Discount

This is where the chapter starts, and the question comes in three simple forms. In all three, the MP is known.

1. Find the SP. "Marked ₹1400, discount 25%." You pay 100% − 25% = 75%. SP = 75100 × 1400 = ₹1050. With the chart, 25% off means MP : SP = 4 : 3, so SP = 34 × 1400 = ₹1050.

2. Find the discount amount (the saving). "Marked ₹2400, discount 15%." Discount = 15% of 2400 = ₹360. Read the question carefully here. If it asks how much does he save, the answer is the discount, not the SP.

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3. Find the discount%. "Marked ₹1250, sold for ₹1100." Discount = 1250 − 1100 = ₹150, and discount% = 1501250 × 100 = 12%. The MP always sits in the denominator.

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When several identical items are sold together, work with the total MP. Fifteen notebooks of ₹80 each have a total MP of ₹1200. If all fifteen go for ₹1000, the discount is 200 on 1200, which is 16 = 1623%.

If an offer changes from one discount to another on the same MP, the extra saving is simply the difference of the two rates, taken on the MP. From 8% to 12% on ₹4500, the extra discount = 4% of 4500 = ₹180.

Radhika sees a kurta marked ₹800 with "15% off". She thinks: 10% of 800 is 80 and 5% is 40, so the discount is ₹120 and she pays ₹680. She did it without picking up a pen.

Now the question is turned around. The SP and the discount are given, and the MP is asked.

Think of it this way. If the discount is 20%, the customer pays 80% of the MP. So the SP is 80% of the MP:
80% of MP = SP ⇒ MP = SP × 10080

In general, MP = SP × 100100 − d. The chart makes it quicker still: 20% off ⇒ MP : SP = 5 : 4. SP = ₹680 ⇒ 4 parts = 680 ⇒ 1 part = 170 ⇒ MP = 5 parts = ₹850.

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This is the step where the usual slip happens. The discount gets added to the SP: "SP is 680 and the discount is 20%, so MP = 680 + 20% of 680 = 816." That is wrong, because the 20% was taken on the MP, not on 680. Never add the discount% to the SP; divide by the paying fraction.

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When the discount rate changes. "At 20% off the SP is ₹4800. What is the SP at 25% off?" The MP has not changed. 80% of MP = 4800, so 1% of MP = 60, and at 25% off the SP = 75% of MP = 75 × 60 = ₹4500. In one step: new SP = old SP × 100 − d₂100 − d₁ = 4800 × 7580.

When the difference of two SPs is given. "Sold at 30% off and at 22% off, the SPs differ by ₹96." The two SPs are 70% and 78% of the same MP, so their difference is 8% of the MP. 8% of MP = 96 gives MP = ₹1200.

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"Which option has MP = ₹5000?" Here four (SP, discount) pairs are given. Work out SP × 100100 − d for each option, or go in reverse: 20% off on ₹5000 gives ₹4000.

Take Radhika's shoes. She paid ₹960 after a 4% discount, which is 96% of the MP. So 1% = 10 and the MP was ₹1000.

A small type, but it catches people. No rupee value is given at all. The question only tells you how the discount, the SP and the MP compare with one another.

Everything rests on one fact: MP = SP + Discount. The MP is the whole, and the SP and the discount are its two parts. Write all three as parts and the question opens up.

Discount as a fraction of MP. "Discount = 320 of MP." Take MP = 20 parts. Then discount = 3 parts and SP = 17 parts. Any ratio can now be read off: discount : SP = 3 : 17, MP : SP = 20 : 17.

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MP as a percentage of the discount. "MP is 160% of the discount." Take discount = 100. Then MP = 160 and SP = 160 − 100 = 60, so MP is 16060 = 26623% of SP.

Read "x percent more than" carefully. "MP is 300 percent more than the discount" is not the same as "MP is 300 percent of the discount". 300% more means MP = 100 + 300 = 400 when the discount is 100. Then SP = 300, and SP is 300400 = 75% of MP. With "300 percent of", MP = 300 and SP = 200, so SP would be only 6623% of MP. You met this difference between "of" and "more than" in Percentage; here it decides the answer.

