Stores use percent change all the time. A discount takes a percent off; sales tax, a tip or a markup adds a percent on. One multiplication can do each job.
Idea clave
With $d\%$ off, you pay $(100 - d)\%$ of the price. With $t\%$ added (tax, tip or markup), you pay $(100 + t)\%$. Write that percent as a decimal and multiply.
Por qué funciona
20% off a price $p$ is $p - 0.20p$. By the distributive property that is $(1 - 0.20)p = 0.80p$: you pay 80%. In the same way, 6% tax gives $p + 0.06p = 1.06p$. When there are two changes, the second percent is taken of the new price, so you multiply twice.
Worked examples
Ejemplo resuelto: A discount
A \$60 jacket is on sale for 20% off. What is the sale price?
- You pay $100\% - 20\% = 80\%$.
- $0.80 \times 60 = 48$.
Respuesta: \$48
Ejemplo resuelto: Sales tax
Shoes cost \$35, and the sales tax is 6%. What is the total?
- You pay $100\% + 6\% = 106\%$.
- $1.06 \times 35 = 37.10$.
- Check: the tax is $0.06 \times 35 = 2.10$, and $35 + 2.10 = 37.10$ ✓.
Respuesta: \$37.10
Ejemplo resuelto: Two discounts in a row
A \$50 backpack is 20% off, and a coupon takes another 10% off the sale price. What is the final price?
- After 20% off: $0.80 \times 50 = 40$.
- The coupon takes 10% off \$40, not off \$50: $0.90 \times 40 = 36$.
- In one step: $50 \times 0.80 \times 0.90 = 50 \times 0.72 = 36$. You pay 72%, which is 28% off, not 30%.
Respuesta: \$36
Error común
Adding the discounts: "20% off and then 10% off is 30% off." That would give \$35, but the second discount comes off a smaller price, so it saves less. Check: 10% of \$40 is \$4, not \$5.
| Situation | Multiply the price by |
|---|---|
| 15% off | $0.85$ |
| 8% sales tax | $1.08$ |
| 20% tip | $1.20$ (for the total) or $0.20$ (for the tip alone) |
| 50% markup | $1.50$ |