Question

Which point on the number line most likely represents $-2\dfrac{5}{8}$ ?
A. on the left of $-3$
B. on the right of $-2$
C. between $-2$ and $-3$
D. in the middle of $-2$ and $-3$

Answer 1

Hint: We first write $-2\dfrac{5}{8}$ as $-\left( 2+\dfrac{5}{8} \right)$ . After that, we show it as $-2\dfrac{5}{8}<-2$ and then again show it as $-2\dfrac{5}{8}>-3$ . By these, we can show that $-2\dfrac{5}{8}$ lies on the left of $-2$ and on the right of $-3$ . Again, we need to check if it is in the middle of them. We do so by taking the equivalent of $\dfrac{1}{2}$ as $\dfrac{4}{8}$ and check if it equals $\dfrac{5}{8}$ .

Complete step by step solution:
The fraction which we are given in this problem is $-2\dfrac{5}{8}$ . This fraction is nothing but a mixed fraction. A mixed fraction is one where an improper fraction is written in the form of $a\dfrac{b}{c}$ where a is a natural number and $\dfrac{b}{c}$ is a proper fraction. A mixed fraction $a\dfrac{b}{c}$ can also be written as $a+\dfrac{b}{c}$ . This means that we can write $-2\dfrac{5}{8}$ as $-\left( 2+\dfrac{5}{8} \right)$ .
Now, we know that,
$\begin{align}
  & \Rightarrow \left( 2+\dfrac{5}{8} \right)>2 \\
 & \Rightarrow -\left( 2+\dfrac{5}{8} \right)<-2 \\
 & \Rightarrow -2\dfrac{5}{8}<-2 \\
\end{align}$
This means that the fraction $-2\dfrac{5}{8}$ lies on the left of $-2$ on the number line.
In a similar manner, we know that,
$\begin{align}
  & \Rightarrow \left( 2+\dfrac{5}{8} \right)<3 \\
 & \Rightarrow -\left( 2+\dfrac{5}{8} \right)>-3 \\
 & \Rightarrow -2\dfrac{5}{8}>-3 \\
\end{align}$
This means that the fraction $-2\dfrac{5}{8}$ lies on the right of $-3$ on the number line.
Again, we can write $\dfrac{1}{2}$ as $\dfrac{1}{2}\times \dfrac{4}{4}=\dfrac{4}{8}$ . So, we can say that $-2\dfrac{5}{8}$ is not equivalent to $-2\dfrac{1}{2}$ and so, it does not lie in the middle of $-2$ and $-3$ .
Thus, we can conclude that $-2\dfrac{5}{8}$ lies on the left of $-2$ and on the right of $-3$ , which is between $-2$ and $-3$ on the number line, option C.

Note: Problems become more problematic when they involve negative numbers. So, for this problem, we must be careful not to commit a mistake by thinking the fraction to be positive. Also, we should take care of the right and left.