Twenty Per Cent Off and Ten More Is Not Thirty Per Cent Off
Learn to find discount and discount per cent from the marked price, work out a marked price that allows both a discount and a profit, combine successive discounts correctly, and add GST to a discounted price.
Is twenty per cent off followed by ten per cent off the same as thirty per cent off?
No — it comes to ****, not .
Take a marked price of . The first discount of brings it to . The second discount of is then taken of ****, not of the original , so it removes only and leaves .
A flat off would have left . So the shopper pays more than the two percentages suggest:
Discounts, like all successive percentages, multiply rather than add. This page covers the second part of the ICSE Class 8 Mathematics chapter: discount, marked price, and the tax added at the till.
Take a marked price of . The first discount of brings it to . The second discount of is then taken of ****, not of the original , so it removes only and leaves .
A flat off would have left . So the shopper pays more than the two percentages suggest:
Discounts, like all successive percentages, multiply rather than add. This page covers the second part of the ICSE Class 8 Mathematics chapter: discount, marked price, and the tax added at the till.
Formula
How do you find the discount and the discount per cent?
The marked price (), also called the list price or printed price, is the price shown on the article. A discount is a reduction allowed on it.
Discount per cent is always calculated on the marked price. This is the counterpart of the previous part's rule that profit per cent is always on the cost price — and keeping the two bases apart is the whole difficulty of the chapter.
Worked example 1. An article marked at is sold for . Find the discount and the discount per cent.
Worked example 2. Find the selling price of an article marked at after a discount of .
Checking: the discount is , and . Correct.
Worked example 3 — finding the marked price. After a discount of an article sells for . Find the marked price.
Checking: . Correct.
The three prices and their bases, set side by side. This is the summary worth memorising:
- Cost price — what the seller paid. Profit and loss percentages are taken on this.
- Marked price — what is printed on the article. Discount percentages are taken on this.
- Selling price — what the buyer actually paid, after any discount and before any tax.
Why the marked price is usually set above the cost price. A shopkeeper marks an article high enough that even after allowing a discount the selling price still exceeds the cost price and leaves a profit. So a discount is not generosity — it is planned into the marked price, which is exactly the calculation of the next section.
A boundary case. If there is no discount, then . Many questions rely on spotting this, since the words sold at the marked price mean the discount is zero and one of the three prices simply disappears from the problem.
Discount per cent is always calculated on the marked price. This is the counterpart of the previous part's rule that profit per cent is always on the cost price — and keeping the two bases apart is the whole difficulty of the chapter.
Worked example 1. An article marked at is sold for . Find the discount and the discount per cent.
Worked example 2. Find the selling price of an article marked at after a discount of .
Checking: the discount is , and . Correct.
Worked example 3 — finding the marked price. After a discount of an article sells for . Find the marked price.
Checking: . Correct.
The three prices and their bases, set side by side. This is the summary worth memorising:
- Cost price — what the seller paid. Profit and loss percentages are taken on this.
- Marked price — what is printed on the article. Discount percentages are taken on this.
- Selling price — what the buyer actually paid, after any discount and before any tax.
Why the marked price is usually set above the cost price. A shopkeeper marks an article high enough that even after allowing a discount the selling price still exceeds the cost price and leaves a profit. So a discount is not generosity — it is planned into the marked price, which is exactly the calculation of the next section.
A boundary case. If there is no discount, then . Many questions rely on spotting this, since the words sold at the marked price mean the discount is zero and one of the three prices simply disappears from the problem.
How do you handle a discount and a profit in the same problem?
Go through the selling price, which is the only quantity both percentages touch. The discount connects to , and the profit connects to .
Worked example 1 — finding the cost price. An article is marked at and sold at a discount of . If the shopkeeper makes a profit of , find the cost price.
First apply the discount to get the selling price:
Then work back through the profit to the cost price:
Checking forwards: selling price, and is of . Correct.
Worked example 2 — finding the marked price, the commonest version. A shopkeeper buys an article for . At what price should he mark it so that after allowing a discount of he still makes a profit of ?
