Selling Two Things at the Same Price Can Still Lose Money
Learn to find profit or loss per cent from cost and selling price, work out the selling price from a required percentage, recover the cost price from the selling price, and handle overhead expenses and two-article problems.
Why does selling two watches at the same price lose money?
Because the same percentage applied to two different cost prices does not give the same amount of money.
Suppose two watches are each sold for , one at a profit and the other at a loss. It looks as though the gain and the loss must cancel. They do not.
The profitable watch cost , so the gain is . The loss-making one cost , so the loss is . Total cost against total sales — an **overall loss of **.
The percentages were equal but the bases were not. This page covers the first part of the ICSE Class 8 Mathematics chapter on profit and loss.
Suppose two watches are each sold for , one at a profit and the other at a loss. It looks as though the gain and the loss must cancel. They do not.
The profitable watch cost , so the gain is . The loss-making one cost , so the loss is . Total cost against total sales — an **overall loss of **.
The percentages were equal but the bases were not. This page covers the first part of the ICSE Class 8 Mathematics chapter on profit and loss.
Formula
How do you find profit or loss per cent?
The cost price () is what the seller paid; the selling price () is what the buyer paid.
Profit and loss percentages are always calculated on the cost price. That single rule decides every question in this chapter, and dividing by the selling price instead is the commonest error there is.
Worked example 1 — a profit. An article costing is sold for .
Worked example 2 — a loss. The same article costing is sold for .
Why the cost price and not the selling price. A percentage measures a change against where it started, exactly as in the chapter on percentage change. For a trader the starting point is what the goods cost, so profit is judged against that. Using the selling price in example 1 would have given , which answers a different question nobody asked.
A quick size check. A profit percentage can exceed — an article bought for and sold for shows a profit. But a loss percentage can never exceed , since the most that can be lost is the whole cost price. A loss of is an arithmetical impossibility and signals a mistake.
The equal-amount asymmetry. A gain of on is , while a loss of on the same cost price is too — because both are measured against the same . The asymmetry in the opening example arose only because the two cost prices were different, which is worth keeping separate from this case.
Profit and loss percentages are always calculated on the cost price. That single rule decides every question in this chapter, and dividing by the selling price instead is the commonest error there is.
Worked example 1 — a profit. An article costing is sold for .
Worked example 2 — a loss. The same article costing is sold for .
Why the cost price and not the selling price. A percentage measures a change against where it started, exactly as in the chapter on percentage change. For a trader the starting point is what the goods cost, so profit is judged against that. Using the selling price in example 1 would have given , which answers a different question nobody asked.
A quick size check. A profit percentage can exceed — an article bought for and sold for shows a profit. But a loss percentage can never exceed , since the most that can be lost is the whole cost price. A loss of is an arithmetical impossibility and signals a mistake.
The equal-amount asymmetry. A gain of on is , while a loss of on the same cost price is too — because both are measured against the same . The asymmetry in the opening example arose only because the two cost prices were different, which is worth keeping separate from this case.
How do you find the selling price from a required percentage?
Multiply the cost price by the appropriate factor:
Think of it as the cost price being by definition. A profit makes the selling price of it; a loss makes it .
Worked example 1. Find the selling price of an article costing to make a profit of .
Checking: profit , and . Correct.
Worked example 2. Find the selling price of the same article at a loss of .
Checking: loss , and . Correct.
Notice that the two selling prices, and , are equally spaced either side of the cost price — above and below — because both percentages were taken of the same cost price.
Worked example 3 — a required profit on a total. A trader buys of rice at per kilogram and wants a profit of . At what price per kilogram should he sell it?
A shortcut works here too: the rate itself rises by , so directly. Since every kilogram carries the same cost and the same margin, the percentage applies equally to the rate and to the total.
The habit worth forming. Convert the percentage into a multiplier before touching the numbers — for a profit, for a loss, for a profit. It removes the two-step find the profit, then add it process and the sign errors that come with it.
Think of it as the cost price being by definition. A profit makes the selling price of it; a loss makes it .
Worked example 1. Find the selling price of an article costing to make a profit of .
Checking: profit , and . Correct.
Worked example 2. Find the selling price of the same article at a loss of .
Checking: loss , and . Correct.
Notice that the two selling prices, and , are equally spaced either side of the cost price — above and below — because both percentages were taken of the same cost price.
Worked example 3 — a required profit on a total. A trader buys of rice at per kilogram and wants a profit of . At what price per kilogram should he sell it?
A shortcut works here too: the rate itself rises by , so directly. Since every kilogram carries the same cost and the same margin, the percentage applies equally to the rate and to the total.
