In Writing All of the Integers From 1 to 199, How Many Times is ‘1’ Used?

In writing all of the integers from 1 to 199, the total number of times ‘1’ is used is 140 times

Step 1: Firstly we have to count the number of times one occurs from 1 to 199

1, 21, 31, 41, 51, 61, 71, 81, 91 - 9 times in unit digit

10, 12, 13, 14, 15, 16, 17, 18, 19 - 9 times in ten’s digit

11 - 2 times i.e. at unit and ten’s digit

On simplifying we get

9 + 9 + 2 = 20 times

Step 2: Now again count the number of times one occurs from 100 to 199

111 will be counted for unit digit, ten's digit and hundred's digit so it will be come 3 times

101, 121, 131, 141, 151, 161, 171, 181, 191 - 9 times at unit digit

110, 112, 113, 114, 115, 116, 117, 118, 119 - 9 times at ten’s digit

100, 102, 103, 104 ….. 198, 199 - 81 times at hundred’s digit

On simplifying we get

81 x 1 + 9 x 2 + 9 x 2 + 3 = 120 times

Step 3: Number of times one occurs from 100 to 199

120 + 20 = 140

Therefore, in all integers from 1 to 199, the total number of times ‘1’ is used is 140 times