PROFIT AND LOSS
Profit and loss problems, one formula for all three.
Enter the total and what is left over or short at two different per-person amounts, and get one unit back with verification. One formula covers profit-one-loss-one, both profit and both loss, a staple primary-school olympiad topic.
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Why one formula covers all three types
| Type | First | Second | Number of units |
|---|---|---|---|
| Profit one, loss one | profit r1 | loss r2 | (r1 + r2) / (n2 - n1) |
| Both profit | profit r1 | profit r2 | (r1 - r2) / (n2 - n1) |
| Both loss | loss r1 | loss r2 | (r1 - r2) / (n2 - n1) |
Write profit as positive and loss as negative, and all three types become the same thing: the difference of the two results divided by the difference of the two per-person amounts. Once you have the number of units, one unit = total / units, and substituting both allocations verifies it.
What is this?
Profit and loss problems share one shape: the same batch of items, or the same group of people, distributed by two different rules, with different outcomes each time. One run leaves a surplus, one run falls short, or both leave a surplus or both fall short. The key is the fixed gap between the two per-person amounts: units = difference of the results / difference of the per-person amounts. Divide the total by that and you have one unit; substitute back to verify. Random practice questions are below.
FAQ
How do I enter profit and loss?
Profit means items are left over, so enter a positive number; loss means the items fall short, so also enter a positive number for how many are missing. The tool applies the sign based on the type you pick: with profit-one-loss-one, enter the profit and the shortfall both as positive numbers.
Why is the number of units not a whole number?
It means the conditions do not describe a consistent situation. The usual cause is a figure copied down wrong, or the two per-person amounts entered the wrong way round. Check the original problem, then let the tool verify it.
How does this differ from chicken-and-rabbit or sum-and-multiple problems?
Chicken-and-rabbit uses the assumption method and sum-and-multiple treats two numbers as parts and parts. A profit-and-loss problem has a total distributed by two different rules with opposite or equal-direction outcomes, so you must pin down the gap between the two per-person amounts before anything else. The three shapes differ, so identify the type first.
Does this upload anything?
No. All arithmetic happens inside your current browser with no network requests and no sign-up, which makes it fine for classroom practice or tutoring at home.