What to charge before fees

A marketplace takes a percentage of whatever you charge, and then a flat fee on top. If you list an item at $p$ pence, what you keep is given by a function of $p$.

Sellers need that function run backwards. Write gross_needed(target, fee_percent, flat), which returns the smallest whole number of pence to list the item at so that the amount kept is at least target.

Worked example, with target = 1000, fee_percent = 10 and flat = 50. Keeping at least 1000 means $p - 0.1p - 50 \ge 1000$, so $0.9p \ge 1050$ and $p \ge 1166.67$. Prices are whole pence, so round up: the answer is 1167. Checking, listing at 1167 keeps $1167 - 116.7 - 50 = 1000.3$ pence, which clears the target, while 1166 keeps 999.4, which does not.

• Constraints: target and flat are between 0 and 100000, and fee_percent is between 0 and 99.
• Work out the fee from the exact price rather than rounding it first: the check above uses 116.7 pence, not 117.
• Round the final price up, never down. Rounding down leaves the seller a fraction of a penny short.
• A target of 0 with no flat fee needs no price at all, so the answer there is 0.
• Whole-number arithmetic avoids every rounding argument here: the condition is $p(100 - f) \ge 100(\text{target} + c)$.