## Fractions and Decimals

Category : 7th Class

FRACTIONS AND DECIMALS

FUNDAMENTALS

• 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.

Various Types of Fractions

1. Improper fraction: A fraction whose numerator is greater than or equal to its denominator is called an improper fraction. If number is written as$\frac{N}{D}$, then $N~\underline{>}\,D$where N = numerator D = Denominator e.g.,$\frac{100}{100},$ $\frac{51}{50},$ $\frac{45}{2},.....$etc.
2. Proper fraction: A fraction whose numerator is less than its denominator is called a proper fraction. e.g.,$\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{8}{18},......$etc.
3. Decimal fraction: A fraction whose denominator is 10, 100, 1000 etc., is called a decimal fraction. e.g., $\frac{2}{10},\,\,\frac{8}{100},\,\,\frac{26}{1000},\,\,\frac{1312}{1000},......$etc.
4. Vulgar fraction: A fraction whose denominator is a whole number other than 10, 100, 1000, etc., is called a vulgar fraction. e.g., $\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{83}{103},\,\,......$etc.
5. Simple fraction: A fraction in its lowest terms is known as a simple fraction. e.g., $\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{8}{17},\,\,......$etc.
6. 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., $5\frac{2}{3},\,\,6\frac{1}{6},\,\,1\frac{3}{4},\,\,107\frac{1}{2},\,\,.......$etc.

$5\frac{2}{3}$can be written as $5+\frac{2}{3}$

1. Like fractions: Fractions having the same denominator but different numerators are called like fractions. e.g.$\frac{6}{17},\,\,\frac{9}{17},\,\,\frac{11}{17},\,\,......$etc.
2. Unlike fractions: Fractions having different denominators are called unlike fractions. e.,$\frac{3}{4},\,\,\frac{6}{7},\,\,\frac{5}{6},\,\,\frac{4}{5},\,\,\frac{8}{13},\,\,.......$etc.
• An important property: If the numerator and denominator of a fraction are both multiplied by the same non zero number, its value is not changed. Thus, $\frac{2}{7}=\frac{2\times 2}{7\times 2}=\frac{2\times 5}{7\times 5}......$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. (See above property) e.g., Equivalent fractions of $\frac{6}{12}$are $\frac{8}{16},\frac{1}{2},\frac{4}{8},\frac{5}{10}$......etc.

• Method of changing unlike fraction to like fraction:

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 fraction.

e.g., Convert the fractions $\frac{1}{6},\,\,\frac{4}{9}$ and $\frac{70}{12}$ into like fractions.

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

Now, $\frac{1}{6}=\frac{1\times 6}{6\times 6}=\frac{6}{36};\frac{4}{9}=\frac{4\times 4}{9\times 4}=\frac{16}{36}$ and$\frac{70}{12}=\frac{70\times 3}{12\times 3}=\frac{21}{36}$.

Clearly, $\frac{6}{36},\frac{16}{36}$ and $\frac{21}{36}$ are like fractions.

Utility: This method is used for comparing 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 fraction: Let $\frac{a}{b}$ and $\frac{c}{d}$ be two given fractions. Then,

(a) $\frac{a}{b}\text{}\frac{c}{d}\text{ }\Leftrightarrow ad>bc~~$                     (b) $\frac{a}{b}\text{=}\frac{c}{d}\text{ }\Leftrightarrow ad=bc~~$                       (c) $\frac{a}{b}\text{}\frac{c}{d}\Leftrightarrow \text{ }ad<bc$

These are important results and should be committed to memory. (or understood as coming out of cross = multiplication).

• Method of comparing more than two fractions;

Step - 1: Convert all the given fractions into like fractions, each having same denominator (follow the method as given above).

Step - 2: Now, if we compare any two of these like fractions, then the one having larger numerator, is larger.

• Addition and subtraction of fractions:

(a) Add / Subtract like fractions:

e.g., (1)  Add $\frac{2}{9}$and $\frac{4}{9}$

$\frac{2}{9}+\frac{4}{9}=\frac{2+4}{9}=\frac{6}{9}$

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

$\frac{6}{7}-\frac{4}{7}=\frac{6-4}{7}=\frac{2}{7}$

To add/subtract unlike fractions, first convert them into like fractions and proceed as above.

e.g., (1) Add $\frac{1}{3},\frac{3}{5}$and$\frac{4}{7}$.

L.C.M of 3, 5 and 7 is 105.

$\therefore \,\,\frac{1}{3}+\frac{3}{5}+\frac{4}{7}=\frac{35}{105}+\frac{63}{105}+\frac{60}{105}=\frac{35+63+60}{105}=\frac{158}{105}=1\frac{53}{105}$

e.g., (2) Subtract$\frac{1}{4}$from$\frac{11}{12}$.

