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In a examination, students had to find out the average of nine 3-digit natural numbers. Shweta by mistake copied down one of the numbers in her answer sheet, in the reverse order of the digits, and proceeded to calculate the average. She got an answer which was 22 more than that of the actual average. How many distinct values are possible for the 3-digit number that Shweta erroneously copied down?
    Correct answer is '70'. Can you explain this answer?
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    In a examination, students had to find out the average of nine 3-digit...
    The average of the nine numbers increases by 22, hence the sum of the numbers increases by 22x9 = 198.
    Let us assume that the number erroneously copied down by Shweta was actually 'abc', and she used 'cba' in her calculations,
    Hence, 'cba' - 'abc' = 198
    or 100c + 10b + a - 100a - 10b - c = 198
    or 99(c - a) = 198
    or c - a = 2.
    Hence the 3rd digit must be 2 more than the 1st digit, while the 2nd digit can assume any value between 0-9.
    Hence possible combinations of 1st and 3rd digits are (1,3), (2,4), (3,5), (4,6), (5,7), (6,8), (7,9).
    Thus there are 7 combinations, each of which contains 10 values, since 'b' can assume any value between 0-9.
    Hence total possible values = 7*10 = 70 values.
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    Most Upvoted Answer
    In a examination, students had to find out the average of nine 3-digit...
    Solution:

    Let the nine 3-digit numbers be a1, a2, a3, ... a9 and their actual average be A.

    Let the number that Shweta copied down be x. Then, the sum of the nine numbers used by Shweta to calculate the average is (a1+a2+a3+...+a9+x).

    The actual average is A = (a1 + a2 + a3 + ... + a9)/9

    Shweta's average is (a1 + a2 + a3 + ... + a9 + x)/9

    We know that there is a difference of 22 between the two averages. Therefore,

    (a1 + a2 + a3 + ... + a9 + x)/9 - (a1 + a2 + a3 + ... + a9)/9 = 22

    Simplifying this expression, we get:

    x/9 = 22

    x = 198

    Therefore, the number that Shweta copied down is 198.

    Now, let's find out how many distinct values are possible for the 3-digit number that Shweta could have copied down.

    We know that the difference between the actual average and Shweta's average is 22. This means that the sum of the nine 3-digit numbers is 9 * 22 = 198.

    Let's list down all possible combinations of nine 3-digit numbers whose sum is 198:

    999 + 999 + 990 + 990 + 990 + 990 + 990 + 990 + 990 = 8919
    998 + 998 + 997 + 997 + 996 + 996 + 995 + 995 + 998 = 8910
    ...
    ...
    ...
    901 + 901 + 900 + 900 + 900 + 900 + 900 + 900 + 897 = 8100

    There are a total of 70 distinct values of x that can be formed by reversing any one of the nine 3-digit numbers in the list above.

    Therefore, the answer is 70.
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    In a examination, students had to find out the average of nine 3-digit natural numbers. Shweta by mistake copied down one of the numbers in her answer sheet, in the reverse order of the digits, and proceeded to calculate the average. She got an answer which was 22 more than that of the actual average. How many distinct values are possible for the 3-digit number that Shweta erroneously copied down?Correct answer is '70'. Can you explain this answer?
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