When 3 digit number is rounded to the nearest hundred . it round to 200. rounded to the nearest ten this number rounds to 200. the sum of the digits of this number is 19. what the number?

Answer 1

The number is 199.

Statement 1: Rounding to the nearest 100 gives 200 → original number is 200 ± 50

Statement 2: Rounding to the nearest 10 gives 200 → original number is 200 ± 5

First statement can be ignored as the second is a closer range, which is a sub-range of the range of the first statement.

So the number is one of 195, 196, ..., 203, 204.

Statement 3: Sum of digits =19

Could check all the 10 possible numbers 195 - 204, but observe:

The last five numbers 200-204 all start 20- so the sum of those digits will be 2+ something, but there is no single digit that is 18, so the number cannot be one of these.

the first five numbers 195-199 all start 19- so the sum of those digits is 10 + something, which is required to be 19; so 19-10 = 9 gives the remaining digit.

Thus the number is 199.