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In an 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 an examination, students had to find out the average of nine 3-dig...
Problem: Finding the number of distinct values of a 3-digit number that Shweta erroneously copied down in an examination for calculating the average of nine 3-digit natural numbers.

Solution:

To solve the problem, we can start by using the formula for the average of n numbers:

Average = (Sum of n numbers) / n

Let the actual sum of the nine 3-digit numbers be S, and let the number that Shweta copied down be x. As per the problem statement, Shweta made a mistake in the digits of x, but the value of x is not specified. We can consider all the possible values of x, and check which ones satisfy the given conditions.

Step 1: Calculation of the actual average

We know that the actual average of the nine 3-digit numbers is given by:

Actual Average = S / 9

Step 2: Calculation of the incorrect average

Shweta calculated the average using the number x with reversed digits. Let the number obtained by reversing the digits of x be y. Then, the sum of the nine numbers used by Shweta would be:

S' = S - x + y

And, the incorrect average would be:

Incorrect Average = S' / 9 = (S - x + y) / 9

As per the problem statement, the incorrect average is 22 more than the actual average. Hence, we have:

(S - x + y) / 9 = (S / 9) + 22

Simplifying this equation, we get:

y = x + 198

Step 3: Checking the possible values of x

Since x is a 3-digit number, its possible values range from 100 to 999. For each value of x, we can calculate the corresponding value of y using the equation y = x + 198. If y is also a 3-digit number, we can consider it as a candidate for the number that Shweta copied down.

If y is not a 3-digit number, it means that the digits of x cannot be reversed to give a valid 3-digit number. In this case, we can discard the value of x as a candidate.

Step 4: Counting the distinct values of x

We can count the number of distinct values of x that satisfy the given conditions. For example, we can use a loop in a programming language to iterate over all possible values of x, and count the number of valid candidates for y. Alternatively, we can use the fact that y = x + 198, and observe that:

- If x ends with 1, then y ends with 9. The possible values of x are 109, 119, ..., 989, which gives 9 candidates for y.
- If x ends with 2, then y ends with 0. The possible values of x are 102, 112, ..., 992, which gives 9 candidates for y.
- Similarly, for each possible digit d (from 0 to 9), there are 9 candidates for y, corresponding to the values of x that end with d.

Therefore, the total number of distinct values of x is:

9 x 10 = 90

However, we need to remove the cases where y is not a
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In an examination, students had to find out the average of nine 3-dig...
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 x 10 = 70 values.
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In an 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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