Chapter 5.3 — Exercise 5.2 — Discounts and GST
Discounts, profit and loss, Goods and Service Tax. This is Lesson 3 of 4 in Chapter 5: Comparing Quantities Using Proportion.
One Formula, Three Real-World Disguises
Exercise 5.2 applies percentage increase and decrease to three connected real-world contexts: discounts on a marked price, profit and loss on a sale, and GST added to a price. Every formula here is really just the same increase/decrease idea from the introduction, renamed to match whichever situation it's describing.
Discount
Discount = discount% × Marked price / 100 | Selling price = Marked price × (100 − discount%)/100A book marked at ₹150 with a 15% discount sells for 150 × 85/100 = ₹127.50. Going the other direction — a gift marked at ₹176 sold for ₹165 — the discount is ₹11, so the discount percentage is (11/176) × 100 = 6.25%.
Profit and Loss
S.P. > C.P. → Profit = S.P. − C.P., Profit% = (Profit/C.P.)×100 | C.P. > S.P. → Loss = C.P. − S.P., Loss% = (Loss/C.P.)×100A shopkeeper buys 200 bulbs at ₹10 each (C.P. = ₹2000), loses 5 to breakage, and sells the remaining 195 at ₹12 each (S.P. = ₹2340). Since S.P. exceeds C.P., this is a gain of ₹340, or (340/2000) × 100 = 17%.
Working Backward from Selling Price to Cost Price
Sometimes the selling price and the profit or loss percentage are given, and the cost price is the unknown. A table sold for ₹2,142 at a 5% gain has cost price = 2142 × 100/105 = ₹2,040. Once the cost price is known, a second selling price for a different target profit follows directly: to gain 10% instead, the new selling price is 2040 × 110/100 = ₹2,244.
A Chain of Transactions
Gopi sold a watch to Ibrahim at a 12% gain, and Ibrahim sold it to John at a 5% loss, with John paying ₹1,330. Working backward one sale at a time: Ibrahim's cost price is 1330 × 100/95 = ₹1,400 (since John's payment was Ibrahim's selling price at a 5% loss from Ibrahim's cost) — and since Ibrahim's cost price was exactly what Gopi sold it to him for, Gopi sold the watch for ₹1,400. Each link in a chain like this is solved the same way, one transaction at a time, from the known end back toward the unknown one.
Combining a Discount with VAT or Tax Already Included
A hotel bill of ₹1,450 already includes 5% VAT, and an 8% discount is then applied to that bill amount. The amount actually payable is 1450 × 92/100 = ₹1,334 — the discount is applied to the bill total as it stands, VAT and all, not recalculated from some pre-VAT amount.
GST — Extracting the Original Price
When a bill amount already includes GST, the original (pre-GST) price is found by reversing the increase formula:
Original price = Bill amount × 100/(100 + GST%)- A bill of ₹10,300 including 3% GST: original price = 10300 × 100/103 = ₹10,000.
- A bill of ₹3,360 including 12% GST: original price = 3360 × 100/112 = ₹3,000.
- A bill of ₹256 including 28% GST: original price = 256 × 100/128 = ₹200.
The reverse situation — GST added on top of a fixed price, rather than already included in a bill — is simpler: a ₹4,500 cellphone with 12% GST added means the dealer pays 12% of 4500 = ₹540 in GST, for a total purchase price of ₹5,040.
A Puzzle: The Smallest Whole-Rupee Result
One question asks something more open-ended: a shop prices an item so that after 4% sales tax is added, the final amount comes out to exactly n whole rupees, with no rounding needed. If the original price is x, then n = x × 104/100 = 26x/25, so x = 25n/26. Since a price in rupees and paise can have at most two decimal digits, 100x must be a whole number of paise: 100x = 2500n/26 = 1250n/13. Because 1250 and 13 share no common factors, 13 must divide n exactly for this to be a whole number — and the smallest positive multiple of 13 is 13 itself. Checking: n = 13 gives x = 25×13/26 = ₹12.50, and 12.50 × 1.04 = ₹13.00 exactly, confirming n = 13 works with no rounding needed. This kind of question rewards rearranging the percentage formula algebraically and asking what constraint makes the result a valid amount of money, rather than guessing numbers to test.
Discount and Profit Are the Same Formula Wearing Different Names
It's worth noticing that "selling price after a discount" and "selling price at a loss" use the exact same structure — multiply the marked or cost price by (100 − percentage)/100 — even though one is about a shop's advertised price and the other is about buying and reselling. Likewise, "profit" and "GST added on top" both use (100 + percentage)/100. There are really only two formulas in this entire exercise, one for percentage increase and one for percentage decrease, just applied to marked prices, cost prices, and bill amounts under different names depending on the context.
Why a Chain of Percentage Changes Doesn't Simply Add Up
When several percentage changes apply one after another — a price rising by different amounts across several days, say — it's tempting to just add the percentages together. That shortcut is wrong, because each new percentage applies to the already-changed amount from the previous step, not to the original value. A share priced at ₹7.50 that rises 6%, then falls 1.5%, then falls another 2% does not end up at 7.50 × (1 + 0.06 − 0.015 − 0.02); each change has to be applied in sequence to the running value, giving a noticeably different final answer than simply summing the percentages would.
Multiplying When You Should Be Dividing, and Vice Versa
- Applying a discount to the wrong base amount. If a bill already includes tax, a discount afterward applies to that full billed amount, not to some earlier pre-tax figure.
- Mixing up which direction a chain of sales runs. In a resale chain, each person's selling price becomes the next person's cost price — track carefully which amount belongs to which stage.
- Using the increase formula when reversing GST. To pull the original price out of a GST-inclusive bill, divide by (100 + GST%)/100 — multiplying by it instead gives a larger, wrong number.
Deciding Whether to Multiply or Divide
A recurring source of confusion in this exercise is knowing whether a given percentage means multiplying by (100±rate)/100, or dividing by it. The rule is really about what's already known versus what's missing: if the original amount is known and you need the changed amount, multiply by (100±rate)/100. If the changed amount is known (a bill, a selling price) and the original is missing, divide by that same factor instead. Every "work backward" question in this exercise — finding a cost price from a selling price, or an original price from a GST-inclusive bill — is this division case; every straightforward "apply the change" question is the multiplication case — deciding which situation you're in, before picking up a calculator, avoids the single most common error in this entire exercise.
From a Single Transaction to Many Time Periods
The same increase-and-decrease percentage reasoning, applied repeatedly over multiple time periods instead of a single transaction, is exactly what Exercise 5.3 covers next with simple and compound interest. For the underlying percentage formulas this exercise applies, revisit the Introduction to Ratios and Proportion.