Rational numbers can be written in different forms.
(1) Rational numbers as fractions:
A rational number is also a fraction, as it can be written in \(\dfrac{a}{b}\) form, where \(b\ne0\).
For example: \(\dfrac{3}{4}\)
(2) Rational numbers as mixed numbers:
Mixed numbers can be converted into improper fractions.
For example: \(3\dfrac{1}{2}\)
\(=\dfrac{(3×2)+1}{2}=\dfrac{6+1}{2}=\dfrac{7}{2}\)
\(\therefore\;\dfrac{7}{2}\) is a rational number.
(3) Rational numbers as ratio:
A ratio can be written as a fraction.
\(\therefore\) It is also a rational number.
For example: \(3:2\) \(=\dfrac{3}{2}\)
\(\dfrac{3}{2}\) is a rational number.
(4) Rational numbers as percent:
A percent can be changed into a fraction by putting it over \(100\).
For example: \(25\,\%\)
\(=\dfrac{25}{100}\)
\(=\dfrac{25\div25}{100\div25}=\dfrac{1}{4}\)
Thus, \(\dfrac{1}{4}\) is a rational number.
A \(65\,\%=\dfrac{13}{20}\)
B \(3:4=\dfrac{6}{2}\)
C \(-9=\dfrac{1}{9}\)
D \(\dfrac{6}{5}=20\,\%\)
Example: \(\dfrac{3}{2}\)
Example: \(\sqrt {15}\)
The types are described as follows:
(1) Terminating decimals: The decimals that end with a finite (limited) number of digits after decimal are known as terminating decimals.
For example: \(0.8,\;0.25,\;0.46374\) etc.
(2) Repeating decimals: The decimals which do not end but one or more digits keep on repeating after decimal are known as repeating decimals or non-terminating but repeating decimals.
Here, 'bar' represents the repetition.
For example: \(0.6262...=0.\overline{62}\)
\(0.888=0.\overline8\)
\(2.1575757...=2.1\overline{57}\)
(3) Non-terminating and non-repeating decimals:
The decimals which neither end with a finite number of digits nor the digits keep on repeating are called non-terminating and non-repeating decimals.
For example: \(2.42152434...\)
Note : Those squares roots which are not perfect square numbers, always give non-terminating and non-repeating decimals.
For example: \(\sqrt 3=1.732.....\)
A \(\sqrt 3\) is an irrational number.
B \(0.\overline2\) is a rational number.
C \(0.232\overline5\) is a non-terminating and non-repeating decimal.
D \(0.2\overline3\) is a repeating decimal.
Case I: When only \(1\) digit after the decimal repeats.
For example: \(\dfrac{16}{45}\)
Denominator \(45\) cannot be changed into \(10\) or \(100\) or \(1000\,.....\), thus, we should use long division method.
Here, \(25\) is the repeating remainder and \(0.3555\,.....\) is a repeating decimal or repetend.
Thus, \(\dfrac{16}{45}=0.3555\,.....\)
\(=0.3\overline5\)
Case II: When more than \(1\) digit repeat.
For example:
\(\dfrac{1}{11}\)
Here, repeating remainder is \(1\) and \(.090909\,.....\) is a repeating decimal or repetend.
Thus, \(\dfrac{1}{11}=0.090909\,.....\)
\(=0.\overline{09}\)
Here, repeating remainder is \(1\) and \(.090909\,.....\) is a repeating decimal or repetend.
Thus, \(\dfrac{1}{11}=0.90909\,.....\)
\(=0.\overline{09}\)
A \(0.67\)
B \(1.\overline{36}\)
C \(1.\overline6\)
D \(1.602\)
Case 1: Terminating decimals on the number line:
Example: \(0.3\)
\(0.3\) means \(3\) tenths.
\(\therefore\) Draw a number line from \(0\) to \(1\) and divide the interval into \(10\) equal segments.
Thus, count \(3\) segments and plot \(0.3.\)
Case 2: Non-terminating decimals on the number line:
Example: \(0.\overline3\)
\(0.\overline3\) means \(0.333\,...\)
\(\therefore\;0.3<0.\overline3<0.4\), which means \(0.\overline3\) lies in between \(0.3\) and \(0.4\) on the number line.
Thus, plotting \(0.\overline3\) between \(0.3\) and \(0.4.\)
For example: \(\dfrac{14}{8}\)
Divide \(14\) by \(8.\)
Thus, \(\dfrac{14}{8}\) gives the quotient as \(1.75\).
Note:
This method can be used in both terminating and non-terminating decimals.
(i) When we have \(10\) or \(100\) or \(1000\,.....\) (multiples of \(10\)) in the denominator.
(ii) Fractions having the denominators that can be written in the form of powers of \(2\,[like\,\,4,8,16,...]\) or \(5\,[like\,\,25,625,...]\) result in terminating decimals.
For example: \(\dfrac{13}{25}\)
For the multiples of \(10\) in the denominator, we multiply both numerator and the denominator by \(4.\)
\(=\dfrac{13×4}{25×4}\)
\(=\dfrac{52}{100}\)
\(=0.52\)
Thus, \(\dfrac{13}{25}\) can be written as \(0.52.\)
Note:
If we have a mixed number to be converted into a decimal, consider the following two methods:
Method 1:
First convert it into an improper fraction and then convert it into a decimal.
