Operation on Rational Numbers
Category : 7th Class
In this topic we will study about addition, subtraction, multiplication and division of rational numbers.
Addition of Rational Numbers
Step 1: Write the rational number in the standard form.
Step 2: Make the denominator same by taking the LCM of denominators.
Step 3: Write all the rational number with the same denominator.
Step 4: Add the numerators
Step 5: Write the numerator getting after addition on the denominator.
Step 6: Reduce the rational number to lowest term.
Add \[2\frac{3}{5},\frac{-15}{13},\frac{-13}{-15},-4\frac{3}{5}\]
Solution:
Step 1: The standard form of given rational numbers are
\[\frac{13}{5},\frac{-15}{13},\frac{13}{15},\frac{-23}{5}\]
Step 2: LCM of 5, 13, 15, 5 is 195
Step3: \[\frac{13}{5}=\frac{13\times 39}{5\times 39}=\frac{507}{195}\]
\[\frac{-15}{13}=\frac{-15\times 15}{13\times 15}=\frac{-225}{195}\]
\[\frac{13}{15}=\frac{13\times 13}{15\times 13}=\frac{169}{195}\]
\[\frac{-23}{5}=\frac{23\times 39}{5\times 39}=\frac{-897}{195}\]
Step 4: The sum of numerators
\[=507+\left( -225 \right)+169+\left( -897 \right)\]\[=-446\]
Step 5: \[\frac{-446}{195}\]
Step 6: \[-2\frac{56}{195}\]
Therefore
\[2\frac{3}{5}+\frac{-15}{13}+\frac{-13}{-15}+\left( -4\frac{3}{5} \right)=-2\frac{56}{195}\]
Subtraction of Two Rational Numbers
Step 1: Write the rational numbers in standard form.
Step 2: Make the denominator same by taking LCM.
Step 3: Subtract numerators.
Step 4: Write the result with denominator and reduce it to lowest terms.
If the sum of two rational numbers is \[\frac{-5}{6}\] and if one of the number is \[-\frac{9}{20}\] then find the other rational number.
Solution:
Sum of Rational number \[=\frac{-5}{16}\]
One number\[=\frac{-9}{20}\], Then the other \[=\frac{-5}{16}-\left( \frac{-9}{20} \right)\]
\[=\frac{-25-\left( -36 \right)}{80}=\frac{-25+36}{80}=\frac{11}{80}\]
Multiplication of Rational Numbers
Product of rational numbers \[=\frac{Product\text{ }of\,the\,numerators}{Product\text{ }of\text{ }the\text{ }denominators}\] and reduce it to the lowest terms.
Reciprocal of Rational Number
For non - zero rational numbers, a rational number is said to be reciprocal of other if the product is 1.
Find the product of \[\frac{13}{6}\times \frac{-18}{91}\times \frac{-5}{9}\times \frac{72}{-125}\]
Solution:
\[\frac{13}{6}\times \frac{-81}{91}\times \frac{-5}{9}\times \frac{72}{-125}=\frac{13\times \left( -18 \right)\times \left( -5 \right)\times 72}{6\times 91\times 9\left( -125 \right)}\]
\[=\frac{-3\times 8}{7\times 25}=\frac{-24}{175}\]
Find the reciprocal of \[\frac{4}{9}\times \frac{-7}{13}\times \frac{-3}{8}\]
Solution:
\[=\frac{4}{9}\times \frac{-7}{13}\times \frac{-3}{8}=\frac{{{\bcancel{4}}^{1}}}{{{\bcancel{9}}_{3}}}\times \frac{-7}{13}\times \frac{-{{\bcancel{3}}^{1}}}{{{\bcancel{8}}_{2}}}=\frac{7}{78}\]
The reciprocal of \[\left( \frac{7}{78} \right)\] is \[\left( \frac{78}{7} \right)\] because \[\frac{7}{78}\times \frac{78}{7}=1\]
Division of Rational Numbers
If a and b are two rational numbers in such a way that \[b\ne 0,\] then \[a\div b\] is same as the product of a and reciprocal of b.
