Tricky percentages: why a 50% margin followed by a 50% discount leaves a 25% loss

What to assume they know

• Percentage increase and decrease, and choosing the right base — that a change of k% is a multiplication by (1 + k/100), and that the base is the value you started from
• Profit, loss and taxes as percentages of a stated amount — cost price, marked price, selling price, margin, discount
• Compounding: why repeated growth multiplies instead of adding — that successive percentage changes multiply
• Comparing two proportions that have different totals — that a percentage discards the base, so it cannot report an amount
• Multiplying two decimals below 1; solving a one-step equation for a fraction

What they should be able to do

• Compute the percentage gain of two offers and rank them, then rank the same two offers by amount, and explain why the two orders can disagree
• Decide when a flat discount beats a percentage discount, and find the purchase value at which they are equal
• Show that two successive discounts of a% and b% are not a discount of (a + b)%, and compute the single discount they are equivalent to
• Model a mark-up followed by a discount as a product of two factors, and read the overall effect off the product
• Explain, in terms of bases, why an equal percentage discount does not undo an equal percentage mark-up
• Solve for the discount that exactly cancels a stated mark-up
• Convert a statement about one quantity being k% of another into the reverse statement, correctly
• Express a "buy n get one free" offer as a percentage discount
• Reason about how a population must change to move a percentage by one point

Where it usually goes wrong

• "30% off and then 20% off is 50% off." It is 44% off. The chapter names the reason — in shopping the two are compounded — and Example 12 prices it out. This is the single most commercially relevant piece of mathematics in the chapter.
• "A 50% mark-up is cancelled by a 50% discount." It leaves a 25% loss, because the discount is taken on the larger number. Surbhi's three-bar model is the whole proof, and it fits on one screen.
• "The bigger percentage gain is the better offer." Option A gains 200% and hands you ₹200; Option B gains 50% and hands you ₹500. Which is better depends on what you had, which is why the chapter refuses to answer it.
• "A percentage discount always beats a flat one." Below ₹250 on the clearance item, the flat ₹50 is better. The break-even point is the answer, not either offer.
• "The club that grew 100% is the bigger club." It may have four members. Growth and size are different questions, and the comic exists to separate them.
• "If mine is 120% of yours, then yours is 80% of mine." Yours is 83.33% of mine. The two statements have different bases, so the two percentages are not complementary.
• "Buy one get one free is a 100% discount on the second item, so 100% off." It is 50% off, because you paid for one of the two items you took home. The item's printed hint — compare free items to total items received — is exactly this correction.
• "To move 99% down to 98%, one or two people leave." Forty-nine do. Students reason about the 99 and should be reasoning about the 1.
• "Percentages never add." They add perfectly well when they share a base — the CGST and SGST lines of the bill in Profit, loss and taxes as percentages of a stated amount add to 18% of one sub-total. The rule is about the base, not about percentages.

Questions to check understanding

• Compute the percentage gain of two offers and choose between them with a stated reason — the board's open-ended format here, where the reason carries the marks
• Find the purchase value at which a flat discount and a percentage discount are equal
• Find the single discount equivalent to two successive discounts, and vice versa
• Given a mark-up percentage and a subsequent discount percentage, determine profit or loss and its percentage, with reasons
• Find the discount that exactly offsets a stated mark-up
• Convert "A is k% of B" into the corresponding statement about B and A
• Express a buy-n-get-one-free offer as a percentage discount, and order several such offers
• Reasoning items where a percentage must be moved by a stated amount and the number who must join or leave is asked for
• Explain in writing why two percentages may not be added

Examples worth working on the board

Values marked arithmetic added here are added here; the chapter prints no answer key for Part II.

