To solve this problem, let's break down the steps that Reuben followed:
1. Reuben puts a number into the machine.
2. The machine multiplies it by 3.
3. The machine subtracts 2 from the result.
4. Reuben takes the new number that came out and puts it back into the machine.
5. The machine multiplies it again by 3.
6. The machine subtracts 2 from the result.
7. Reuben takes the new number that came out and puts it back into the machine.
8. The machine multiplies it by 3.
9. The machine subtracts 2 from the result.
10. The final number that came out was 271.
Now, let's work backward step by step to find the original number Reuben put into the machine. We will reverse each of the operations that were done by the machine.
Step 10: The final number that came out was 271.
To backtrack, we need to add 2 since the machine subtracted 2. So, 271 + 2 = 273.
Step 9: The number before this was 273.
To reverse the "multiplied by 3" operation, we need to divide by 3. So, 273 ÷ 3 = 91.
Step 8: The number before this was 91.
Also, divide by 3 here: 91 ÷ 3 = 30.33 (rounded to 2 decimal places).
Step 7: The number before this was 30.33.
Again, divide by 3: 30.33 ÷ 3 = 10.11 (rounded to 2 decimal places).
Step 6: The number before this was 10.11.
Once more, divide by 3: 10.11 ÷ 3 ≈ 3.37 (rounded to 2 decimal places).
Step 5: The number before this was 3.37.
Now, add 2 to it since the machine subtracted 2: 3.37 + 2 = 5.37.
Step 4: The number before this was 5.37.
Divide by 3: 5.37 ÷ 3 ≈ 1.79 (rounded to 2 decimal places).
Step 3: The number before this was 1.79.
Again, add 2: 1.79 + 2 = 3.79.
Step 2: The number before this was 3.79.
Dividing by 3: 3.79 ÷ 3 = 1.26 (rounded to 2 decimal places).
Step 1: The number before this was 1.26.
Finally, we have reached the beginning. So, the original number that Reuben put into the machine was approximately 1.26.
Therefore, the original number that Reuben put into the machine was approximately 1.26.