The POTW Input/Output Machine takes a number as input and adds \(10\) to the number. The machine then takes this sum and multiplies it by \(2\). Finally, the machine takes this product, subtracts \(30\) from the number, and outputs this new number.
Anala and Mei each input a positive integer into the machine. If the sum of their two outputs is \(130\), how many possibilities are there for the positive integer that Anala input into the machine?
We will work backward from the final sum, \(130\), by ‘undoing’ each of the three operations to determine the sum of their two numbers before any operations were performed.
The final operation performed by the machine on each number was to subtract \(30\). Subtracting \(30\) from each number decreases their sum by \(60\). Therefore, the sum of the two numbers immediately before the third operation was performed was \(130 + 60 = 190\).
Multiplying each of their numbers by \(2\) increases the sum of the two numbers by a factor of \(2\). Since the second sum of their two numbers was \(190\), the sum of their two numbers immediately before the second operation was performed must have been \(180 \div 2 = 95\).
Finally, the first operation performed by each of Anala and Mei was to add \(10\) to their number. Adding \(10\) to each of their numbers increases the sum by \(20\), and so the sum of their numbers before the first operation must have been \(95 - 20 = 75\).
Each of their original integers are positive and the two integers have a sum of \(75\).
Therefore, Anala’s original integer could be any integer from \(1\) to \(74\), inclusive. Thus, there are \(74\) possibilities for Anala’s original integer.