Question

Rational number \[\dfrac{-18}{5}\] lies between the following two consecutive integers: -
(a) -2 and -3
(b) -3 and -4
(c) -4 and -5
(d) -5 and -6

Answer 1

Hint: Convert the given rational number into decimal form by multiplying it with \[\dfrac{2}{2}\]. Now, check from the options that the obtained decimal number is greater than which integer and less than which integer and that will be our answer.

Complete step-by-step solution
Here, we have been provided with the rational number \[\dfrac{-18}{5}\] and we have to check from the options that, between which two consecutive integers this rational number lies.
Now, multiplying the given rational number with \[\dfrac{2}{2}\], we get,
\[\begin{align}
  & \Rightarrow \dfrac{-18}{5}=\left( -1 \right)\times \dfrac{18}{5}\times \dfrac{2}{2} \\
 & \Rightarrow \dfrac{-18}{5}=\dfrac{-36}{10} \\
 & \Rightarrow \dfrac{-18}{5}=-3.6 \\
\end{align}\]
Therefore, in the decimal form \[\dfrac{-18}{5}\] can be written as -3.6.
We already know that,
\[\Rightarrow \] 3 < 3.6 < 4
Now, multiplying the above terms with (-1) and changing the direction of inequality, we get,
\[\Rightarrow \] -3 > -3.6 > -4
\[\Rightarrow \] -4 < -3.6 < -3
Hence, -3.6 is greater than -4 but less than -3. So, \[\dfrac{-18}{5}\] is greater than -4 but less than -3. Therefore, the given rational number lies between -4 and -3.
Hence, option (b) is the correct answer.

Note: One may note that when we were comparing the numbers and multiplied them with (-1) then the direction of inequality got reversed. This is due to the reason that whenever we multiply some positive numbers with some negative number then it pushes back that positive number on the number line to an extent up to which it was on the positive side. The direction of inequality also changes when we take reciprocal numbers. There can be another method to solve the question. What we can do is we will multiply each option with \[\dfrac{5}{5}\] and then compare all the numerators and check that between which two numbers in the numerator -18 lies.