To determine how many months the cell phone bill of $412 covers, we need to account for the activation fee and the monthly cost.
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Identify total cost without the activation fee: Since the activation fee of $100 is a one-time payment, we first subtract it from the total bill: \[ 412 - 100 = 312 \] This gives us the total amount that is attributed to the monthly bill.
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Determine the number of months covered: Now, we divide the remaining balance ($312) by the monthly cost ($52): \[ 312 \div 52 = 6 \] This indicates that the bill covers 6 months.
The operations needed to solve this problem are subtraction and division.
So, the correct response is: subtraction and division.