Answer

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**Hint:**Here we need to round off to the nearest thousand to find the smallest and the greatest number. So if we have to find the smallest number, we have to check in the range of $8000$ to $9000$and if we want to find the largest number we have to find in the range of $9000$ to $10000$.

**Complete step by step answer:**

In the above question, we have few examples, if the number at the thousand’s place is followed by \[{\mathbf{5}},{\text{ }}{\mathbf{6}},{\text{ }}{\mathbf{7}},{\text{ }}{\mathbf{8}},{\text{ }}{\mathbf{or}}{\text{ }}{\mathbf{9}},\]the number is rounded up.

\[\left( i \right){\text{ }}3846 \to 4000\;\]

We see the digit in the hundred’s place is $8$, we round to the nearest multiple of thousand which is greater than the number. Hence, \[3846\] is nearer to \[4000\] than \[3000\].If the number at the hundred’s place is followed by \[{\mathbf{0}},{\text{ }}{\mathbf{1}},{\text{ }}{\mathbf{2}},{\text{ }}{\mathbf{3}}{\text{ }}{\mathbf{or}}{\text{ }}{\mathbf{4}}\], the number is rounded up.

\[\left( {ii} \right){\text{ }}8039 \to 8000\]

We see the digit in the hundred’s place is $0$, we round to the nearest multiple of thousand which is smaller than the number. Hence, \[8039\] is nearer to \[8000\] than \[9000\].

Here, we have to round off to the thousand. So, if we want to find the smallest number, then we have to check in the range of $8000$ to $9000$. So, according to the above definition the closer number is equal to $8500$.Also, if we want to find the greater number, then we have to check in the range of $9000$ to $10000$.

**Therefore, according to the above definition the closer number is equal to $9499$.**

**Note:**While rounding off to the nearest thousand, if the digit in the hundreds place is between \[0{\text{ }}-{\text{ }}4\] i.e., \[ < {\text{ }}5\], then the hundred place is replaced by $0$. If the digit in the hundreds place is $ = $ to or \[ > {\text{ }}5\], then the hundreds place is replaced by and the thousands place is increased by $1$.

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