Question

Which of the following numbers lies between \[\dfrac{-5}{2}\] and $\dfrac{3}{4}$ ?
$\begin{align}
  & a)1 \\
 & b)0 \\
 & c)-3 \\
 & d)3 \\
\end{align}$

Answer 1

Hint: Now to compare the numbers, first the denominators must be the same. Hence we will take LCM of both the denominators which are 2 and 4 and then rewrite the given fraction with denominator = LCM obtained. Now we will check for each option if it lies or not between the given numbers. Again to compare the options with given numbers, we will equate the denominators.

Complete step-by-step solution:
Now first consider the given numbers \[\dfrac{-5}{2}\] and $\dfrac{3}{4}$ .
Now to compare the number we must have a denominator equal. To do so we must first find the LCM of the current denominators. The given numbers have denominators 2 and 4 and we know that LCM of 2 and 4 is 4. Now we want to write each fraction in a way such that its denominator is 4.
Now we will keep $\dfrac{3}{4}$ as it is and for the number \[\dfrac{-5}{2}\] we will multiply its numerator and denominator by 2 Hence we get $\dfrac{-5\times 2}{2\times 2}=\dfrac{-10}{4}$ .
Hence Now we have the given two numbers as $\dfrac{-10}{4}$ and $\dfrac{3}{4}$.
Now let us consider each option.
Again to compare the options we must have each number with the same denominators. Hence we will multiply the numerator and denominator by 4 in each case to compare.
(i). Now let us consider 1.
Now 1 is nothing but $\dfrac{1}{1}$
Multiplying the numerator and denominator with 4 we get.
$\dfrac{1\times 4}{1\times 4}=\dfrac{4}{4}$ . Now we know that $\dfrac{3}{4}<\dfrac{4}{4}$
Hence 1 does not lie between $\dfrac{-10}{4}$ and $\dfrac{3}{4}$ .
(ii). Now let us consider 0.
Now 0 is nothing but $\dfrac{0}{1}$
Multiplying the numerator and denominator with 4 we get.
$\dfrac{0\times 4}{1\times 4}=\dfrac{0}{4}$ . Now we know that $\dfrac{-10}{4}<\dfrac{0}{4}<\dfrac{3}{4}$ . Hence 0 lies between the given numbers.
(iii)Now let us consider – 3.
Now -3 is nothing but $\dfrac{-3}{1}$
Multiplying the numerator and denominator with 4 we get.
$\dfrac{-3\times 4}{1\times 4}=\dfrac{-12}{4}$ . Now we know that $\dfrac{-12}{4}<\dfrac{-10}{4}$
Hence –3 does not lie between the given numbers.
(iv)Now let us consider 3.
Now 3 is nothing but $\dfrac{3}{1}$
Multiplying the numerator and denominator with 4 we get.
$\dfrac{3\times 4}{1\times 4}=\dfrac{12}{4}$. Now we know that $\dfrac{3}{4}<\dfrac{12}{4}$
Hence 3 does not lie between $\dfrac{-10}{4}$ and $\dfrac{3}{4}$ .
Hence we get out of all the given options only 0 lies between the given numbers. Hence option b is the correct option.

Note: Now we can also solve this question by converting the fractions into decimals. We know \[\dfrac{-5}{2}=-2.5\] and $\dfrac{3}{4}=0.75$ .Now out of 1, 0, -3, 3 we know only 0 lies between – 2.5 and 0.75. Hence we can easily solve this by converting the fractions into decimals.