Asima and Nile each think of an integer greater than 0. Each of them performs the following operations on their integer: they double it, then subtract 10 , and then multiply it by 4 . The sum of their results is 440 . How many possibilities are there for Asima's original integer?
(A) 64
(B) 44
(C) 65
(D) 45
(E) 66
We proceed to work backward from the final sum, 440, 'undoing' each of the three operations to determine the sum of their two numbers before any operations were performed.

The final operation performed by each of Asima and Nile was to multiply their number by 4 . Multiplying each of their numbers by 4 increases the sum of the two numbers by a factor of 4 . That is, the final sum of their two numbers was 440, and so the sum of their two numbers immediately before the last operation was performed was $440 \div 4=110$.

The second operation performed by each of Asima and Nile was to subtract 10 from their
number.

Subtracting 10 from each of their numbers decreases the sum of the two numbers by 20 .

That is, the sum of their two numbers immediately following the second operation was 110 , and so the sum of their two numbers immediately before the second operation was performed was $110+20=130$.

Finally, the first operation performed by each of Asima and Nile was to double their number. Doubling each of their numbers increases the sum of the two numbers by a factor of 2 .

That is, the sum of their two numbers immediately following the first operation was 130, and so the sum of their two numbers before the first operation was performed was $130 \div 2=65$.

Each of their original integers is greater than 0 and the two integers have a sum of 65 .

Therefore, Asima's original integer could be any integer from 1 to 64, inclusive.

Thus, there are 64 possibilities for Asima's original integer.

ANSWER: (A)

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