Question

Express \[\dfrac{3}{4} + \dfrac{2}{3} + \dfrac{{ - 4}}{5}\] in the form of \[\dfrac{p}{q}\] .

Answer 1

Hint: We are asked to express the given problem in the form \[\dfrac{p}{q}\] . It is the standard form of a rational number where \[q\] can never be zero. To solve the given question, we will use the LCM method as the denominator in each term is different. At first, we will get the common denominator using LCM of denominators. And then simply add the numerators. Now, we will simplify to the standard form \[\dfrac{p}{q}\] where \[p,q\] should not have any common divisor except for \[1\] .

Complete step-by-step answer:
Given problem,
 \[\dfrac{3}{4} + \dfrac{2}{3} + \dfrac{{ - 4}}{5}\]
To solve the given problem, at first we have to get the common denominator which we will take using LCM of the denominators. So, we will calculate the LCM of the denominators.
LCM of \[4,3,5\] is \[60\] .
Multiply the numerator and denominator of each term by \[60\] to make the common denominator.
 \[\dfrac{3}{4} \times \dfrac{{60}}{{60}} + \dfrac{2}{3} \times \dfrac{{60}}{{60}} + \dfrac{{ - 4}}{5} \times \dfrac{{60}}{{60}}\]
Taking \[\dfrac{1}{{60}}\] common from each term, we get,
 \[\dfrac{1}{{60}}\left( {\dfrac{{3 \times 60}}{4} + \dfrac{{2 \times 60}}{3} - \dfrac{{4 \times 60}}{5}} \right)\]
Simplifying it, we obtain,
 \[
  \dfrac{1}{{60}}\left( {45 + 40 - 48} \right) \\
   \Rightarrow \dfrac{{37}}{{60}} \\
 \]
Hence \[\dfrac{3}{4} + \dfrac{2}{3} + \dfrac{{ - 4}}{5}\] in the form of \[\dfrac{p}{q}\] can be expressed as \[\dfrac{{37}}{{60}}\] .

Note: If we perform any of the four mathematical operations i.e. addition, subtraction, multiplication or division, to numbers in the form of \[\dfrac{p}{q}\] , we will always get the result in the same form.
Real numbers can be classified into two categories: rational numbers and irrational numbers. The numbers in the form of \[\dfrac{p}{q}\] are rational ones and others are the irrational numbers. The decimal expansion of rational numbers will either be terminating or non-terminating recurring numbers while the decimal expansion of irrational numbers is non-terminating, non-recurring.