I keep messing up multi-step word problems where a bunch of things happen in sequence. My brain wants to smash the discounts together and call it a day, but apparently math is picky about the order.
Here’s the kind of problem I’m talking about, with simple numbers:
– Sticker price: $100
– First discount: 20% off the sticker price
– Then another 10% off the discounted price
– Then a $5 coupon (it says the coupon is after the percentage discounts)
– Then 8% sales tax (it says tax is on the final discounted price after the coupon)
– I pay using a $20 gift card at the very end
My attempt (which I’m not confident about): I combined the two percents as 20% + 10% = 30%, so I took 30% off $100 to get $70. Then I subtracted the $5 coupon to get $65. Then I added 8% tax to get $70.20. Then I took off the $20 gift card and got $50.20 out of pocket. This feels too neat, and I know that 20% + 10% isn’t actually the same as taking 20% off and then 10% off – the second 10% is on a smaller number. Also, I’m never sure where the fixed $5 goes relative to tax, and whether the gift card should change the taxable amount or not in these textbook problems.
Can someone show me a clean, no-nonsense way to set this up so I don’t mix steps? Like: when can I combine percentages into one multiplier, where do fixed-dollar coupons slot in, and exactly what number do I apply the tax to? A short step-by-step checklist or a mental trick would be great. Please use the $100 example above so I can see the pattern.
Why I’m confused: I try to shortcut by adding percentages, I’m fuzzy on whether tax hits before or after the fixed coupon, and I don’t know if the gift card counts as a discount or just payment at the end. I’m fine doing arithmetic – I’m just botching the order.
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3 Responses
Think in this fixed order: percentage changes multiply, fixed-dollar coupons subtract, tax multiplies, and payments (like gift cards) subtract. Successive percent discounts never add; they combine by multiplying their factors. A good mental mantra is “multiply, subtract, multiply, subtract”: multiply by (1 − discount1), then by (1 − discount2), subtract any fixed coupon (if the problem says coupon comes before tax), multiply by (1 + tax), then subtract any gift card. On your $100 example: percentage stage gives 100 × 0.80 × 0.90 = 72.00 (overall factor 0.8 × 0.9 = 0.72, i.e., 28% off, not 30%). Coupon stage: 72.00 − 5 = 67.00. Tax stage: 67.00 × 1.08 = 72.36. Payment stage: 72.36 − 20 = 52.36 out of pocket. In symbols, when the coupon is pre-tax: final cash = ((price × Π(1 − discount rates) − coupon) × (1 + tax)) − gift card. The gift card is just payment at the end; it does not change the taxable amount.
Totally get the urge to mash the percentages together-I used to do that too! The clean way is: turn every percent step into a multiplier, subtract fixed-dollar coupons in between where stated, apply tax with a multiplier after all discounts/coupons, and treat a gift card as just how you pay at the very end (it doesn’t change tax in these textbook problems). Think of it like resizing a photo and then taking a bite: percentage discounts and tax “resize” (multiply), while a $5 coupon is a literal chunk you cut off (subtract), and the gift card is simply the money you use to pay. For your $100 example: first 20% off means multiply by 0.80; then 10% off means multiply by 0.90, so 100 × 0.8 × 0.9 = 72. Then the $5 coupon comes off: 72 − 5 = 67. Then 8% tax on that: 67 × 1.08 = 72.36. Finally, use the $20 gift card: 72.36 − 20 = 52.36 out of pocket. A handy mental checklist is “multiply, multiply, subtract, multiply, subtract” here: 0.8, 0.9, −5, ×1.08, −20. When can you combine percentages? Only when they’re all percentage multipliers in a row with nothing else in between-then you multiply them: 0.8 × 0.9 = 0.72, which is an overall 28% off (not 30%). For more practice with discounts and tax together, this Khan Academy walkthrough is spot on: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-ratios-prop-topic/cc-6th-percent-word-problems/v/sales-tax-and-discount-word-problem
You’re right that order matters. I like to turn each step into a simple operation, then just read left to right. As I go, I sanity‑check by keeping the “type” of each step straight:
– A percent discount is a multiplier (1 − rate).
– A fixed-dollar coupon is a subtraction.
– Sales tax is a multiplier (1 + rate).
– A gift card is just a final subtraction (payment), not a discount.
When you can combine percentages
– If there are several percentage discounts in a row (with no fixed-dollar coupons or tax in between), you can combine them into one multiplier by multiplying their factors. Do not add the percentages.
– Example: 20% off then 10% off → multiply by 0.80 then 0.90 → overall factor 0.8 × 0.9 = 0.72 (which is 28% off, not 30%).
Where fixed-dollar coupons go
– Put them exactly where the problem states. If it says the coupon is after the percent discounts but before tax, subtract it after the multipliers for discounts and before the tax multiplier.
What number to apply tax to
– Tax is computed on the taxable selling price. In textbook problems, that almost always means: after all store discounts and coupons that reduce price, before any payment method like a gift card.
Gift cards
– Treat a gift card as money you use to pay at the register. It does not change the taxable price; subtract it at the very end.
Now, your $100 example, step by step
I’ll write the whole process as a single line first, then compute:
Total you owe = ((100 × 0.80 × 0.90 − 5) × 1.08) − 20
– Start: $100
– 20% off → × 0.80 → $100 × 0.80 = $80
– Then 10% off → × 0.90 → $80 × 0.90 = $72
– Quick check: 0.8 × 0.9 = 0.72, so this is a 28% net discount, not 30%.
– $5 coupon (after percent discounts) → $72 − $5 = $67
– 8% sales tax on that final discounted price → $67 × 1.08 = $72.36
– Pay $20 with gift card at the end → $72.36 − $20 = $52.36 out of pocket
So the correct out‑of‑pocket amount is $52.36.
A tiny worked example to lock in the “don’t add percents” idea
– Price $50, 10% off then 10% off:
– Multiply: $50 × 0.9 × 0.9 = $50 × 0.81 = $40.50
– If you (incorrectly) add them to 20%: $50 × 0.80 = $40. That’s too low by $0.50.
Compact checklist you can reuse
1) Start with price P.
2) For each percent discount d, multiply by (1 − d). You can combine all back‑to‑back percent discounts into one factor by multiplying them.
3) Subtract any fixed-dollar coupons that occur before tax.
4) Multiply by (1 + tax rate).
5) Subtract gift cards or other payments at the end.
General template (if you like a formula)
If you have P, percent discounts d1, d2, …, a total pre‑tax fixed coupon amount C, tax rate t, and a gift card G:
Amount you pay = ((P × (1 − d1) × (1 − d2) × … − C) × (1 + t)) − G
Notes
– The order of the percent multipliers themselves doesn’t matter; they commute. But their position relative to fixed-dollar coupons and tax does matter.
– In real life, some stores round at each step. Textbook problems usually assume you compute exactly then round at the very end.