This is example 1 run in reverse. First find the selling price he needs:
Then find the marked price that gives that selling price after a discount:
**Why the marked price is not simply above the selling price.** A discount is of the marked price, so the selling price is of it — and recovering the whole from means **dividing by **, not multiplying by . Multiplying by would have given , and a discount on that leaves , short of the needed.
That mistake is the same finding the whole from a part error met in the percentage chapter, and it is worth recognising as an old enemy in a new setting.
Worked example 3 — a required discount. A dealer marks an article at which cost him . What discount per cent can he allow and still make a profit of ?
The selling price he needs is
so the discount he can afford is
The order that never fails. Write down which of the three prices you are given and which you want, then move one arrow at a time. Trying to combine a discount percentage and a profit percentage into a single figure does not work, because they are percentages of different bases.
Worked example 1 — finding the cost price. An article is marked at and sold at a discount of . If the shopkeeper makes a profit of , find the cost price.
First apply the discount to get the selling price:
Then work back through the profit to the cost price:
Checking forwards: selling price, and is of . Correct.
Worked example 2 — finding the marked price, the commonest version. A shopkeeper buys an article for . At what price should he mark it so that after allowing a discount of he still makes a profit of ?
This is example 1 run in reverse. First find the selling price he needs:
Then find the marked price that gives that selling price after a discount:
**Why the marked price is not simply above the selling price.** A discount is of the marked price, so the selling price is of it — and recovering the whole from means **dividing by **, not multiplying by . Multiplying by would have given , and a discount on that leaves , short of the needed.
That mistake is the same finding the whole from a part error met in the percentage chapter, and it is worth recognising as an old enemy in a new setting.
Worked example 3 — a required discount. A dealer marks an article at which cost him . What discount per cent can he allow and still make a profit of ?
The selling price he needs is
so the discount he can afford is
The order that never fails. Write down which of the three prices you are given and which you want, then move one arrow at a time. Trying to combine a discount percentage and a profit percentage into a single figure does not work, because they are percentages of different bases.
How do you combine two or more successive discounts?
Multiply the multipliers, exactly as for successive percentage changes. Each discount applies to whatever the price is at that stage.
Worked example 1. An article marked at is offered at successive discounts of and . Find the selling price.
Setting it out stage by stage: after the first discount, then after the second.
The single equivalent discount. For two successive discounts of and :
Checking it here:
And confirms a single discount of would have the same effect. The subtracted term is exactly what makes two discounts fall short of their sum.
The order makes no difference.
the same answer, because multiplication is commutative. So * then and then * cost the shopper exactly the same — which is worth knowing, since advertisements sometimes imply otherwise.
Worked example 2 — three discounts. Find the selling price of an article marked after successive discounts of , and .
The single equivalent discount is , well short of the that adding would give. With more discounts the gap widens, because each new percentage is taken of an ever smaller amount.
Worked example 3 — comparing two offers. Which is better for the buyer on an article marked : a single discount of , or successive discounts of and ?
They are identical, since . So a comparison question can come out level, and the only way to know is to work both out — guessing from the totals against would have given the wrong answer.
The general rule for comparisons. Successive discounts are always worth less than their sum as a single discount, so * and * beats neither a single discount nor even a one. Converting every offer to a single multiplier makes any number of offers directly comparable.
Worked example 1. An article marked at is offered at successive discounts of and . Find the selling price.
Setting it out stage by stage: after the first discount, then after the second.
The single equivalent discount. For two successive discounts of and :
Checking it here:
And confirms a single discount of would have the same effect. The subtracted term is exactly what makes two discounts fall short of their sum.
The order makes no difference.
the same answer, because multiplication is commutative. So * then and then * cost the shopper exactly the same — which is worth knowing, since advertisements sometimes imply otherwise.
Worked example 2 — three discounts. Find the selling price of an article marked after successive discounts of , and .
The single equivalent discount is , well short of the that adding would give. With more discounts the gap widens, because each new percentage is taken of an ever smaller amount.
