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?

$13$

$26$

$102$

$103$

$110$

Solution:

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.

2016 AMC 10A Problem 6

Solution:

Problem 6 in Other Years