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For the distributions given below : Which of the following is correct for the above distributions?

1. Standard deviation of $A$ is significantly lower than standard deviation of $B$
2. Standard deviation of $A$ is slightly lower than standard deviation of $B$
3. Standard deviation of $A$ is same as standard deviation of $B$
4. Standard deviation of $A$ is significantly higher than standard deviation of $B$

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Can anyone elaborate the maths behind it?

Answer: C) Standard deviation of A is the same as the standard deviation of B. This should be answered intuitively from the graph otherwise it would take too much time.
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could you please tell how to answer it intuitively? Is it because the difference in height of two consecutive bars same in both graphs?
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Yes, the mean of both graphs is the same and all the deviations are also the same.  Thus both equal.
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how are the means same?

isnt mean =sigma(value x frequency)/5?

70,110 are the means if calculated in that way.
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If you look at the graph carefully,u will notice that the graph has been rotated around 30,and standard deviation does NOT change when graph is been rotated,however note that mean may change if the graph is not symmetrical ( which is true for this example,anyway that's not asked) .

Note  :- Standard deviation doesn't change when graph is shifted right or left or moved up & down,or rotated around any point. It may change when the graph is sketched or squeezed