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Suppose Radhika's shop gives a discount equal to 14 of the SP. Take SP = 4 and discount = 1. Then MP = 5 and the discount% = 15 = 20%. The same discount is 14 of the SP but only 15 of the MP, because the base changes the fraction.

"25% off, and then an extra 10% off." Most students add the two and answer 35%. That is wrong; this page shows the correct method.

In successive discounts, the second discount is given on the price already reduced by the first one, not on the original MP. On ₹2000:

• After 25% off, the price is 75% of 2000 = ₹1500.
• The next 10% is taken on ₹1500, so it is ₹150, not ₹200. Final price = ₹1350.

The total discount is ₹650, which is 32.5%, not 35%. The second discount came out smaller because its base was smaller.

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This is the same successive percentage change you met in Percentage. Each discount is a multiplying factor, and successive discounts simply multiply:
SP = MP × 100 − x100 × 100 − y100 × 100 − z100 …

Single equivalent discount. This is the one discount that gives the same final price. For two discounts x% and y%:
Equivalent discount = x + y − xy100 %
For 25% and 10%: 25 + 10 − 250100 = 32.5%. For 15% and 12%: 27 − 1.8 = 25.2%.

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For three discounts, use the formula twice: combine two of them first, then combine that result with the third. For 20%, 15% and 10%: 20 and 15 give 32, then 32 and 10 give 32 + 10 − 3.2 = 38.8%.

Ratio view. Every discount is an MP : SP ratio, so the ratios multiply. 25% off = 4 : 3 and 20% off = 5 : 4. Together MP : SP = 20 : 12 = 5 : 3, and the equivalent discount is 5 − 35 = 25 = 40%.

Two facts are worth keeping:

• The order does not matter. 10% then 20% gives the same price as 20% then 10%, because multiplication does not depend on order.
• Two successive discounts always give less than their sum, because the second one works on a smaller base.

A pen is marked ₹200, and Radhika's favourite shop gives 10% and then another 10%. After the first, ₹180. The second takes 10% of 180, that is ₹18, so she pays ₹162. Equivalent discount = 10 + 10 − 1 = 19%.

Here one piece of the chain is hidden. Either one of the two discounts is missing, or the MP is.

The second discount is missing. "MP ₹1200, first discount 20%, final SP ₹816. Find the second discount." Walk along the chain step by step:

• After the first discount, the price = 80% of 1200 = ₹960.
• The second discount takes ₹960 down to ₹816, so it is ₹144.
• Second discount% = 144960 × 100 = 15%.

The second discount% is calculated on the reduced price (₹960), not on the MP (₹1200). Dividing 144 by 1200 is exactly the wrong answer the options are waiting for.

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The first discount is missing. Walk the chain backwards. "MP ₹6400, second discount 25%, final ₹4200." Before the second discount the price was 4200 × 10075 = ₹5600. So the first discount took 6400 to 5600, which is 8006400 = 12.5%.

The MP is missing. Divide by both factors. "25% and 20% off, SP ₹1500." SP = MP × 0.75 × 0.8 = 0.6 MP, so MP = 15000.6 = ₹2500.

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The equivalent discount is given and one discount is missing. "A single discount of 30% = successive discounts of 20% and x%." The paying fractions must match: 0.70 = 0.80 × (1 − x100). So 1 − x100 = 7080 = 78, and x = 18 = 12.5%.

Radhika's coat is marked ₹1000. After a first discount of 20% and a second one, she pays ₹720. The second discount is ₹80 on ₹800, which is 10%, not 8%.

Real shops rarely stop at one percentage. They add "₹2000 cashback", "extra 5% for cash payment", "18% GST extra" or "₹1000 off above ₹15000". One rule covers all of them: apply each step in the order written, on the price you have at that step.

Percentage after percentage works like successive discounts. "20% off, and an additional 5% for cash" on ₹8000: first ₹6400, then 95% of 6400 = ₹6080.

A flat rupee discount or cashback after a percentage. Subtract the rupees at that step. ₹50000 with 15% off = ₹42500, then ₹1500 cashback leaves ₹41000.

Tax on the discounted price. GST or VAT is an increase, and it is charged on the price after the discount. ₹2500 with 20% off = ₹2000; 18% GST on 2000 = ₹360, so the bill = ₹2360. As factors: 2500 × 0.8 × 1.18.

