Profit, loss and taxes as percentages of a stated amount

What to assume they know

• Computing a percentage in your head by splitting it — a percentage of a quantity, and recovering the quantity
• Percentage increase and decrease, and choosing the right base — percentage increase and decrease, and that the base is the value you started from
• The FDP trio: fraction, decimal and percentage as one object — moving between a percentage and its decimal multiplier
• Adding and subtracting money amounts, and multiplying money by a decimal
• Reading a table of items, quantities, rates and amounts

What they should be able to do

• Define cost price, marked price and selling price, and identify each in a stated situation
• Explain why one amount can be a selling price and a cost price at the same time, and locate an example in the chapter's own supply chain
• Compute percentage profit and percentage loss on the cost price
• Work backwards from a stated profit or loss margin to the selling price, or to the cost price
• Compute a selling price after a stated percentage discount on a marked price, and then the profit percentage on the cost
• Distinguish gross profit from net profit, and say which expenses belong to which
• Compute a profit percentage on revenue and on cost for the same transaction, and say which question each answers
• Verify the tax lines on a bill, and account for a single tax rate that is printed as two halves
• Explain why two percentage profits on unequal cost prices cannot be added or averaged

Where it usually goes wrong

• "Marked price and selling price are the same thing." Only when nobody bargains and there is no discount. The sweater figure separates them at every stage.
• "Cost price is what the customer pays." It is what the seller paid. The chapter's chain exists to break this: ₹300 is what the retailer paid and what the wholesaler received.
• "Profit percentage is profit divided by selling price." That is a legitimate figure — Manisha's 33.33% — and it is not what "profit percentage" conventionally means. The chapter prints both and says which question each answers. Teach the convention as a convention.
• "A 20% margin means selling price times 0.8." It means cost price times 1.2. Students routinely apply the discount move to a margin.
• "To find the cost price from a 12% loss, take 12% of the selling price and add it." The base is the cost price, which is unknown, so it is a division by 0.88 — not a multiplication by 1.12. This is the commonest error in the whole section and it produces answers that look almost right.
• "Loss per kilogram and loss on the whole lot are different percentages." They are identical, because both parts of the ratio scale together. Raghu's item asks this directly.
• "Gross profit and net profit are the same when there are no obvious costs." Transport, salaries and electricity are the chapter's own examples of costs that never touched the goods.
• "5% profit and 10% loss on two items at the same selling price cancel to a 5% loss." The direction is right by accident; the size is wrong, and that is the point. The two cost prices differ — ₹80,000/1.05 ≈ ₹76,190 against ₹80,000/0.90 ≈ ₹88,889 — so the outlay is ≈ ₹1,65,079 against ₹1,60,000 received, a loss of ≈ ₹5,079, or about 3.08% of outlay, smaller than the naive 5%. Percentages sitting on different bases cannot be added or averaged in either direction, and a class that "cancels" them lands on the wrong number even when it guesses the right sign.
• "The bill must be wrong because 9% + 9% is not 18% of anything on it." Two percentages of the same sub-total add perfectly well. The bill is arithmetically correct in every line.

Questions to check understanding

• Compute profit or loss percentage on cost price from a stated cost and selling price
• Given cost price and a stated margin, find the selling price; given selling price and a stated margin, find the cost price
• Given MRP and a discount percentage, find the selling price, then the profit percentage against a stated cost — two bases in one item, which is how the board sets it
• Recover the original price from a price paid after a stated discount
• Distinguish gross from net profit given a list of expenses, and compute a net profit percentage on revenue
• Verify or correct the tax lines of a bill, with the rate split into two halves
• Select the correct expression for a price inclusive of a percentage tax, from a list of near-misses
• Two-transaction items where equal selling prices and opposite percentages must not be added — and a written justification of why not
• A ratio-only item with no money named, such as the pencils item, which tests whether the student needs numbers to compute a percentage

Examples worth working on the board

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

• The three names (Part II, §1.3, p.16). The section opens on the ordinary experience of a shopkeeper quoting a price and a customer paying a negotiated one. The price quoted is the marked price, and the page notes it is sometimes the MRP; the amount handed over once a discount is applied is the selling price; what the shopkeeper paid to acquire the item is the cost price. The page's own framing is that these labels are relative to who is speaking, and the figure that follows is there to prove it.
• The sweater's journey (Part II, §1.3, p.16, figure). Four panels left to right, captioned Manufacturing Unit, Wholesale Store, Retail Store, Customer, with arrows between them. Three price boxes sit beneath, each straddling the gap between two panels. The figures are lettering inside the artwork and do not extract at all — read from the printed page:

| Stage | CP | MP | SP | |---|---|---|---| | Manufacturing Unit | ₹230 | ₹255 | ₹253 | | Wholesale Store | ₹253 | ₹310 | ₹300 | | Retail Store | ₹300 | ₹480 | ₹430 |

The Customer panel carries no price box. Note the chain: each stage's cost price is the previous stage's selling price, and the assignment above is confirmed by Example 4 on the next page, which gives the retailer exactly ₹300, ₹480 and ₹430. This figure is the whole argument of section 2 and it is worth a slow minute: ₹253 is a selling price to the manufacturer and a cost price to the wholesaler, and it is the same ₹253.

