Determine the number of months required for $\$ 780$ to earn $\$ 24.65$ simple interest at $4.1 \%$ per-annum. a) 10 months b) 9 months c) 7 months d) 11 months e) 8 months

Step 1 ：Given that the principal amount (P) is $780, the interest (I) is $24.65, and the rate of interest (R) is 4.1% or 0.041.

Step 2 ：We can use the formula for simple interest, I = PRT, and rearrange it to solve for time (T) in years: T = I / (PR).

Step 3 ：Substitute the given values into the formula: T = 24.65 / (780 * 0.041) = 0.7707942464040024 years.

Step 4 ：However, we want the time in months, not years. Therefore, we need to convert the time from years to months by multiplying by 12: months = 0.7707942464040024 * 12 = 9.249530956848028 months.

Step 5 ：Since the number of months should be an integer, we need to round the result to the nearest whole number: months_rounded = round(9.249530956848028) = 10 months.

Step 6 ：\(\boxed{10}\) months is the number of months required for $780 to earn $24.65 simple interest at 4.1% per-annum.