Worked example 3 — comparing two offers. Which is better for the buyer on an article marked : a single discount of , or successive discounts of and ?
They are identical, since . So a comparison question can come out level, and the only way to know is to work both out — guessing from the totals against would have given the wrong answer.
The general rule for comparisons. Successive discounts are always worth less than their sum as a single discount, so * and * beats neither a single discount nor even a one. Converting every offer to a single multiplier makes any number of offers directly comparable.
How do you add GST to a discounted price?
Tax is charged on the selling price — that is, after the discount has been deducted.
where is the rate of GST, or of sales tax or VAT in older questions. All three work identically in the arithmetic.
The order is fixed and matters: discount first, tax second. Tax is levied on what the buyer actually pays for the goods, not on the printed price.
Worked example 1. An article is sold for and GST is charged at . Find the amount payable.
So the GST itself is .
Worked example 2 — the full chain. An article is marked at . A discount of is allowed and GST is charged at . Find the amount a customer pays.
Discount first:
Then tax:
Had the tax been charged on the marked price instead, the customer would have paid as well — the same, since multiplication commutes. But the tax collected would be recorded differently, and examination questions expect the discount applied first.
Worked example 3 — working backwards from the bill. A customer pays for an article, including GST at . Find the selling price before tax.
Checking: . Correct.
The temptation is to take **of ** — which is — and subtract it, giving . That is wrong, because the was a percentage of the pre-tax price, not of the total. The check exposes it: , not .
Worked example 4 — recovering the marked price from a bill. A customer pays including GST at , after a discount of on the marked price. Find the marked price.
Undo the tax, then undo the discount:
The four prices in a full question, in order. Cost price, then marked price, then selling price after discount, then amount payable after tax. Each step has its own base: profit on the CP, discount on the MP, and tax on the SP. Getting a question right is almost entirely a matter of using the right base at each arrow — which is why writing the chain down before calculating is worth the ten seconds it takes.
where is the rate of GST, or of sales tax or VAT in older questions. All three work identically in the arithmetic.
The order is fixed and matters: discount first, tax second. Tax is levied on what the buyer actually pays for the goods, not on the printed price.
Worked example 1. An article is sold for and GST is charged at . Find the amount payable.
So the GST itself is .
Worked example 2 — the full chain. An article is marked at . A discount of is allowed and GST is charged at . Find the amount a customer pays.
Discount first:
Then tax:
Had the tax been charged on the marked price instead, the customer would have paid as well — the same, since multiplication commutes. But the tax collected would be recorded differently, and examination questions expect the discount applied first.
Worked example 3 — working backwards from the bill. A customer pays for an article, including GST at . Find the selling price before tax.
Checking: . Correct.
The temptation is to take **of ** — which is — and subtract it, giving . That is wrong, because the was a percentage of the pre-tax price, not of the total. The check exposes it: , not .
Worked example 4 — recovering the marked price from a bill. A customer pays including GST at , after a discount of on the marked price. Find the marked price.
Undo the tax, then undo the discount:
The four prices in a full question, in order. Cost price, then marked price, then selling price after discount, then amount payable after tax. Each step has its own base: profit on the CP, discount on the MP, and tax on the SP. Getting a question right is almost entirely a matter of using the right base at each arrow — which is why writing the chain down before calculating is worth the ten seconds it takes.
Exam tip
Exam tip: discount is on the marked price, tax on the selling price
Three different bases, and each percentage has only one: profit on the cost price, discount on the marked price, tax on the selling price. Write the chain amount payable before you start.
Apply the discount first and the tax second. Tax is charged on what the buyer pays for the goods, not on the printed price.
For successive discounts, multiply the multipliers — never add the percentages. then gives , a single equivalent of ****.
Use as a check on the multiplier working, not as a replacement for it.
Remember the order of discounts makes no difference to the final price.
To find a marked price from a selling price, divide by — do not multiply by . A discount means the SP is of the MP, so recovering the MP means dividing by .
