two whole numbers are multiplied.}
\displaystyle \text{2. In the product, place the decimal point so that the product has the same}
\displaystyle \text{number of decimal places as the given decimal.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Multiply }74.53\text{ by }16.
\displaystyle \textbf{Solution:}
\displaystyle \text{Ignoring the decimal point, multiply }7453\text{ by }16.
\displaystyle \begin{array}{r} 7453\\ \times\phantom{0}16\\ \hline 44718\\ 74530\\ \hline 119248\\ \hline \end{array}
\displaystyle \text{Since }74.53\text{ has }2\text{ decimal places, the product also has }2.
\displaystyle \therefore 74.53\times16=1192.48.
\displaystyle \\

\displaystyle \textbf{MULTIPLICATION OF TWO OR MORE DECIMALS}
\displaystyle \text{1. Ignoring the decimal points, multiply the decimals as whole numbers.}
\displaystyle \text{2. In the product, place the decimal point so that the number of decimal}
\displaystyle \text{places equals the sum of the decimal places in all the given decimals.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Find the product of }9.76\text{ and }1.2.
\displaystyle \textbf{Solution:}
\displaystyle \text{Ignoring the decimal points, we get:}
\displaystyle 976\times12=11712.
\displaystyle \text{The total number of decimal places is }2+1=3.
\displaystyle \therefore 9.76\times1.2=11.712.
\displaystyle \\

\displaystyle \textbf{DIVISION OF DECIMALS}

\displaystyle \textbf{DIVIDING A DECIMAL BY }10,\ 100,\ 1000,\textbf{ ETC.}
\displaystyle \text{On dividing a decimal by }10,\ 100,\ 1000,\text{ etc., move the decimal}
\displaystyle \text{point to the left by as many places as there are zeros in the divisor.}
\displaystyle \\

\displaystyle \textbf{EXAMPLES}
\displaystyle \text{Divide: (i) }91.5\text{ by }10\qquad\text{(ii) }662.19\text{ by }100.
\displaystyle \textbf{Solution:}
\displaystyle \text{(i) }91.5\div10=9.15
\displaystyle \text{The decimal point is moved one place to the left.}

\displaystyle \text{(ii) }662.19\div100=6.6219
\displaystyle \text{The decimal point is moved two places to the left.}
\displaystyle \\

\displaystyle \textbf{DIVIDING A DECIMAL BY A WHOLE NUMBER}
\displaystyle \text{Divide as in ordinary division. In the quotient, place the decimal point}
\displaystyle \text{directly above the decimal point in the dividend. If required, annex zeros}
\displaystyle \text{to the right of the decimal part of the dividend to continue the division.}
\displaystyle \\

\displaystyle \textbf{DIVIDING A DECIMAL BY A DECIMAL}
\displaystyle \text{1. Convert the divisor into a whole number by multiplying both the dividend}
\displaystyle \text{and the divisor by the same suitable power of }10.
\displaystyle \text{2. Divide the new dividend by the resulting whole-number divisor.}
\displaystyle \\

\displaystyle \textbf{SIMPLIFICATION OF EXPRESSIONS INVOLVING DECIMALS}
\displaystyle \text{Use the BODMAS rule to simplify expressions involving decimals.}
\displaystyle \text{The operations are performed in the following order:}
\displaystyle \text{1. Brackets}\qquad\text{2. Of}\qquad\text{3. Division}
\displaystyle \text{4. Multiplication}\qquad\text{5. Addition}\qquad\text{6. Subtraction}
\displaystyle \\

\displaystyle \textbf{REMOVAL OF BRACKETS}
\displaystyle \text{When brackets are nested, simplify the innermost grouping first and then}
\displaystyle \text{proceed outwards. Common grouping symbols are:}
\displaystyle \text{1. Bar or Vinculum}\qquad\text{2. Parentheses }( )
\displaystyle \text{3. Curly Brackets }\{\,\}\qquad\text{4. Square Brackets }[\,]
\displaystyle \\

\displaystyle \textbf{CONVERSION OF A FRACTION INTO A DECIMAL}

\displaystyle \textbf{TO CONVERT A FRACTION INTO A DECIMAL BY DIVISION}
\displaystyle \text{1. Divide the numerator by the denominator.}
\displaystyle \text{2. If a non-zero remainder is obtained, place a decimal point in the}
\displaystyle \text{quotient and annex a zero to the remainder.}
\displaystyle \text{3. Continue the division in the same way as division of whole numbers.}
\displaystyle \text{4. Repeat the process until the remainder becomes zero, the digits begin}
\displaystyle \text{to repeat, or the required number of decimal places has been obtained.}
\displaystyle \\