• Example 4, Kishanlal's sweater (Part II, §1.3, p.17). He buys from the wholesaler at ₹300, marks it at ₹480, and after bargaining sells at ₹430. Selling price above cost price gives a profit of ₹430 − ₹300 = ₹130; the page adds that a selling price below cost price would be a loss. Then the base is chosen explicitly: the cost price is taken as 100%, and the percentage profit is (130/300) × 100 = 43.3%. Two small rough bars accompany it — one scaled 0, 300, 430 with the buying price marked, and one scaled 0 to 100% with the 130 shown running past the 100% mark and a question mark at its end. Set for the student on the same page: the profit percentages of the wholesaler and the manufacturer, using the figure's numbers. Arithmetic added here: the wholesaler makes ₹47 on ₹253, about 18.58%; the manufacturer makes ₹23 on ₹230, exactly 10%.
• Example 5, Raghu's rice (Part II, §1.3, p.17). Stock is getting old. He had bought at ₹35 per kg and clears 10 kg for ₹300. So the cost price of that stock is ₹350, the loss is ₹350 − ₹300 = ₹50, and the percentage loss is (50/350) × 100 = 14.28%. Two rough bars again, cost price above selling price against a 0-to-100% scale. The question printed under it is the good one: could the loss percentage have been computed per kilogram instead, and would it be the same? Arithmetic added here: yes — ₹5 lost on ₹35 is the same fraction, so 14.28% either way. A percentage is a ratio, and scaling both parts of a ratio by 10 changes nothing. Section 5 exists for that argument.
• Example 6, Shyamala's vases (Part II, §1.3, pp.17–18). She bought them at ₹2650 each and sells a slightly damaged one at a loss of 18%. The student is told to estimate first and draw a rough diagram. Two printed methods:
  • with the buying price as 100%, the selling price is 18% less, so 82%, and 82% of 2650 = 0.82 × 2650 = ₹2173;
  • the loss is 18% of 2650, that is (18/100) × 2650 = 477, and 2650 − 477 = 2173. The two methods are the additive and multiplier forms of one change — the same pair that Percentage increase and decrease, and choosing the right base proves equal.
• Three items set for the student in the running text (Part II, §1.3, pp.17–18), each running the calculation in a different direction:
  • Shambhavi, a stationery shop: she buys 200-page notebooks at ₹36 each and sells at a 20% profit margin — find the selling price. Arithmetic added here: ₹43.20.
  • Shambhavi again: she sells crayon boxes at ₹50 with a 25% margin — find what she paid the wholesaler. Arithmetic added here: ₹40. (Note the direction: the base here is the unknown.)
  • Snehal, whose strawberries could not leave Panchgani for Hyderabad because of heavy rain: he sells at ₹80 per kg at a 12% loss — find the cost price. Arithmetic added here: ₹90.91, to the nearest paisa.
• Discount (Part II, §1.3, p.18). The page states plainly what 30% off means — the shop is willing to cut the price by 30% — and then sets the compound item: a utensil store offers a 35% discount on a cooker with MRP ₹1800; find the selling price; and if the cost price was ₹900, find the percentage profit after the sale. Arithmetic added here: selling price ₹1170, profit ₹270 on ₹900, so 30%. Two percentages appear in one item and they have different bases — 35% of 1800 and 30% of 900 — which is exactly the confusion section 7 must clear.
• Gross and net profit (Part II, §1.3, p.18, tinted box). Kishanlal's sales last month were ₹80,000; what he spent buying the goods he sold was ₹48,000; the difference, ₹32,000, is the gross profit. Other expenses — transport, employees' salaries, the electricity bill — came to ₹8,000, leaving ₹24,000 as the net profit. A note in the same box fixes the chapter's convention: from here on, profit means gross profit.
• Manisha's fertiliser — the same profit, two bases (Part II, §1.3, pp.18–19). She buys 50 kg bags at ₹500 and sells at ₹750, a profit of ₹250. Printed both ways: (250/500) × 100 = 50% against what she paid, and (250/750) × 100 = 33.33% against what she took in. Two rough bar diagrams accompany it, both on Part II p.18 — one scaled 0 to 500 with the 250 and the 50% marked, and one scaled 0, 500, 750 with the 250 and the 33.33% marked. Two is the right count because the point is the same ₹250 measured twice. The page then states the rule of use, which is the heart of this topic: the cost-price base answers how much profit was made compared with what was invested in buying the goods, and the revenue base answers how much net profit was made on overall revenue. The box continues onto Part II p.19 with the monthly version: sales ₹1,50,000, cost of goods ₹1,00,000, so gross profit ₹50,000; monthly expenses ₹5,000, so net profit ₹45,000; and the net profit percentage on revenue is (45000/150000) × 100 = (3/10) × 100 = 30%.
• Taxes and the printed bill (Part II, §1.3, p.19). The page notes that tax rates — GST, income tax — are specified as percentages, that GST appears on bills, and that the tax is part of what the buyer pays and goes to the government. A small receipt is printed beside the text. Its contents are artwork and do not extract — read from the printed page:

| | | | | |---|---|---|---| | XY ELECTRICALS — SALES RECEIPT | | | Date: 06/07/2025 | | Item | Qty. | Price | Amount | | CFL Bulb | 3 | ₹150.00 | ₹450.00 | | | | Sub Total | ₹450.00 | | | | CGST 9% | ₹40.50 | | | | SGST 9% | ₹40.50 | | | | TOTAL | ₹531.00 |

