Percentages
Scenario
Flipkart shows '20% off, then an extra 10% off'. Your friend says 'nice, 30% off'.
Is it actually 30% off? And does it matter which discount comes first?
Mental model
A percentage is a multiplier, not an amount.
+20% means x1.2, -20% means x0.8. Once every % is a multiplier, a chain of changes (discounts, interest, growth, depreciation) collapses to one multiplication - and you stop adding them.
Formula
Final = Original x product of (1 +/- r_i). Successive a% then b% gives net = a + b + ab/100.
The single idea behind discounts, simple/compound interest, growth and depreciation: a % change is a multiplier; chain them by multiplying, not adding.
Pattern discovery
Apply two discounts back to back and watch the final price.
- Rs100, 20% off then 10% off -> Rs72 (not Rs70)
- Rs100, 10% off then 20% off -> Rs72 (same)
- Rs100, 50% off then 50% off -> Rs25 (not Rs0)
Why is it never the simple sum? What is the second discount taken OF?
Rule: Each discount applies to what's LEFT, not the original - so changes MULTIPLY: 0.8 x 0.9 = 0.72, i.e. 28% off, never 30%. Order doesn't matter because multiplication commutes.
Worked example
Problem: A jacket is listed at Rs2000. The store gives 25% off, then an extra 10% off at checkout. What do you pay?
- Turn each discount into a multiplier: 25% off = x0.75, 10% off = x0.90.
- Discounts apply to what is LEFT, so chain them by multiplying: 0.75 x 0.90 = 0.675.
- Apply to the list price: 2000 x 0.675 = 1350.
- Cross-check with the net-% shortcut: -25 - 10 + (25 x 10 / 100) = -32.5%, and 2000 x 0.675 = 1350. Matches.
Answer: Rs1350 - an effective 32.5% off, not 35%.
Shortcuts
- 10% is your base unit: Find 10% (just move the decimal), then build any percentage from it. (35% of 260 -> 26 + 26 + 26 + 13 = 91.)
- Successive % in one step: a% then b% gives net = a + b + ab/100 (signs included). (-20% then -10% -> -20 -10 + (200/100) = -28% -> x0.72.)
Common mistakes
- Adding successive percentages: 20% then 10% is NOT 30% off - the second % is taken of the already-reduced amount, so it's 28%.
- Percentage of the wrong base: Profit% is on COST price, discount% is on MARKED price. Same number, different whole - mixing the base is the #1 error.
Glossary
- multiplier
- A number you multiply another number by to scale it up or down.
- marked price
- The original price shown on an item before any discounts are applied.
- cost price
- The original price the shopkeeper paid for the item, before marking it up for sale.
Recall questions
- Is a 20% then 10% discount the same as 30% off?
- Does the order of two successive discounts change the final price?
- Fast mental math: 35% of 260 using the 10% base unit.
Questions & answers
A shopkeeper marks up his goods 20% above cost, then offers a 20% discount. Profit or loss %?
4% loss.
Approach: x1.2 x 0.8 = 0.96, i.e. 4% below cost. The markup base (cost) and the discount base (marked price) differ, so the two 20%s never cancel.
A salary rises 10% one year and falls 10% the next. Net change over two years?
1% lower than the start.
Approach: x1.1 x 0.9 = 0.99 -> down 1%. A rise then equal fall always loses a little (the fall is on a bigger base).
Continue learning
Next: Profit, Loss & Discount