7th Class Mathematics Decimals and Fractions Fractions and Decimals

Fractions and Decimals

• A fraction is a part of a whole.
• A number of the form \[\frac{p}{q}\], where p and q are whole numbers and q\[\ne \]0 is known as a fraction.
• In the fraction\[\frac{p}{q}\], p is called the numerator and q is called the denominator.
• The numerator tells us how many parts are considered of the whole.
• The denominator tells us how many equal parts the whole is divided into.

            Note: Usually fractions are written in their lowest terms.

            The numerator and the denominator of a fractions in its lowest are coprime.

            That is, their H. C.F. is 1.

• Types of fractions:

            (i)  Simple fraction: A fraction in its lowest terms is known as a simple fraction.

            e.g.,\[\frac{12}{25},\frac{5}{7},-\frac{4}{3}\,\,etc.,\]

            (ii) Decimal fraction: A fraction whose denominator is 10, 100, 1000 etc., is called a decimal fraction.

            e.g.,\[\frac{3}{10},\frac{7}{100},\frac{24}{1000},\frac{131}{1000}\,etc.\]

            (iii) Vulgar fraction: A fraction whose denominator is a whole number other than 10, 100, 1000, etc., is called a vulgar fraction.

            e.g.,\[\frac{2}{9},\frac{4}{13},\frac{11}{20},\frac{27}{109}etc.,\]

            (iv) Proper fraction: A fraction whose numerator is less than its denominator is called a proper fraction.                                                                                            

            e.g.,\[\frac{3}{7},\frac{5}{11},\frac{23}{40},\frac{73}{100}etc.,\]                                                   

            (v) Improper fraction: A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.         

            e.g.,\[\frac{11}{7},\frac{25}{12},\frac{41}{36},\frac{53}{53}etc.,\]

            (vi) Mixed fraction: A number which can be expressed as the sum of a natural number and a proper fraction is called a mixed fraction.

            e.g.,\[1\frac{3}{4},4\frac{5}{7},7\frac{3}{13},12\frac{6}{5}etc.,\]

• Like fractions: Fractions having the same denominator but different numerators are called like fractions.

            e.g.,\[\frac{5}{14},\frac{9}{14},\frac{11}{14},etc.,\]

• Unlike fractions: Fractions having different denominators are called unlike fractions,

            e.g.,\[\frac{2}{5},\frac{5}{7},\frac{9}{13},etc.,\]

• An important property: If the numerator and denominator of a fraction are both multiplied by the same none zero number, its value is not changed.

            Thus,\[\frac{3}{4},=\frac{3\times 2}{4\times 2}=\frac{3\times 3}{4\times 3}=\frac{3\times 4}{4\times 4}\,\,etc.,\]

• Equivalent fractions: A given fraction and the fraction obtained by multiplying (or dividing) its numerator and denominator by the same non-zero number, are called equivalent fractions.

            E.g., Equivalent fractions of \[\frac{9}{12}\]are \[\frac{3}{4},\frac{6}{8},\frac{12}{16}\] etc.,

• Method of changing unlike fractions to like fractions:

            Step 1: Find the L.C.M. of the denominators of all the given fractions.

            Step 2: Change each of the given fractions into an equivalent fraction having denominator equal to the L.C.M. of the denominators of the given fractions.

• g., convert the fraction \[\frac{5}{6},\frac{7}{9}\,\,and\,\frac{11}{12}\] into like fractions.

            L.C.M. of 6, 9 and 12 = 3\[\times \]2 \[\times \]3 \[\times \] 2 = 36

            Now,\[\frac{5}{6}=\frac{5\times 6}{6\times 6}=\frac{30}{36};\,\,\,\,\,\,\frac{7}{9}=\frac{7\times 4}{9\times 4}=\frac{28}{36}\] and

            \[\frac{11}{12}\times \frac{11\times 3}{12\times 3}=\frac{33}{36}.\]

            Clearly, \[\frac{30}{36},\frac{28}{36}\]and \[\frac{33}{36}\] are like fractions.

• Irreducible fractions: A fraction \[\frac{a}{b}\]is said to be irreducible or in lowest terms, if the H.C.F of a and b is 1. They are also called simple fractions.

            If H.C.F. of a and b is not 1, then\[\frac{a}{b}\]is said to be reducible.

