2016 AMC 10A Problem 6

Question

Ximena lists the whole numbers

1

through

30

once. Emilio copies Ximena’s numbers, replacing each occurrence of the digit

2

by the digit

1.

Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena’s sum than Emilio’s?

13

26

102

103

110

Changing a units digit

2

to

1

loses

1

Changing a tens digit

2

to

1

loses

10

When Emilio changes a units digit

2

to

1

, his copied number is smaller by

1

. This happens in

2,12,22

, for a loss of

3

.

When he changes a tens digit

2

to

1

, his copied number is smaller by

10

. This happens for

20,21,\ldots,29

, for a loss of

100

.

Thus Emilio’s sum is

103

less than Ximena’s sum, so Ximena’s sum is

103

larger.

Thus, the correct answer is D.

Problem 6 in other years