$\frac{11}{12}-\frac{1}{4}=\frac{11-3}{12}=\frac{8}{12}=\frac{2}{3}$

• Multiplication of fraction, by a whole number:

(a) Proper / Improper fraction: In order 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., $6\times \frac{16}{17}=\frac{96}{17}=5\frac{11}{17}$

(b) Multiplying a whole number by a mixed fraction: In order multiply a whole number by a mixed fraction, first covert the mixed fraction to an improper fraction and then multiply.

e.g.,$21\times 6\frac{3}{7}=21\times \frac{45}{7}=145$

(c) Multiplying a fraction by a fraction: In order to multiply a fraction by another fraction, we take the product of their numerators and divide it by the product of their denominators. The result is then 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).

Elementary question

Express in lowest terms the following number $\frac{10}{23}\times \frac{46}{47}$

Ans.     $\frac{10\times 46}{23\times 47}=\frac{460}{1081}$; this is expressed m lowest terms as$\frac{20}{47}$.

• Reciprocal of a fraction:

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

e.g., $\frac{3}{7}$ and $\frac{7}{3}$ are the reciprocals of each other, since$\left( \frac{3}{7}\times \frac{7}{3} \right)=1.$

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

Example: Reciprocal of ${}^{6}/{}_{7}$ is $\frac{1}{{}^{6}/{}_{7}}=1\times \frac{7}{6}=\frac{7}{6}$

Note: Reciprocal of 0 does not exist.

• Division of fractions:

(a) Division of a whole number by any fraction: In order to divide a whole number by a fraction, we have to multiply the whole number by the reciprocal of the given fraction.

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

(b) Division of a fraction by a whole number: In order 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.

(c) Division of a fraction by another fraction: In order to divide a fraction by another fraction, we have to multiply the first fraction by the reciprocal of the second.

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

Decimals

• The number expressed in decimal form are called decimal numbers or simply decimals.

e.g., 6.0, 16.53, 26.67, 0.012 etc.

• A decimal number has two parts, namely

(i) Whole - number part and                     (ii) Decimal part

These parts are separated by a dot (.), called the decimal point.

Hence, read as one hundred twenty three point two three four

• Representing decimals on number line: Let us represent 0.6 on the number line.

0.6 lies between 0 and 1.

0.6 is 6 tenths.

So, divide the unit length between 0 and 1 into 10 equal parts & take 6 parts as shown.

• The number of digits in the decimal part of a decimal number given the number of decimal places.

e.g., 6.29 has two decimal places and 7.123 has three decimal places.

• Decimals having the same number of decimal places are called like decimals,

e.g., 6.66, 7.03, 18.21 etc.,

• Decimal having different number of decimal places are called unlike decimals,

e.g., 333.1, 226.02, 111.111 etc.,

Note: Adding any number of zeros to the extreme right of the decimal part of a decimal number does not change its value.

e.g., 3.13 = 3.130 = 3.1300 etc.

Comparing decimals.

Step - 1: Convert the given decimal numbers into like decimals.

Step - 2: First compare the whole - number part.

Step - 3: 'If the whole number parts are equal, compare the tenths digit. The decimal number with the greater digit in the tenths place is greater.

Step - 4: If the tenths digits are also equal, compare the hundredths digits, and so on.

e.g.,      (i) 66.27 > 63.27

(ii) 21.65 > 21.45

(iii) 202.013 < 202.018

• Conversion of decimals to fractions:

e.g., express 6.16 as a fraction.

Step - 1:  In the numerator, write the give decimal number without the decimal point.

$6.16=\frac{616}{\left( \,\,\,\,\, \right)}$

Step - 2:   From 'right to left', count the number of decimal places, say 'x'. Here x = 2.

Step - 3:  hi the denominator, write 1 followed by 'x' number of zeros. $6.16=\frac{616}{100}$

Step - 4: Simplify the obtained fraction into its lowest terms. $\frac{616}{100}=\frac{154}{25}$

• Conversion of fractions to decimals:

(i) Conversion of decimal fraction to decimal.

Express $\frac{27153}{1000}$ as a decimal

Step - 1: Write the numerator as it is, 27153.

Step - 2: Count the number of zeros in the denominator. Let it be 'x'. Here x = 3.

Step - 3: From 'right to left' in the number, place the decimal point after 'x' places.

$\therefore \,\,\,\frac{27153}{1000}=27.153$

(ii) Conversion of a non-decimal fraction into a decimal.

Express $\frac{618}{7}$ as a decimal.

Step - 1: Divide the number as usual.

Step - 2: If dividend is less than divisor, place the decimal point in quotient.

Step - 3: Place a zero m the dividend.

Step - 4: Repeat the division process till remainder is zero or as per the required number of decimal places.

Note:-   (a) Express $\frac{618}{7}$ as a decimal (upto 3 decimal) $\frac{618}{7}=88.285$

(b) Express $\frac{618}{7}$ as a decimal (upto 2 decimals) $\frac{618}{7}=88.28$

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