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Slab offers. "20% off above ₹5000, and ₹500 more off if the amount after discount exceeds ₹8000." Check each condition on the right amount. On ₹12000, 20% off leaves ₹9600, which is above ₹8000, so the ₹500 also applies: ₹9100.

Price raised and then discounted. A shop raises the price by 3313% and then gives 25% off. The factors are 43 × 34 = 1, so the final price equals the original. A rise of 3313% and a fall of 25% cancel exactly, as you saw in Percentage.

To go backwards and find the original price, divide by all the factors. If there is a flat rupee step, add it back first and then divide.

Radhika buys a ₹1500 jacket at 20% off and pays 5% GST: 1500 × 0.8 = 1200, 5% of 1200 = 60, and the bill is ₹1260. Had the shop added GST first and given the discount afterwards, the bill would still be 1500 × 1.05 × 0.8 = ₹1260, because only multiplication is involved. A flat ₹ discount is different: when it comes changes the answer. So always follow the order given.

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From here the main part of the chapter begins. The shopkeeper marks the item above CP and then gives a discount. Does he gain or lose, and by how much?

Take CP = 100, and the picture becomes clear at once.
"Marked 50% above CP, discount 30%." MP = 150. Discount = 30% of 150 = 45. SP = 105. A profit of 5 on a CP of 100 means 5% profit.

The markup was 50% and the discount 30%, yet the profit is not 50 − 30 = 20%. The reason is the base: the 50% is on CP, while the 30% is on the bigger MP, so the discount cuts more rupees than it seems to.

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This is again a successive percentage change: first +M% (markup), then −D% (discount). So the net-change formula from Percentage works directly:
Profit% = M − D − M × D100
For 50% and 30%: 50 − 30 − 1500100 = 5%. A negative answer means a loss.

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With successive discounts, multiply all the factors. "Marked 100% above CP, discounts of 25% and 20%." 100 → 200 → 150 → 120, so the profit is 20%.

When rupee values are given (say CP ₹900 and MP ₹1200), no formula is needed. Find the SP from the MP, then profit% = SP − CPCP × 100.

The ratio view, which is where the quick method begins. CP : MP : SP can be built as a chain. Markup 50% ⇒ CP : MP = 2 : 3. Discount 30% ⇒ MP : SP = 10 : 7. Make the MP common: CP : MP : SP = 20 : 30 : 21, and profit = 21 − 2020 = 5%. Types 8 and 9 use the same idea in reverse.

Radhika marks her handmade bags 60% above cost and gives 15% off. 1.6 × 0.85 = 1.36, so she still earns 36%. If she gives 37.5% off, 1.6 × 58 = 1 and she earns nothing. That is the "no profit, no loss" point.

This type comes up again and again, and this is where the ratio method really earns its place.

Recall the master relation from the foundation: the SP seen from the CP and the SP seen from the MP are the same number.
CP × 100 + P100 = SP = MP × 100 − D100

So MPCP = 100 + P100 − D (use 100 − L for a loss).

Read it as a ratio. "Discount 20%, profit 12%." MP : CP = 112 : 80 = 7 : 5. That one ratio answers every question about this item:

• MP = ₹1400 ⇒ 7 parts = 1400 ⇒ 1 part = 200 ⇒ CP = 5 × 200 = ₹1000.
• CP = ₹650 ⇒ 5 parts = 650 ⇒ 1 part = 130 ⇒ MP = 7 × 130 = ₹910.

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Why does it work? Because 7 : 5 only says that 112% of CP = 80% of MP. You saw this in Percentage: if x% of A = y% of B, then A : B = y : x.

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The basic method for the same question goes through the SP. MP ₹1400, 20% off ⇒ SP = ₹1120. Profit 12% ⇒ CP = 1120 × 100112 = ₹1000. This is perfectly correct and you must know it; the ratio method simply joins the two steps into one.

When the SP is given directly. "Sold for ₹690 after a 25% discount, profit 15%. Find the CP." The MP is not needed at all: CP = 6901.15 = ₹600. The discount is extra information here, and finding the MP first only wastes effort.

