Profit, Loss & Discount • Topic 4 of 6

Equivalent Discount

The "equivalent" or "single" discount is the one discount that produces the same selling price as a chain of successive discounts. For two discounts d1 and d2, the equivalent single discount is d1 + d2 − d1·d2/100 — always LESS than the simple sum, because the second cut works on a smaller amount. This is the discount mirror of the successive-percentage formula, only the cross term is subtracted instead of added. CAT uses it two ways: (1) "which offer is better — flat 40%, or 30% then 15%?" and (2) "two successive discounts of x% equal a single discount of what?". For the equal-rate case, the single equivalent is 2x − x²/100. For three discounts, collapse two with the formula, then combine the result with the third — or just multiply the three (1 − d/100) factors and convert the product back to a percentage off.

✅ Solved examples

1. Successive discounts of 20% and 30% equal what single discount?
Net = 20 + 30 − (20 × 30)/100 = 50 − 6 = 44%.
2. Which is the better offer: a flat 40% discount, or successive 25% and 20%?
Successive = 25 + 20 − 5 = 40%. Both equal 40%, so they are identical.
3. Two successive discounts of 10% each equal what single discount?
2(10) − 10²/100 = 20 − 1 = 19%.
4. Find the single discount equivalent to 10%, 20% and 25% in succession.
Multiplier = 0.90 × 0.80 × 0.75 = 0.54 ⇒ 1 − 0.54 = 0.46 ⇒ 46%.

✏️ Practice — try these, take hints as needed

1. Single discount equal to 25% and 20% successive?
d1 + d2 − d1·d2/100.
25 + 20 − 500/100.
45 − 5.
40%
2. Two equal successive discounts give a single 19%. Find each.
2x − x²/100 = 19.
Try x = 10.
20 − 1.
10% each
3. Better deal: flat 30% or successive 20% then 12%?
Successive = 20 + 12 − 2.4.
= 29.6%.
Compare with 30%.
Flat 30% (gives more off)
4. Single discount equivalent to 50% and 40% successive?
50 + 40 − 2000/100.
90 − 20.
Subtract the cross term.
70%
5. Single discount equivalent to 10%, 10%, 10% in succession?
0.9 × 0.9 × 0.9.
= 0.729.
1 − 0.729.
27.1%

📝 Topic test — 8 questions

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