• The two puzzle clubs — a four-panel comic (Part II, §1.3, p.26, across the top of the page). Every word and number in it is lettering inside the artwork and none of it extracts — read from the printed page. The exchange, panel by panel: a student recruits for the Puzzle-Solvers Club, calling it the fastest growing in the school; a companion backs the claim by saying they grew by 80% this year; a listener remarks that "fastest-growing" seems suspicious. In the next panel a second student invites Bittu to a Puzzle-Creators Club, and Bittu agrees it sounds interesting. Then the second student says their club just grew by 100% this year. In the last panel the Puzzle-Solvers — the same 80%-growth pair from the first panel, identifiable by the red book one of them carries — reply that their club has 678 members, and the Creators' side retorts that it hopes those members are all right with taking second place on the fastest-growing list. There are two clubs in this comic, not three, and the 678 belongs to the 80% club. That is the joke, and none of the lettering extracts, so this brief is the teacher's only source for it. The argument, which is added here: 100% growth on a club of two is two new members; 80% growth on a club of 678 would be more than five hundred. The percentage ranks the proportion grown, and the panel with 678 members is there to show that ranking by proportion can invert the ranking by size. This comic is the best hook in the chapter and it is invisible to anyone working from the text.
• "Would You Rather?" (Part II, §1.3, p.25, bold subheading under "Tricky Percentages", flagged Math Talk). You have won a contest and the organisers offer two options: Option A, deposit ₹100 and get back ₹300; Option B, deposit ₹1000 and get back ₹1500. Find the percentage gain of each; you may take one option only once; which would you choose and why. Arithmetic added here: A gains ₹200 on ₹100, a 200% gain; B gains ₹500 on ₹1000, a 50% gain. The percentages and the rupees point in opposite directions, and the question deliberately has no single right answer — it depends on whether what you have is ₹100 or ₹1000.
• The caution the chapter prints (Part II, §1.3, p.26, tinted box directly under the comic): when comparing percentages, remember that what is being compared are proportions rather than plain amounts.
• The stock clearance sale (Part II, §1.3, p.26). A provision store lets a customer choose one of two offers on any purchase above ₹150: a 20% discount or a ₹50 discount. Decide which to take for a basket worth ₹180, one worth ₹225, and one worth ₹300. Arithmetic added here: 20% of 180 is ₹36, of 225 is ₹45, of 300 is ₹60 — so the flat ₹50 wins on the first two and the percentage wins on the third, and the two offers are exactly equal at a purchase of ₹250. That break-even is the thing to show: below it a fixed amount is the better discount, above it a proportion is.
• Example 12, Cakely and Cakify (Part II, §1.3, p.26). Cakely advertises 30% + 20% off all cakes; Cakify advertises 50% off. Which is cheaper. The page concedes that both seem as if they should give the same benefit, and says plainly that although 30% + 20% is arithmetically 50%, the shop's usage of "30% + 20%" means compounding. Worked on a ₹200 cake: applying 30% takes it to ₹200 − ₹60 = ₹140; applying 20% to that takes it to ₹140 − ₹28 = ₹112. Cakify's flat 50% gives ₹100. Arithmetic added here, for section 6: 0.7 × 0.8 = 0.56, so the pair of discounts is really a 44% discount, whatever the price of the cake. And the order does not matter — 0.8 × 0.7 is the same product — which is a good thing to test, because students expect taking the big one first to matter.
• Example 13, "A Mishap" (Part II, §1.3, p.27, with its own bold subheading). Surbhi buys cookware from a wholesaler and keeps a 50% profit margin on everything. To clear the remaining stock she offers a 50% discount, expecting to come out level. Three questions are printed: (i) had she in fact escaped a loss; (ii) supposing the stock she cleared had originally been priced at ₹12,000 before the discount, what was the loss in rupees and what was it as a percentage; (iii) which discount percentage would have let her sell at the figure she had herself paid, with neither profit nor loss. The model, printed as three stacked bars with a double-headed arrow above each: the top bar is x, labelled as the price bought at; the middle bar is 1.5x, labelled as the selling price including profit; the bottom bar is 0.75x, labelled as after discount. The printed answers:
  • (i) the selling price is 3/4 of what the goods cost her, so a 25% loss;
  • (ii) 0.75x = 12,000 gives x = ₹16,000, and she lost ₹4000;
  • (iii) 1.5x − d × (1.5x) = x gives d = 1/3 = 0.33, so the discount should have been 33.33%. Arithmetic added here: the percentage loss in (ii) is ₹4000 on ₹16,000, which is the same 25% part (i) already established — worth showing, because it demonstrates the percentage does not depend on the rupee figure at all.
• The argument sections 8 and 9 must carry (not in the book, from the chapter's own model). The mark-up adds 50% of x, that is 0.5x. The discount removes 50% of 1.5x, that is 0.75x. The two percentages are equal and the amounts are not, because the second base is one and a half times the first. Generally, a mark-up of m followed by a discount of d leaves a factor of (1 + m)(1 − d), and breaking even needs (1 + m)(1 − d) = 1, so d = m/(1 + m). With m = 0.5 that is 1/3 — the chapter's own 33.33%, arrived at by an argument rather than by an equation. Show both.
• Ariba and Arun (Part II, §1.3, p.27, flagged Try This). Ariba says the number of marbles she has is 120% of the number Arun has; what would be an appropriate statement for Arun to make comparing his marbles with hers. Arithmetic added here: Arun has 100/120 = 5/6 of Ariba's, that is 83.33% of hers — so he has about 16.67% fewer, not 20% fewer. This is the base asymmetry of Percentage increase and decrease, and choosing the right base as a one-line puzzle, and it is the cleanest item in the chapter for the point that reversing a comparison reverses the base too.
• Figure it Out items on this topic.
  • Part II p.28, Q5: a shopkeeper first prices an item at a 35% margin, then, because sales are poor, knocks 30% off that marked figure — profit or loss, with reasons. Arithmetic added here: 1.35 × 0.70 = 0.945, so a 5.5% loss. Surbhi's mishap with less alarming numbers, and the item asks for the reason rather than the number.
  • Part II p.29, Q13: three shops stock identical goods at identical prices and offer Shop A: buy 1 get 1 free; Shop B: buy 2 get 1 free; Shop C: buy 3 get 1 free. (i) with one item at ₹100, the effective price per item at each shop, and the shops ordered cheapest to costliest; (ii) the percentage discount at each, with a printed hint to compare the free items against the total items received; (iii) which shop to use if you need exactly 4 items, and why. Arithmetic added here: A works out at ₹50 an item, B at ₹66.67, C at ₹75; the discounts are 1 free in 2, 1 in 3 and 1 in 4, so 50%, 33.33% and 25%. Part (iii) has a wrinkle worth keeping: four items come to ₹200 at A, ₹300 at C, and — buying one deal plus one loose item — ₹300 at B, so A wins, but B and C tie: the strict ordering from (i) does not survive for exactly four items.
  • Part II p.30, Q14, flagged Try This: 99% of the 100 occupants of a room are left-handed; how many of the left-handers must walk out before that share falls to 98%. Arithmetic added here: the one right-handed person must become 2% of the room, so the room must shrink to 50, so 49 left-handers leave. Half the room walks out to move the figure by one percentage point — and the reason is that the fixed quantity is the other group. This is the best item in the chapter, and section 12 should let the class guess first.
  • Part II p.29, Q7 — the x% of y against y% of x item — is printed in this same set but is handed to Computing a percentage in your head by splitting it, which uses it as a mental computation shortcut.