The habit worth forming. Convert the percentage into a multiplier before touching the numbers — for a profit, for a loss, for a profit. It removes the two-step find the profit, then add it process and the sign errors that come with it.
How do you recover the cost price from the selling price?
Divide by the same factor you would have multiplied by:
Worked example 1. An article is sold for at a profit of . Find the cost price.
Checking forwards: . Correct.
Worked example 2. An article is sold for at a loss of . Find the cost price.
Checking: . Correct.
The error this question is designed to catch. Given a selling price of and a profit, it is tempting to take **of **, getting , and subtract it to get . That is wrong, because the was a percentage of the cost price, not of the selling price.
The check exposes it at once: , which is not . So the answer fails its own test, and that is why checking forwards matters so much on this question type.
Worked example 3 — a two-stage question. By selling an article for a shopkeeper loses . At what price must he sell it to gain ?
First recover the cost price:
Then apply the required gain:
The cost price is the bridge between the two selling prices, and there is no way to get from to without finding it. Every sold at X per cent loss, what price for Y per cent gain question works this way.
**Why the answer is not .** Moving from a loss to a gain is a percentage-point shift — but of the cost price, not of the old selling price. Twenty per cent of is , and , which does agree. Taking of instead would have given — close enough to look right and still wrong.
Worked example 1. An article is sold for at a profit of . Find the cost price.
Checking forwards: . Correct.
Worked example 2. An article is sold for at a loss of . Find the cost price.
Checking: . Correct.
The error this question is designed to catch. Given a selling price of and a profit, it is tempting to take **of **, getting , and subtract it to get . That is wrong, because the was a percentage of the cost price, not of the selling price.
The check exposes it at once: , which is not . So the answer fails its own test, and that is why checking forwards matters so much on this question type.
Worked example 3 — a two-stage question. By selling an article for a shopkeeper loses . At what price must he sell it to gain ?
First recover the cost price:
Then apply the required gain:
The cost price is the bridge between the two selling prices, and there is no way to get from to without finding it. Every sold at X per cent loss, what price for Y per cent gain question works this way.
**Why the answer is not .** Moving from a loss to a gain is a percentage-point shift — but of the cost price, not of the old selling price. Twenty per cent of is , and , which does agree. Taking of instead would have given — close enough to look right and still wrong.
How do overhead expenses and two-article problems work?
Overhead expenses are any extra costs the seller pays — repairs, transport, packing, labour. They are added to the cost price:
Every percentage is then worked out on that total, since that is what the seller actually spent.
Worked example 1. A man buys a cycle for , spends on repairs and sells it for . Find his profit per cent.
Using as the cost price would have given — a much flattering figure that ignores money genuinely spent.
Worked example 2 — two articles at equal percentages. Two watches are each sold for , one at a profit and the other at a loss. Find the overall result.
Find each cost price separately:
**An overall loss of , despite the two percentages appearing to cancel.
The shortcut for this standard question. When two articles are sold at the same price**, one at profit and the other at loss, the result is always a loss of
For that gives , matching the full calculation. For it gives .
It is the same result as two opposite percentage changes in the previous chapter, arriving by a different route — and it is always a loss, never a gain, whatever is.
Worked example 3 — two articles, different percentages. A trader buys two articles for each, selling one at a profit and the other at a loss. Find the overall profit per cent.
Here the cost prices are equal, so the percentages can be combined directly on the common base:
The distinction between the two problem types. When the selling prices are equal, the cost prices differ and the percentages cannot be combined. When the cost prices are equal, they can. Reading which is given is the first thing to do, and it decides the whole method.
Every percentage is then worked out on that total, since that is what the seller actually spent.
Worked example 1. A man buys a cycle for , spends on repairs and sells it for . Find his profit per cent.
Using as the cost price would have given — a much flattering figure that ignores money genuinely spent.
Worked example 2 — two articles at equal percentages. Two watches are each sold for , one at a profit and the other at a loss. Find the overall result.
Find each cost price separately:
**An overall loss of , despite the two percentages appearing to cancel.
The shortcut for this standard question. When two articles are sold at the same price**, one at profit and the other at loss, the result is always a loss of
For that gives , matching the full calculation. For it gives .
It is the same result as two opposite percentage changes in the previous chapter, arriving by a different route — and it is always a loss, never a gain, whatever is.
Worked example 3 — two articles, different percentages. A trader buys two articles for each, selling one at a profit and the other at a loss. Find the overall profit per cent.
Here the cost prices are equal, so the percentages can be combined directly on the common base:
The distinction between the two problem types. When the selling prices are equal, the cost prices differ and the percentages cannot be combined. When the cost prices are equal, they can. Reading which is given is the first thing to do, and it decides the whole method.