\displaystyle \textbf{ROUNDING OFF OF DECIMALS}
\displaystyle \text{The process of obtaining the value of a decimal correct to the required}
\displaystyle \text{number of decimal places is called rounding off. The value obtained is}
\displaystyle \text{called the rounded-off value of the decimal.}
\displaystyle \\

\displaystyle \textbf{TO ROUND OFF A DECIMAL TO THE REQUIRED DECIMAL PLACES}
\displaystyle \text{1. Retain as many digits after the decimal point as are required and examine}
\displaystyle \text{the first digit immediately to the right of the last retained digit.}
\displaystyle \text{2. If this digit is }5\text{ or more, increase the last retained digit by }1.
\displaystyle \text{If it is less than }5,\text{ leave the last retained digit unchanged.}
\displaystyle \text{3. Omit all the digits to the right of the last retained digit.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Write }2.6483\text{ correct to }2\text{ decimal places.}
\displaystyle \textbf{Solution:}
\displaystyle \text{In }2.6483,\text{ retain the first }2\text{ digits after the decimal point.}
\displaystyle \text{The first omitted digit is }8,\text{ which is greater than }5.
\displaystyle \text{Therefore, increase the last retained digit, }4,\text{ by }1.
\displaystyle \therefore 2.6483\text{ rounded off to }2\text{ decimal places is }2.65.
\displaystyle \\

\displaystyle \textbf{TERMINATING DECIMALS}
\displaystyle \text{When a fraction is expressed as a decimal by division, if the division}
\displaystyle \text{terminates after a finite number of steps, the decimal is called a}
\displaystyle \text{terminating decimal.}
\displaystyle \text{Examples: }\frac{1}{2}=0.5\qquad\text{and}\qquad\frac{1}{4}=0.25.
\displaystyle \\

\displaystyle \textbf{NON-TERMINATING DECIMALS}
\displaystyle \text{When a fraction is expressed as a decimal by division, if the division}
\displaystyle \text{continues indefinitely and a zero remainder is never obtained, the decimal}
\displaystyle \text{is called a non-terminating decimal.}
\displaystyle \text{Example: }\frac{1}{3}=0.333\ldots
\displaystyle \\

\displaystyle \textbf{REPEATING OR RECURRING DECIMALS}
\displaystyle \text{If a digit or a group of digits in the decimal part repeats indefinitely,}
\displaystyle \text{the decimal is called a repeating or recurring decimal.}
\displaystyle \text{Examples: }\frac{1}{3}=0.333\ldots=0.\overline{3}
\displaystyle \text{and }\frac{5}{6}=0.8333\ldots=0.8\overline{3}.
\displaystyle \\

\displaystyle \textbf{PURE RECURRING DECIMALS}
\displaystyle \text{A recurring decimal in which all the digits after the decimal point form}
\displaystyle \text{the repeating block is called a pure recurring decimal.}
\displaystyle \text{Examples: }\frac{1}{3}=0.\overline{3}
\displaystyle \text{and }\frac{3}{7}=0.428571428571\ldots=0.\overline{428571}.
\displaystyle \\

\displaystyle \textbf{MIXED RECURRING DECIMALS}
\displaystyle \text{A recurring decimal in which one or more digits after the decimal point}
\displaystyle \text{do not repeat before the repeating block begins is called a mixed}
\displaystyle \text{recurring decimal.}
\displaystyle \text{Examples: }\frac{5}{6}=0.8333\ldots=0.8\overline{3}
\displaystyle \text{and }0.5833\ldots=0.58\overline{3}.
\displaystyle \\

\displaystyle \textbf{CONVERSION OF A DECIMAL INTO A FRACTION (RATIONAL NUMBER)}

\displaystyle \textbf{TO CONVERT A TERMINATING DECIMAL INTO A FRACTION}
\displaystyle \text{To convert a terminating decimal into a fraction, follow these steps:}
\displaystyle \text{1. Write the given decimal without the decimal point as the numerator.}
\displaystyle \text{2. Take }1\text{ followed by as many zeros as there are decimal places in}
\displaystyle \text{the given decimal as the denominator.}
\displaystyle \text{3. Reduce the resulting fraction to its simplest form.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Convert }13.484\text{ into a fraction.}
\displaystyle \textbf{Solution:}
\displaystyle 13.484=\frac{13484}{1000}=\frac{3371}{250}=13\frac{121}{250}.
\displaystyle \\