Figures to have open

• The four-panel comic of Part II p.26, retold rather than reproduced: two clubs, their growth percentages, and the 678-member count. The chapter's own hook, and the data has to come from this brief because it does not extract.
• Surbhi's three-bar model of Part II p.27 — x, 1.5x, 0.75x with the printed labels. The chapter's own figure and the centre of the topic. Redraw as a schematic.
• A crossing-point figure for the two clearance discounts, purchase value along the axis and discount amount up it, the flat ₹50 as a horizontal line and the 20% as a sloping one. Not in the book; it turns three separate answers into one picture.
• A side-by-side of 0.5x added and 0.75x removed on one scale. Not in the book, and it is what makes section 8 land.
• Items-received against items-paid-for for the three shop offers. Standard schematic.
• No photograph is needed. The chapter's Cakely and Cakify shopfront illustrations carry the two offers as signage; the offers themselves are in this brief.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 1, "Fractions in Disguise", §1.3 "Using Percentages", the unnumbered bold subheading "Tricky Percentages", Part II pp.25–27, which itself contains two further bold subheadings: "Would You Rather?" (Part II p.25) and "A Mishap" (Part II p.27). The four-panel comic and the caution box are at the top of Part II p.26; the Try This on the marbles closes Part II p.27.
• Exercise items: Part II p.28 no. 5; Part II p.29 no. 13; Part II p.30 no. 14. Part II p.29 no. 7 is in the same set and belongs to Computing a percentage in your head by splitting it.
• The base asymmetry this topic exploits is established at Part II pp.15–16 (Percentage increase and decrease, and choosing the right base); the multiplying of successive factors at Part II pp.21–23 (Compounding: why repeated growth multiplies instead of adding); the price vocabulary at Part II pp.16–18 (Profit, loss and taxes as percentages of a stated amount).
• The chapter's SUMMARY, Part II p.31, does not carry a separate bullet for this section; read on the printed page.