With successive discounts, multiply the discount factors: MP × 0.9 × 0.95 = CP × 1.14.

Radhika allows 15% off and still earns 2%. MP : CP = 102 : 85 = 6 : 5, so if her CP is ₹750, the MP must be ₹900.

The same master relation, now solved for a percentage instead of a price.

Finding the markup. "25% discount, still 5% profit. How far above CP is the MP?"
MPCP = 100 + 5100 − 25 = 10575 = 75
So CP = 5 and MP = 7. The MP is 2 parts above the CP, that is 25 = 40% above CP.

Finding the discount. "Marked 40% above CP, profit 19%. Find the discount."
CP = 100, MP = 140, SP = 119. The discount is 21 on an MP of 140, which is 15%.
Divide the discount by the MP, not by the CP. Writing 21100 = 21% is the wrong answer you will always find among the options.

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In formula form: (100 + M)(100 − D) = 100(100 + P). With M = 40 and P = 19: 140 × (100 − D) = 11900 ⇒ 100 − D = 85 ⇒ D = 15%.

The maximum discount without loss. No loss means SP = CP. If the MP is 50% above CP, then CP = 100 and MP = 150, and the discount can go up to 50 on 150, that is 13 = 3313%. As a formula, maximum discount = M100 + M × 100%. This is the "back to the original" rule from Percentage: after a rise of 12, a fall of 13 brings you back.

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When rupee values are given (CP ₹4000, MP ₹6000, profit 20%), find the SP from the CP and then the discount from the MP: SP = 4800, so the discount = 1200 on 6000 = 20%.

Radhika marks a scarf 25% above cost and finally earns 10%. CP = 100, MP = 125, SP = 110, so the discount is 15 on 125 = 12%. She gave 12% off, not 15%.

Here the markup and the discount come as percentages, but the profit (or loss) is given in rupees. Or a percentage discount is mixed with a flat rupee discount. The plan never changes: first build the percentage picture with CP = 100, then match the rupee value to its correct part.

Profit in rupees. "Marked 50% above CP, 30% discount, gain ₹90. Find the CP." With CP = 100: MP = 150, SP = 105, profit = 5. So 5% of CP = ₹90 and CP = ₹1800. Note that the ₹90 belongs to the 5, not to the 50 or the 30.

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Loss in rupees works the same way. If the net factor is 0.9, the loss is 10% of CP; a loss of ₹400 then means CP = ₹4000.

A flat rupee discount after a percentage discount. A rupee step cannot be turned into a factor, so write the SP in two ways and equate them:
SP = CP × (markup factor) × (discount factor) − flat discount
SP = CP × (1 + profit fraction)
"Marked 50% above, 20% off, then a ₹120 cash discount, profit 12%." With CP = x: 1.5x × 0.8 − 120 = 1.12x ⇒ 1.2x − 1.12x = 120 ⇒ 0.08x = 120 ⇒ x = ₹1500.

Note that the percentage steps alone would give a 20% profit, but the shopkeeper ends with only 12%. The missing 8% of CP is exactly the ₹120.

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Extra cost on the buyer's side. If a person buys at a discount and then spends on repairs, his total cost = purchase price + repairs, and the profit% is taken on this total cost.

Radhika marks a lamp 50% above cost, gives 20% off and earns ₹80. With CP = 100, SP = 120, so the profit is 20% of CP = ₹80. That gives CP = ₹400, MP = ₹600 and SP = ₹480.

In this type the same item is imagined in two situations: "with discount" and "without discount", or "at 20% off" and "at 5% off". What stays fixed? The CP and the MP do not change between the two situations; only the SP does.

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So the method has two steps:
1. Use the situation with full information to find the link between MP and CP (the ratio method of Type 8 fits perfectly here).
2. Apply that link to the second situation.

Profit without discount. "20% discount, profit 8%. What is the profit if no discount is given?" MP : CP = 108 : 80 = 27 : 20. Without a discount SP = MP = 27 when CP = 20, so the profit = 720 = 35%.

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Selling at a different % of MP. "At 75% of MP, loss 10%. Profit at 90% of MP?" CP = 100 ⇒ SP = 90 = 75% of MP ⇒ MP = 120. Then 90% of 120 = 108, a profit of 8%. In one step, the SP grows in the ratio 90 : 75, so 90 × 9075 = 108.