To work backwards from a bill, undo the tax first (multiply by ), then undo the discount. Never take the tax percentage of the total.
Check every answer forwards through the whole chain, and keep the rupee symbol on every money line.
Apply the discount first and the tax second. Tax is charged on what the buyer pays for the goods, not on the printed price.
For successive discounts, multiply the multipliers — never add the percentages. then gives , a single equivalent of ****.
Use as a check on the multiplier working, not as a replacement for it.
Remember the order of discounts makes no difference to the final price.
To find a marked price from a selling price, divide by — do not multiply by . A discount means the SP is of the MP, so recovering the MP means dividing by .
To work backwards from a bill, undo the tax first (multiply by ), then undo the discount. Never take the tax percentage of the total.
Check every answer forwards through the whole chain, and keep the rupee symbol on every money line.
Did you know
Why do shops advertise two discounts instead of one bigger one?
A sign reading * off, plus a further off sounds more generous than off*. The two offers are identical.
The reason the first sounds better is that most people add the percentages in their head and arrive at . The arithmetic gives , because the second discount is taken of the already reduced price and so removes less money than it appears to.
The gap grows with the size of the discounts. Two offers of each sound like off — free — and in fact leave the shopper paying a quarter of the marked price, a single equivalent discount of . Three offers of sound like and actually come to .
Which is why converting every offer to a single multiplier is the only safe way to compare them. Two discounts of and come to a multiplier of , so they are worth exactly as much as a single — and less than a single , however the sign is worded.
The reason the first sounds better is that most people add the percentages in their head and arrive at . The arithmetic gives , because the second discount is taken of the already reduced price and so removes less money than it appears to.
The gap grows with the size of the discounts. Two offers of each sound like off — free — and in fact leave the shopper paying a quarter of the marked price, a single equivalent discount of . Three offers of sound like and actually come to .
Which is why converting every offer to a single multiplier is the only safe way to compare them. Two discounts of and come to a multiplier of , so they are worth exactly as much as a single — and less than a single , however the sign is worded.
Key takeaways
Discount, marked price and GST: quick revision
- , and — always on the marked price.
- . So marked and sold for is a discount, and less gives .
- Finding MP from SP: . So after off came from .
- Three bases: profit on the cost price, discount on the marked price, tax on the selling price.
- Discount and profit together go through the SP: . marked, off gives , and a profit means .
- Reversed: needing a profit gives , and allowing a discount gives — by dividing by , not multiplying by .
- A dealer with and wanting a profit needs , so he can allow a ** discount.
- Successive discounts multiply**: , a single equivalent of — not .
- Single equivalent: , so . The order makes no difference.
- Three discounts: , an equivalent of .
- Comparison: a single and successive with both give — identical.
- GST: amount payable , with the discount applied first. plus gives ; and less then plus gives .
- Backwards from a bill: undo the tax first. including came from ; and including after off came from a marked price of .
- Never take the tax percentage of the total bill.
Work one full chain from cost price to final bill and then back again — if the two directions do not meet at the same numbers, one of the three bases was used in the wrong place.
- . So marked and sold for is a discount, and less gives .
- Finding MP from SP: . So after off came from .
- Three bases: profit on the cost price, discount on the marked price, tax on the selling price.
- Discount and profit together go through the SP: . marked, off gives , and a profit means .
- Reversed: needing a profit gives , and allowing a discount gives — by dividing by , not multiplying by .
- A dealer with and wanting a profit needs , so he can allow a ** discount.
- Successive discounts multiply**: , a single equivalent of — not .
- Single equivalent: , so . The order makes no difference.
- Three discounts: , an equivalent of .
- Comparison: a single and successive with both give — identical.
- GST: amount payable , with the discount applied first. plus gives ; and less then plus gives .
- Backwards from a bill: undo the tax first. including came from ; and including after off came from a marked price of .
- Never take the tax percentage of the total bill.
Work one full chain from cost price to final bill and then back again — if the two directions do not meet at the same numbers, one of the three bases was used in the wrong place.