\displaystyle \textbf{TO CONVERT A RECURRING DECIMAL INTO A FRACTION}
\displaystyle \text{The method of converting a recurring decimal into a fraction is explained}
\displaystyle \text{in the following example.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Express }0.666\ldots\text{ as a fraction.}
\displaystyle \textbf{Solution:}
\displaystyle \text{Let }x=0.666\ldots
\displaystyle \therefore 10x=6.666\ldots
\displaystyle \text{Subtracting the first equation from the second equation, we get:}
\displaystyle 10x-x=6.666\ldots-0.666\ldots
\displaystyle \Rightarrow 9x=6
\displaystyle \Rightarrow x=\frac{6}{9}=\frac{2}{3}.
\displaystyle \therefore 0.666\ldots=\frac{2}{3}.
\displaystyle \\

\displaystyle \textbf{TO CONVERT A PURE RECURRING DECIMAL INTO A FRACTION}
\displaystyle \textbf{(SHORT-CUT METHOD)}
\displaystyle \text{Write the repeating digits only once in the numerator and write as many}
\displaystyle \text{nines in the denominator as there are digits in the repeating block.}
\displaystyle \text{For a decimal with a whole number part, retain the whole number part and}
\displaystyle \text{convert only the recurring decimal part into a fraction.}
\displaystyle \\

\displaystyle \textbf{EXAMPLES}
\displaystyle \text{(i) }0.\overline{4}=\frac{4}{9}.

\displaystyle \text{(ii) }0.\overline{54}=\frac{54}{99}=\frac{6}{11}.

\displaystyle \text{(iii) }1.\overline{074}=1+\frac{74}{999}=1\frac{74}{999}.
\displaystyle \\

\displaystyle \textbf{TO CONVERT A MIXED RECURRING DECIMAL INTO A FRACTION}
\displaystyle \textbf{(SHORT-CUT METHOD)}
\displaystyle \text{In the numerator, take the difference between the number formed by all the}
\displaystyle \text{digits in the decimal part, taking the repeating digits only once, and the}
\displaystyle \text{number formed by the non-repeating digits.}
\displaystyle \text{In the denominator, write as many nines as there are repeating digits,}
\displaystyle \text{followed by as many zeros as there are non-repeating digits.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Convert }0.5\overline{3}\text{ into a fraction.}
\displaystyle \textbf{Solution:}
\displaystyle 0.5\overline{3}=\frac{53-5}{90}=\frac{48}{90}=\frac{8}{15}.
\displaystyle \\

\displaystyle \textbf{REMARK}
\displaystyle \text{The order relation between fractions can also be established by converting}
\displaystyle \text{the fractions into decimals and then comparing their decimal values.}
\displaystyle \\

\displaystyle \textbf{H.C.F. AND L.C.M. OF DECIMALS}
\displaystyle \text{To find the H.C.F. or L.C.M. of given decimals, follow these steps:}
\displaystyle \text{1. Convert the given decimals into like decimals.}
\displaystyle \text{2. Remove the decimal points and find the H.C.F. or L.C.M. of the resulting}
\displaystyle \text{whole numbers, as required.}
\displaystyle \text{3. In the result, place the decimal point so that it has the same number of}
\displaystyle \text{decimal places as each of the like decimals obtained in Step 1.}
\displaystyle \\

\displaystyle \textbf{EXAMPLE}
\displaystyle \text{Find the H.C.F. and L.C.M. of }0.54,\ 1.8\text{ and }7.2.
\displaystyle \textbf{Solution:}
\displaystyle \text{Converting the given decimals into like decimals, we get:}
\displaystyle 0.54,\qquad1.80,\qquad7.20.
\displaystyle \text{Each of these decimals has }2\text{ decimal places.}
\displaystyle \text{Removing the decimal points, we get }54,\ 180\text{ and }720.
\displaystyle \text{H.C.F. of }54,\ 180\text{ and }720=18.
\displaystyle \therefore \text{H.C.F. of }0.54,\ 1.8\text{ and }7.2=0.18.
\displaystyle \text{L.C.M. of }54,\ 180\text{ and }720=2160.
\displaystyle \therefore \text{L.C.M. of }0.54,\ 1.8\text{ and }7.2=21.60=21.6.
\displaystyle \\

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