It closes with a thank-you line. The printed instruction is to check whether the calculations on the bill are correct. Arithmetic added here: every line holds — 3 × 150 = 450; 9% of 450 = 40.50, twice; 450 + 40.50 + 40.50 = 531. A Try This on the same page asks students to bring bills from home, look at what elements they contain, and compare them.

• Where the 18% went (section 11, an added reading). The two tax lines are 9% each and total 18% of the sub-total — the standard GST rate that Q3 of the exercise set names explicitly (Part II p.28). The bill itself never prints the figure 18% — checked on the printed page. That gap is worth a section: a single rate reaching the customer as two half-rates is exactly the kind of thing that makes a bill look wrong when it is right, and it is also a clean instance of two percentages of the same base being safe to add.
• Figure it Out items on this topic.
  • Part II p.19, Q1: a geometry box bought for ₹75, sold for ₹110 — profit margin with respect to cost. Arithmetic added here: 46.66%.
  • Part II p.19, Q2: a carpenter's chair, materials ₹475, wanted margin 50% — the selling price. Arithmetic added here: ₹712.50.
  • Part II p.19, Q3: a company's revenue last year ₹2.5 crore with a profit margin of 25% — the total expenditure. See Notes: the item does not say whether the margin is on cost or on revenue, and the two readings give ₹2 crore and ₹1.875 crore.
  • Part II p.19, Q4: a shirt originally ₹300 with a 25% discount — what Anwar pays. Arithmetic added here: ₹225.
  • Part II p.20, Q7: a milkman sells two buffaloes at ₹80,000 each, making 5% profit on one and 10% loss on the other; find his overall position. Arithmetic added here: the cost prices were about ₹76,190 and about ₹88,889, so about ₹1,65,079 in, ₹1,60,000 out, a loss of about ₹5,079 — roughly 3.08%. The naive answer, a 5% loss from 5 minus 10, is wrong because the two percentages sit on different cost prices. Section 12 is built on this item.
  • Part II p.28, Q3: a phone priced ₹8,250 with 18% GST added — which of seven printed expressions gives the final price. The options include 8250 + 18, 8250 + 1800, 8250 + 18/100, 8250 × 18, 8250 × 1.18, 8250 + 8250 × 0.18, and 1.8 × 8250. Arithmetic added here: the two that work are 8250 × 1.18 and 8250 + 8250 × 0.18, both giving ₹9,735 — and they are the multiplier and additive forms of one change.
  • Part II p.29, Q9, flagged Try This: a shopkeeper prices pencils so that the selling price of 3 pencils equals the cost of 5 pencils — profit or loss, and what percentage. Arithmetic added here: each pencil sells for five-thirds of its cost, so a profit of two-thirds, that is 66.66%. No rupee figure is given anywhere in the item, which is the point: the percentage does not need one.

Figures to have open

• The sweater's journey of Part II p.16, redrawn as four labelled stages with the three CP/MP/SP boxes and arrows between. The chapter's own figure; the numbers are load-bearing and must be taken from the table in this brief, since they do not extract.
• The receipt of Part II p.19, rebuilt as a clean bill with the same seven lines. Rebuild it; do not reproduce the printed artwork.
• Kishanlal's and Raghu's rough bars (Part II p.17), redrawn: an amount scale above a percentage scale, with the profit or loss segment marked past or short of the 100% point.
• Manisha's two bars (both on Part II p.18), redrawn — the same ₹250 measured twice. This is the single most important image in the topic.
• Two unequal cost-price bars ending at one selling price, for section 12. An added figure; the chapter sets the item and draws nothing.
• No photograph is needed. The chapter's sweater, cooker, strawberry and bill illustrations are decorative apart from the price boxes and the bill lines.

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 subheadings "Profit and Loss" (Part II p.16, running to Part II p.19) and "Taxes" (Part II p.19), with the gross/net tinted box spanning Part II pp.18–19 and the Try This on Part II p.19.
• Exercise items: Part II p.19 nos. 1–4; Part II p.20 no. 7; Part II p.28 no. 3; Part II p.29 no. 9.
• The chapter's SUMMARY, Part II p.31, records that tax rates, and the profit or the loss made on a deal, are all reportable as percentages.
• The additive-and-multiplier equivalence used in Example 6 and in the GST item is proved at Part II p.16, Example 3 (Percentage increase and decrease, and choosing the right base).