For example: \(1\dfrac{1}{2}\)
\(=\dfrac{(2×1)+1}{2}\)
\(=\dfrac{2+1}{2}\)
\(=\dfrac{3}{2}\)
\(=1.5\)
Method 2:
Keep aside the whole number and convert only the fraction part into a decimal.
For example: \(-2\dfrac{2}{3}=-\left(2+\dfrac{2}{3}\right)\)
\(=-2-\dfrac{2}{3}\)
\(\implies-\dfrac{2}{3}=-0.66\)
Write the whole number and the decimal part together.
\(=-2.66\)
A \(2.05\)
B \(0.05\)
C \(-2.05\)
D \(1.25\)
Case 1: Terminating decimal as a rational number:
For example: \(0.36\)
According to the place value chart, \(0.36\) is thirty-six hundredths.
So, it can be written as-
\(=\dfrac{36}{100}\)
Now, simplify the fraction obtained.
\(=\dfrac{36\div4}{100\div4}\)
\(=\dfrac{9}{25}\)
\(9\) and \(25\) do not have any common factor other than \(1\).
\(\therefore\;\dfrac{9}{25}\) is in its simplest form.
Thus, \(0.36\) can be written as \(\dfrac{9}{25}\) in the rational form.
Case 2: Terminating decimal as mixed numbers:
For example:
\(6.05\)
According to the place value chart, \(6.05\) is six and five hundredths.
So, it can be written as-
\(6.05=6\dfrac{5}{100}\)
\(=6+\dfrac{5}{100}\)
Now, we simplify only the fraction part i.e. \(\dfrac{5}{100}.\)
\(=\dfrac{5\div5}{100\div5}\)
\(=\dfrac{1}{20}\)
Thus,
\(6+\dfrac{5}{100}=6+\dfrac{1}{20}\)
or \(6\dfrac{5}{100}=6\dfrac{1}{20}\)
or \(6.05=6\dfrac{1}{20}\)
A \(\dfrac{61}{4}\)
B \(\dfrac{161}{8}\)
C \(15\dfrac{1}{8}\)
D \(\dfrac{45}{26}\)
Example: Compare \(1\dfrac{3}{4}\) and \(\dfrac{5}{12}\)
(1) First, convert them into the same form i.e. rational number.
\(1\dfrac{3}{4}=1+\dfrac{3}{4}\)
\(=\dfrac{(1×4)+3}{4}\)
\(=\dfrac{4+3}{4}\)
\(=\dfrac{7}{4}\)
\(\dfrac{5}{12}\) is already a rational number.
(2) For comparing \(\dfrac{5}{12}\) and \(\dfrac{7}{4}\), convert them into their equivalent fractions having the same denominator.
\(\dfrac{7}{4}=\dfrac{7×3}{4×3}=\dfrac{21}{12}\)
\(\dfrac{5}{12}\) already has the denominator
(3) \(21>5\)
\(\therefore\;\dfrac{21}{12}>\dfrac{5}{12}\)
Note:
On the number line, the number which is to the left of another number is always smaller than the other, even if it looks bigger.
For example:
\(-11\) is at the left of \(-9\).
\(\therefore\;-11<-9\)
A \(3.20>2\dfrac{4}{5}\)
B \(3.20<2\dfrac{4}{5}\)
C \(3.20=2\dfrac{4}{5}\)
D \(3.20\geq2\dfrac{4}{5}\)
\(3:4,\;50\,\%,\;-12.5,\;2^3\) and \(\dfrac{-8}{6}\)
\(3:4\) can be written as \(\dfrac{3}{4}\)
\(50\,\%\) can be written as \(\dfrac{50}{100}=\dfrac{1}{2}\)
\(-12.5\) can be written as \(\dfrac{-125}{10}=\dfrac{-25}{2}\)
\(2^3\) can be written as \(2^3=\dfrac{8}{1}\)
\(\dfrac{-8}{6}\) can be written as \(\dfrac{-8}{6}=\dfrac{-4}{3}\)
\(\dfrac{3}{4}=\dfrac{3×3}{4×3}=\dfrac{9}{12}\)
\(\dfrac{1}{2}=\dfrac{1×6}{2×6}=\dfrac{6}{12}\)
\(\dfrac{-25}{2}=\dfrac{-25×6}{2×6}=\dfrac{-150}{12}\)
\(\dfrac{8}{1}=\dfrac{8×12}{1×12}=\dfrac{96}{12}\)
\(\dfrac{-4}{3}=\dfrac{-4×4}{3×4}=\dfrac{-16}{12}\)
\(\dfrac{9}{12},\;\dfrac{6}{12},\;\dfrac{-150}{12},\;\dfrac{96}{12},\;\dfrac{-16}{12}\)
\(\dfrac{-150}{12}<\dfrac{-16}{12}<\dfrac{6}{12}<\dfrac{9}{12}<\dfrac{96}{12}\)
\(-12.5<\dfrac{-8}{6}<50\,\%<3:4<2^3\)
A Alex
B Cooper
C Carl
D None of these