A difference in profit given in rupees. "At 15% off and at 6% off, the profits differ by ₹432." The CP is the same in both, so the difference in profit = the difference in SP = 9% of MP, which gives MP = ₹4800. The CP is not even needed.

Profit in rupees in one situation, SP asked in the other. Find the CP from the rupee profit, then the SP, then the MP, and finally the new SP.

Radhika sells a bag at 25% off and earns 20%. MP : CP = 120 : 75 = 8 : 5. With no discount she would sell for 8 what cost her 5, a profit of 60%. At 37.5% off she would sell at exactly 5, with no profit and no loss.

"Buy 4, Get 1 Free" is also a discount, written in a different language. The customer takes 5 items and pays for 4. So he pays 45 of the MP of what he takes home, and the discount is 15 = 20%.

In general, with "Buy x, Get y free" the customer receives x + y items and pays for x:
Discount% = yx + y × 100
The denominator is the total number of items taken, not the number bought. "Buy 4, Get 1" is 20%, not 25%.

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A part discount on one item. "Buy 3 and get the 4th at 50% off." On 4 items the discount is half of one item, that is 0.5 item, so the discount% = 0.54 = 12.5%.

Effective price per item. "Buy 3 at ₹600 each, get the fourth free." You pay 1800 for 4 items ⇒ ₹450 per item.

Free items with cost and profit. When a company gives some items free, they are still part of its cost. Count the items that are actually paid for, find the revenue and compare it with the total cost. "1 free for every 4 bought" means that in every group of 5, only 4 are paid for.

Different discounts on different items. "The MPs of a shirt and a trouser are in the ratio 2 : 3; 20% off on the shirt; total discount 14%." Take the MPs as 2 and 3 (total 5). Total discount = 14% of 5 = 0.7, and the shirt's discount = 20% of 2 = 0.4. So the trouser gets 0.3 on 3, which is 10%. This is a weighted average, the same idea you use in Alligation.

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Radhika's bookshop runs "Buy 2, Get 1 Free". A customer takes 3 books and pays for 2, so the discount is 13 = 3313%. If she changes it to "Buy 3, Get 2 Free", the discount becomes 25 = 40%.

This is the last and the longest type. Nothing new is taught here; everything you have learnt so far is joined into one chain. The questions come in two kinds.

1. A chain of people. Factory → wholesaler → retailer → customer. Each person's SP becomes the next person's CP, and every step is a multiplying factor:

• markup or profit r% ⇒ × 100 + r100
• discount d% ⇒ × 100 − d100
• a fee or tax added on a price ⇒ × 100 + f100

Multiply all the factors, then go forwards (multiply) or backwards (divide) to the price that is asked.

Take a wholesaler with a 25% profit and a retailer who marks up 40% and gives 10% off, so that the customer pays ₹630. Factor = 1.25 × 1.4 × 0.9 = 1.575, so the wholesaler's cost = 6301.575 = ₹400.

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When the profit of someone in the middle is given in rupees (the showroom earns ₹3000 at 25% profit), first find that person's cost (3000 ÷ 0.25 = 12000), and then go back along the chain.

2. Prices written as letters. "He sells for ₹P at 25% profit. Next, he marks it at ₹1.2P and gives 5% off." The letter is nothing to fear. The first sentence tells you CP = P1.25. The new SP = 1.2P × 0.95 = 1.14P. Profit factor = 1.14PP ÷ 1.25 = 1.14 × 1.25 = 1.425, so the profit is 42.5%. The letter always cancels; only the factors matter.

Another of the same family: "MP is ₹X; bought at 20% off; sold at 10% more than X." Cost = 0.8X, SP = 1.1X, profit = 1.10.8 − 1 = 37.5%.

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Radhika buys a dupatta at 30% off the MP and sells it at the full MP. Cost = 0.7, SP = 1, profit = 0.30.7 = 4267%. The discount was 30%, yet her profit is more than 30%, because profit is counted on her smaller cost.

Don't hurry in this type. Write the chain on paper with one arrow per step, and calculate only once at the end.