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.
(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 more...