We know from the information given, that Cheryl's income elasticity is not only negative but strongly so, at -2.5; meaning, this is a very inferior good for her. As her income increases, her expenditures for this good will decrease at a rate 2.5 times greater in percentage terms.

Specifically, we know that = -2.5 or %Q = -2.5 x %I (where I is income).

Cheryl's income rose from $100,000 to $110,000; an increase of 10%. From the formula above, we know that this means her consumption of good A will fall by 25%; in other words, it will be only 0.75 times (3/4) her purchases before her income increase. Since she was originally consuming 100 units of A, she will now purchase 100 x 0.75 = 75 units.

7. Cheryl's income elasticity for good A is -2.5. Her current income is $100,000 and she normally buys 100 units of good A per year. If her income increases to $110,000 how many units of good A will she buy?

250. This would be correct if her elasticity were +15; meaning, the good is a luxury and purchases increase with income. In particular, an increase of 10% would lead to increased purchases of 150%. However, here her income elasticity is negative.
75. As shown above, this is correct.
125. This would require that the good be a luxury, since it represents an increase of 25% due to an increase in income. Her elasticity would have to be a +2.5 not -2.5
100. This would require that her income elasticity be zero; purchases aren't affected by income.
82.5. This is consistent with a negative income elasticity, but not -2.5. Instead, it would have to be -1.75.