Financial Percentages: Sales Tax, Tips, and Discounts
Warm-up
Show a real receipt with a discount line and a tax line and ask: "The sticker said \$49.99 — why did I pay \$42.36?" Reconstructing the receipt (25% off → \$37.49; 13% tax → \$42.36) IS the warm-up.
The unit turns percent fluency into consumer power: tax, tips, discounts, and their compositions, computed mentally where possible and exactly where it counts.
Explore
Mental-percent bootcamp first: 10% by point-shift; 5% as half of that; 1% by two shifts; 15% as 10+5; 20% as double-10. Each student builds a tip table for a \$36 restaurant bill (10%: 3.60; 15%: 5.40; 20%: 7.20) — no calculator, under a minute.
Then composition stations: (1) discount-then-tax on the same item, computed as chained multipliers ( then ); (2) the multiplier merge: — one number that does both jobs; (3) the percent-of-a-percent trap: "30% off, then EXTRA 20% off" — is that 50% off? ( — it's 44% off. The stacked-discount myth, busted by multiplication.)
Formalize
Formalize the multiplier view of every percent move:
The multiplier framing solves three problems at once: composition (multiply the multipliers), the stacked-discount myth (multipliers multiply, percents don't add), and reversibility questions ("what price BEFORE the 13% tax?" — divide by 1.13, never subtract 13%).
Practice
Practice: a tip table for a new bill; two discount-tax chains; one "pre-tax price" reversal; one stacked-discount audit (store's claim vs multiplier truth); one tip-splitting problem with rounding to quarters.
Exit ticket: "$80 item, 40% off, 10% tax: final price via ONE multiplier." (.)
Exit ticket
Practice: a tip table for a new bill; two discount-tax chains; one "pre-tax price" reversal; one stacked-discount audit (store's claim vs multiplier truth); one tip-splitting problem with rounding to quarters.
Exit ticket: "$80 item, 40% off, 10% tax: final price via ONE multiplier." (.)
The purchase: jeans listed \$65, on sale 30% off, tax 12%.
Step 1: Discount multiplier: pay 70% → .
Step 2: Tax multiplier: .
Step 3: One-multiplier version: → ✓ — the whole receipt is one number.
Step 4: Mental-check layer: 30% off \$65 is roughly \$20 off → \$45ish; tax adds a bit over a tenth → low fifties ✓.
Step 5: Read the multiplier as a sentence: "after everything, you pay 78.4% of sticker." That single-number summary is what comparison shopping actually needs.
The receipt shows a FINAL total of $28.25 including 13% tax. A friend claims the pre-tax price was . Audit this.
Step 1: Test the friend's claim forward: — NOT $28.25. The claim fails its own check.
Step 2: The correct reversal: the total IS pre-tax × 1.13, so pre-tax .
Step 3: Verify: ✓.
Step 4: Diagnose the friend's error: they took 13% OF THE TOTAL, but the tax was 13% of the (smaller) pre-tax price — wrong base. Percent moves are multipliers; undoing a multiplier is DIVISION. "Subtract the same percent" undershoots every time, and now the class knows by exactly how much and why.
Two stores, same \$120 jacket. Store A: "50% off!" Store B: "30% off, plus an EXTRA 30% off at the register!"
Step 1: The bait: 30 + 30 = 60 > 50, so B looks better. Half the class bites; good.
Step 2: Multiplier truth: A → → $60. B → → $58.80.
Step 3: Verdict: B wins — but barely (\$1.20), nothing like the "60% vs 50%" the signs implied. B's true discount is 51%.
Step 4: Why stacking underperforms: the second 30% applies to the ALREADY REDUCED price — a smaller base. Sequential percents always compound on shrinking bases; only the multipliers tell the truth.
Step 5: The consumer reflex, installed: any "extra % off" sign triggers the multiplier product before the wallet opens. Marketing arithmetic and actual arithmetic are different subjects; this unit teaches the second.