• Comparing fractions: Let \[\frac{a}{b}\] and \[\frac{c}{d}\] be two given fractions. Then,

            (a)\[\frac{a}{b}>\frac{c}{d}\Leftrightarrow ad>bc\]          (b)\[\frac{a}{b}=\frac{c}{d}\Leftrightarrow ab=bc\]

            (iii) \[\frac{a}{b}>\frac{c}{d}\Leftrightarrow ad<bc\]

• Method of comparing more than two fractions:

            Step 1: Find the LC.M. of the denominators of the given fractions. Let it be m.

            Step 2: Convert all the given fractions into like fractions, each having m as denominator.

            Step 3: Now, if we compare any two of these like fractions, then the one having larger numerator is larger.

• Addition and subtraction of fractions:

            (i) Add/Subtract like fractions:

            To add/subtract like fractions, add/subtract the numerators and place the sum/ difference on the same denominator as that of the given fractions.

            e.g.,(1) Add\[\frac{2}{7}\] and \[\frac{3}{7}\].

            \[\frac{2}{7}+\frac{3}{7}=\frac{2+3}{7}=\frac{5}{7}\]

            e.g., (2) Subtract \[\frac{4}{7}from\frac{6}{7}\].

            (ii) Add/Subtract unlike fractions:

            To add/subtract unlike fractions, first convert them into like fractions and proceed as in (i).

            e.g., (1) Add \[\frac{1}{3},\frac{2}{5}and\frac{3}{7}\]

            LC.M of 3, 5 and 7 is 105.

            e.g., (2) Subtract \[\frac{3}{4}from\frac{7}{12}\].

• Multiplication of fractions:

            Multiplying a whole number with a proper or an improper fraction: To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.

            e.g.,\[14\times \frac{4}{9}=\frac{12}{9}=\frac{12}{9}=\frac{4}{3}\]

• Multiplying a whole number by a mixed fraction: To multiply a whole number by a mixed fraction, first convert the mixed fraction to an improper fraction and then multiply.

            e.g.,\[14\times 2\frac{3}{7}=14\times \frac{17}{7}=37\]

• Multiplying a fraction by a fraction: The product of two or more fractions is the product of their numerators divided by the product of their denominators expressed

            in lowest terms i.e., if\[\frac{a}{b}\] and\[\frac{c}{d}\] are the fractions then their product is\[\frac{ac}{bd}\]expressed in the lowest terms.

• Calculating fractional part of a quantity:

            To know the fractional part of a quantity, the fraction and the quantity are multiplied.

            e.g., \[\frac{1}{3}\]of Rs. 90 =\[Rs.\,\frac{1}{3}\times 90\]= Rs.30 

• Reciprocal of a fraction:

            Two fractions are said to be the reciprocal of each other, if their product is 1.

            e.g.,\[\frac{4}{9}\,and\frac{9}{4}\] and are the reciprocals of each other, since\[\left( \frac{4}{9}\times \frac{9}{4} \right)=1\].

            In general, if\[\frac{a}{b}\]is a non-zero fraction, then its reciprocal is.\[\frac{b}{a}\].

            Note: Reciprocal of o does not exist.

• Division of fractions:

            Division of a whole number by any fraction: To divide a whole number by a fraction, we have to multiply the whole number by the reciprocal of the given fraction.

            e.g.,\[3\div 2\frac{2}{5}=3\div \frac{12}{5}=\frac{3\times 5}{12}=\frac{5}{4}\]

• Division of a fraction by a whole number: To divide a fraction by a whole number, we have to multiply the given fraction by the reciprocal of the whole number.

            e.g.\[4\frac{2}{5}\div 11=\frac{22}{5}\times \frac{1}{11}=\frac{2}{5}\]

            While dividing a mixed fraction by a whole number, convert the mixed fraction into an improper fraction and then divide.

• Division of a fraction by another fraction: To divide a fraction by another fraction, we have to multiply the first fraction by the reciprocal of the second.

            e.g.,\[3\frac{1}{4}\div 2\frac{1}{5}=\frac{13}{4}\div \frac{11}{5}=\frac{13}{4}\times \frac{5}{11}=\frac{65}{44}=1\frac{21}{44}\]