The product of two rational numbers is \[\frac{-23}{88}\] If one of the number is\[\frac{5}{22}\] then find the other rational number.
Solution:
Product of two number \[=\frac{-23}{88}\]
one number \[=\frac{5}{22},\] So the other number \[=\frac{-23}{88}\div \frac{5}{22}=\frac{-23}{88}\times \frac{22}{5}=\frac{-23}{20}\]
If \[p=\frac{-4}{5}\times \left( \frac{10}{9} \right)\times \left( \frac{3}{4} \right)\] and \[Q=(-6)\times \frac{7}{5}\times \frac{2}{3}\]then P + Q is equal to:
(a) \[-2\frac{14}{15}\]
(b) \[-4\frac{14}{15}\]
(c) \[-3\frac{14}{15}\]
(d) \[-1\frac{14}{15}\]
(e) None of these
Answer: (b)
Solution:
Here \[P=\underset{1}{\overset{-1}{\mathop{\left( \frac{-\cancel{4}}{\cancel{5}} \right)}}}\,\times \underset{3}{\overset{-2}{\mathop{\left( \frac{-\cancel{10}}{\cancel{9}} \right)}}}\,\times \frac{\cancel{3}}{\cancel{4}}=\frac{2}{3}\]and Q \[=\cancel{\overset{2}{\mathop{6}}\,}\times \frac{7}{5}\times \frac{2}{\cancel{3}}=\frac{-28}{5}\]
\[\Rightarrow P+Q=\frac{2}{3}+\left( \frac{-28}{5} \right)=\frac{10+\left( -84 \right)}{15}=\frac{-74}{15}=-4\frac{14}{15}\]
Simplify: \[\left( \frac{5}{13}\times \frac{6}{15} \right)\div \left( \frac{9}{12}\times \frac{4}{3} \right)-\left( \frac{3}{11}\times \frac{5}{6} \right)\]
(a) \[\frac{42}{286}\]
(b) \[\frac{109}{286}\]
(c) \[\frac{-21}{286}\]
(d) \[\frac{21}{286}\]
(e) None of these
Answer: (c)
Explanation
\[\left( \frac{\cancel{5}}{13}\times \frac{\overset{2}{\mathop{\cancel{6}}}\,}{\underset{\begin{smallmatrix} \cancel{3} \\ 1 \end{smallmatrix}}{\mathop{\cancel{15}}}\,} \right)\div \left( \underset{\cancel{3}}{\overset{\cancel{3}}{\mathop{\frac{\cancel{9}}{\cancel{12}}}}}\,\times \frac{\cancel{4}}{\cancel{3}} \right)-\left( \frac{\cancel{3}}{11}\times \frac{5}{\underset{2}{\mathop{\cancel{6}}}\,} \right)\]
\[=\frac{2}{23}\div 1-\frac{5}{22}\Rightarrow \frac{2}{13}-\frac{5}{22}=\frac{44-65}{286}=\frac{-21}{286}\]
Find the value of \[\frac{-1}{2}+\frac{1}{3}+\frac{-5}{6}+\frac{5}{3}+\frac{-7}{4}.\]
(a) \[\frac{23}{12}\]
(b) \[\frac{15}{12}\]
(c) \[\frac{-13}{12}\]
(d) \[\frac{-11}{12}\]
(e) None of these
Answer: (c)
The value of \[{{(17\times 12)}^{-1}}\]Is equal to :
(a) \[{{17}^{-1}}\times {{12}^{1}}\]
(b) \[17\times {{\left( \frac{1}{12} \right)}^{-1}}\]
(c) \[{{\left( \frac{1}{17} \right)}^{-1}}\times {{12}^{-1}}\]
(d) \[\left( \frac{1}{17} \right)\times {{12}^{-1}}\]
(e) None of these
Answer: (d)
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