Question

Assertion (A): $\left[ {\dfrac{1}{2}} \right] + \left[ {\dfrac{1}{2} + \dfrac{1}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{2}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{3}{{1000}}} \right] + ...\left[ {\dfrac{1}{2} + \dfrac{{999}}{{1000}}} \right] = 500$
(where $\left[ . \right]$ denotes G.I.F)
Reason (R): \left[ {\dfrac{1}{2} + \dfrac{r}{{1000}}} \right] = \left\\{ \begin{gathered} 0;\,\,if\,0 \leqslant r < 500 \\\ 1;\,\,if\,\,500 \leqslant r \leqslant 999 \\\ \end{gathered} \right.
A. Both Assertion and Reason are individually true and Reason is the correct explanation of Assertion.
B. Both Assertion and Reason are individually true and Reason is not the correct explanation of Assertion.
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true.

Answer 1

The question is related to verbal reasoning. First we consider the reason, on substituting the value of r randomly we verify the reason whether it is true or false. If the reason is true means we are going to check if the assertion is true or false. Then we are choosing the option from the given options.

Complete step by step answer:
Here first we consider the reason and we check whether it is true or false.
Now consider the reason.
Reason (R): \left[ {\dfrac{1}{2} + \dfrac{r}{{1000}}} \right] = \left\\{ \begin{gathered} 0;\,\,if\,0 \leqslant r < 500 \\\ 1;\,\,if\,\,500 \leqslant r \leqslant 999 \\\ \end{gathered} \right.
Consider the value of r as 2
$\Rightarrow \left[ {\dfrac{1}{2} + \dfrac{2}{{1000}}} \right]$
On dividing the number 1 by 2 and on dividing the number 2 by 1000 we have
$\Rightarrow \left[ {0.5 + 0.002} \right]$
Adding 0.5 and 0.002
$\Rightarrow \left[ {0.502} \right]$
where $\left[ . \right]$ denotes G.I.F, The greatest integer function is a function that returns a constant value for each specific interval. These values are the rounded-down integer values of the expression found inside the brackets.
Therefore $\left[ {\dfrac{1}{2} + \dfrac{2}{{1000}}} \right] = 0$

Consider the value of $r$ as 555
$\Rightarrow \left[ {\dfrac{1}{2} + \dfrac{{555}}{{1000}}} \right]$
On dividing the number 1 by 2 and on dividing the number 555 by 1000 we have
$\Rightarrow \left[ {0.5 + 0.555} \right]$
Adding 0.5 and 0.555
$\Rightarrow \left[ {1.555} \right]$
Therefore $\left[ {\dfrac{1}{2} + \dfrac{{555}}{{1000}}} \right] = 1$
Hence the reason is true.
Now we check the assertion.
Consider Assertion (A): $\left[ {\dfrac{1}{2}} \right] + \left[ {\dfrac{1}{2} + \dfrac{1}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{2}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{3}{{1000}}} \right] + ...\left[ {\dfrac{1}{2} + \dfrac{{999}}{{1000}}} \right]$
This can be written as
$\Rightarrow \left[ {\dfrac{1}{2}} \right] + \left[ {\dfrac{1}{2} + \dfrac{1}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{2}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{3}{{1000}}} \right] + ... + \left[ {\dfrac{1}{2} + \dfrac{{500}}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{{501}}{{1000}}} \right] + ... + \left[ {\dfrac{1}{2} + \dfrac{{999}}{{1000}}} \right]$

From the reason the above inequality is written as
$\Rightarrow 0 + 0 + 0 + 0 + ... + 1 + 1 + ... + 1$
When $r$ ranges from 0 to 499, the value will be zero. When r ranges from 500 to 999, the value will be 1.
$\Rightarrow 1 + 1 + ... + 1$
There are 500 terms from the number 500 to the number 999
$\Rightarrow \sum\limits_{n = 1}^{500} 1$
On applying the formula $\Rightarrow \sum\limits_{i = 1}^n 1 = n$, so we have
$\Rightarrow \left[ {\dfrac{1}{2}} \right] + \left[ {\dfrac{1}{2} + \dfrac{1}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{2}{{1000}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{3}{{1000}}} \right] + ...\left[ {\dfrac{1}{2} + \dfrac{{999}}{{1000}}} \right] = 500$
Therefore the assertion is true. Therefore both Assertion and Reason are individually true and Reason is the correct explanation of Assertion.

Hence option A is the correct answer.

Note: In verbal reasoning there are different kinds. The assertion is a simple statement and the reason is the explanation for the assertion. To understand the greatest integer function we have to know about the rounding off numbers to its nearest place value. When we are rounding the number if the number is less than number 5 we write the number as it is otherwise we are adding one number to the previous place value.