Exam tip
Exam tip: every percentage here is on the cost price
Profit per cent and loss per cent are always calculated on the cost price. Dividing by the selling price is the error that ruins more answers in this chapter than any other.
Use multipliers: a profit means , and a loss means . To go backwards, divide by the same factor.
Never take the percentage of the selling price when recovering a cost price. Given at profit, the answer is , not of .
Add overhead expenses to the cost price before calculating anything, and say so in the working.
For a sold at a loss, what price for a gain question, find the cost price first — it is the bridge, and no shortcut avoids it.
In a two-article problem, read whether the selling prices or the cost prices are equal. Equal selling prices means you must find each cost price separately.
Remember that two articles at the same price with gain and loss always give a loss of — and quote it as a check, not as a substitute for the working.
A **loss percentage can never exceed ; a profit percentage can.
And check every answer forwards** — apply your percentage to your cost price and confirm the given selling price returns.
Use multipliers: a profit means , and a loss means . To go backwards, divide by the same factor.
Never take the percentage of the selling price when recovering a cost price. Given at profit, the answer is , not of .
Add overhead expenses to the cost price before calculating anything, and say so in the working.
For a sold at a loss, what price for a gain question, find the cost price first — it is the bridge, and no shortcut avoids it.
In a two-article problem, read whether the selling prices or the cost prices are equal. Equal selling prices means you must find each cost price separately.
Remember that two articles at the same price with gain and loss always give a loss of — and quote it as a check, not as a substitute for the working.
A **loss percentage can never exceed ; a profit percentage can.
And check every answer forwards** — apply your percentage to your cost price and confirm the given selling price returns.
Did you know
Why is the equal-percentage trick always a loss and never a gain?
Sell two articles at the same price, one at profit and one at loss, and the outcome is a loss every single time — whatever is and whatever the selling price is.
The reason is which article costs more. To sell at a profit you must have bought cheaply, and to sell at a loss you must have bought dearly. So the loss-making article is always the more expensive one, and a fixed percentage of a bigger amount is a bigger amount of money.
Take and a selling price of . The gain is of , which is . The loss is of , which is . The loss wins by , and it wins because is the larger base.
That argument does not depend on the numbers at all, which is why the result is universal. And the size of the shortfall, per cent, grows as the square of the percentage — so at the loss is , at it is , and at it is .
The reason is which article costs more. To sell at a profit you must have bought cheaply, and to sell at a loss you must have bought dearly. So the loss-making article is always the more expensive one, and a fixed percentage of a bigger amount is a bigger amount of money.
Take and a selling price of . The gain is of , which is . The loss is of , which is . The loss wins by , and it wins because is the larger base.
That argument does not depend on the numbers at all, which is why the result is universal. And the size of the shortfall, per cent, grows as the square of the percentage — so at the loss is , at it is , and at it is .
Key takeaways
Profit, loss and cost price: quick revision
- and .
- and — always on the cost price.
- with gives a profit; with it gives an loss.
- A profit percentage can exceed ; a loss percentage can never exceed .
- Finding SP: or . So at profit gives , and at loss gives .
- at per kg for a profit: total , total , rate per kg.
- Finding CP: or . So at profit gives , and at loss gives .
- Never take the percentage of the selling price when recovering a cost price.
- Two-stage: at a loss gives , so a gain needs . The cost price is the bridge.
- Overhead expenses are added: . A cycle at plus repairs sold for gives a profit on .
- Equal selling prices — find each cost price separately. Two watches at with gain and loss: s of and , total against , a ** loss**.
- The shortcut: same selling price with gain and loss always gives a loss of .
- Equal cost prices — percentages can be combined: each at profit and loss gives on , a profit.
Work a set of find the cost price questions and check each one forwards — the check is what separates the right method from the tempting one.
- and — always on the cost price.
- with gives a profit; with it gives an loss.
- A profit percentage can exceed ; a loss percentage can never exceed .
- Finding SP: or . So at profit gives , and at loss gives .
- at per kg for a profit: total , total , rate per kg.
- Finding CP: or . So at profit gives , and at loss gives .
- Never take the percentage of the selling price when recovering a cost price.
- Two-stage: at a loss gives , so a gain needs . The cost price is the bridge.
- Overhead expenses are added: . A cycle at plus repairs sold for gives a profit on .
- Equal selling prices — find each cost price separately. Two watches at with gain and loss: s of and , total against , a ** loss**.
- The shortcut: same selling price with gain and loss always gives a loss of .
- Equal cost prices — percentages can be combined: each at profit and loss gives on , a profit.
Work a set of find the cost price questions and check each one forwards — the check is what separates the right method from the tempting one.