Question

A positive integer which when added to \[1000\] gives a sum which is greater than when it is multiplied by\[1000\]. That positive integer is:
a) \[1\]
b) \[5\]
c) \[7\]
d) \[3\]

Answer 1

Hint: We are given above that we have to find a positive integer which when added to \[1000\] gives a sum which is greater than the product obtained, when it is multiplied by thousand. We solve it by assuming that number to be \[x\]. We then form an equation\[1000 + x < 1000x\]. We now proceed to solve this inequality to reach our result.

Complete step-by-step answer:
To solve the given question we assume positive integers to be a variable. Let \[x\] be the positive integer such that, this when added to \[1000\] gives a sum which is greater than when it is multiplied by a thousand. Then the sum of \[x\] with \[1000\] will be \[1000 + x\] and the product of \[x\] with \[1000\] will be \[1000x\]. Then according to the question,
\[1000 + x < 1000x\]
\[
   \Rightarrow 1000 > 1000x - x \\
   \Rightarrow 1000 > 999x \\
   \Rightarrow \dfrac{{1000}}{{999}} > x \\
   \Rightarrow x < 1.001 \\
 \]
We know that the only possible positive integer which is less than \[1.001\] is\[1\], hence our answer for the given question is option a).

So, the correct answer is “Option A”.

Note: This answer might look weird one because when you do equation by putting the value of \[1\] in place of \[x\], you end up getting equality, that is both the sum and product comes out to be equal. This indeed is the answer as we have proved this above. Hence, such questions should be proved mathematically instead of using hit and trial method or trying out the given options.