Lesson 5: Percent Applications — Tax, Tip, and Discount

Here is a satisfying fact that this lesson will let you check for yourself: an $84 restaurant bill, with 7% tax and an 18% tip added, comes to exactly $105.00. Shops, restaurants and phone companies all communicate in percents: discounts slash prices, tax adds a slice, and tips reward good service. If you can compute all three quickly and accurately, no bill will ever surprise you. This lesson builds on the 10% building block from the previous lesson and puts it to work in real shops and restaurants.

Discounts: paying less than the tag

A discount is a percent taken off the original price. There are two ways to compute the sale price, and you should know both.

Worked example 1. A jacket costs $80 with 25% off. Method 1: the discount is 25% of 80 = $20, so the sale price is 80 − 20 = $60. Method 2 (the multiplier): if 25% comes off, you pay 75%, and 0.75 × 80 = $60 directly. Method 2 is faster and handles stacked steps neatly.

Worked example 2. Trainers cost $120, reduced by 30%, and then 8% sales tax is added to the sale price. Sale price: 0.70 × 120 = $84. Tax: 8% of 84 = $6.72. Total paid: 84 + 6.72 = $90.72. Notice the tax is charged on the reduced price, not the original — a detail shops rely on you missing.

Tax and tips: adding a percent on

Sales tax is a percent added by the government: 8% tax on a $46 item is 0.08 × 46 = $3.68. A tip is a percent you choose to add for service: a 20% tip on a $46 meal is 0.20 × 46 = $9.20, making the diner's total 46 + 9.20 = $55.20 before tax. Quick mental tips: 10% is found by moving the decimal point one place left ($46 → $4.60); double it for 20%; halve it for 5%; add 10% and 5% together for 15%.

Worked example 3. A video game costs $60. It is on sale at 10% off, and your state charges 6% tax on the sale price. Sale price: 0.90 × 60 = $54. Tax: 0.06 × 54 = $3.24. Total: 54 + 3.24 = $57.24. (Equivalently, 54 × 1.06 = 57.24 — multiplying by 1.06 adds 6% in one step.)

Percent increase and decrease

Percents also measure change. When a $40 game rises to $50, the increase is $10, and the percent increase is 10 ÷ 40 × 100 = 25% — always divide by the original amount. A town whose population falls from 2,000 to 1,800 has shrunk by 200 ÷ 2,000 = 10%. Dividing by the new amount instead is one of the most common errors in all of percent work.

In the real world: reading a receipt

A receipt bundles everything in this lesson: subtotal, discount lines, tax, and a suggested tip printed as a helpful percent. Restaurants often print tip suggestions calculated on the after-tax total, which quietly inflates the tip. Knowing that a 20% tip on an $84 pre-tax bill should be 0.20 × 84 = $16.80 lets you spot when the "helpful" suggestion of $18 is quietly charging you 21.4%. Percent fluency is consumer armour.

Common mistakes

• Subtracting the percent number from the price. 25% off $80 does not mean $80 − 25 = $55. The discount is 25% of 80 = $20, so the price is $60.
• Charging tax on the original price after a discount. Tax applies to what you actually pay: $6.72 on the $84 sale price in Example 2, not $9.60 on $120.
• Adding percents that belong to different bases. A 10% discount followed by a 10% tax does not cancel out: $60 → $54 → $57.24. Each percent acts on a new amount.
• Dividing by the new amount for percent change. A rise from $40 to $50 is 10 ÷ 40 = 25%, not 10 ÷ 50 = 20%.

Exam Experience

1. A $40 meal needs a 15% tip. How much is the tip? A) $4.00 B) $6.00 C) $9.00 D) $15.00

2. An item priced at $50 is discounted by 20%. What is the sale price? A) $30 B) $40 C) $45 D) $60

3. A state charges 5% sales tax. What is the tax on a $200 purchase? A) $5 B) $10 C) $20 D) $25

4. (Constructed response) A family's restaurant bill comes to $84 before tax and tip. The tax rate is 7%. They decide to leave an 18% tip, calculated on the $84 before tax. (a) Calculate the tax. (b) Calculate the tip. (c) Calculate the total amount the family pays. (d) The restaurant prints a "suggested tip" of $17.50. Is that more or less than 18% of the bill before tax? Show how you know.

Answer key and explanations

• 1. B — 10% of 40 is 4 and 5% is 2, so 15% is $6.00.
• 2. B — The discount is 20% of 50 = $10, so the sale price is 50 − 10 = $40.
• 3. B — 5% of 200 = 0.05 × 200 = $10.
• 4. (a) 0.07 × 84 = $5.88. (b) 0.18 × 84 = $15.12. (c) 84 + 5.88 + 15.12 = $105.00. (d) More: 18% of 84 is $15.12, and $17.50 > $15.12 (it is about 20.8% — the restaurant based it on the after-tax total).

Scoring guide for question 4 (4 points)

• 1 point for the tax, $5.88.
• 1 point for the tip, $15.12, calculated on the pre-tax amount.
• 1 point for the total, $105.00.
• 1 point for a correct verdict (more) supported by comparing $17.50 with the correctly computed $15.12.

Going further: stacked percents and percents in reverse

Real pricing stacks effects, and the results defy first instincts — which is exactly why shops love stacking.

Worked example 4. A $200 coat is reduced by 40%: you pay 0.60 × 200 = $120. Staff get a further 10% off the sale price: 0.90 × 120 = $108. Total saving: 200 − 108 = $92, which is 92 ÷ 200 = 46% — not the 50% that "40% + 10%" suggests. Percents multiply; they do not add, because each acts on a new, smaller base.

Worked example 5. Savings earn interest: $500 in an account paying 4% simple interest for one year earns 0.04 × 500 = $20, growing the balance to $520. Interest is a percent increase the bank pays you — the same machinery as tax, running in your favour.

Worked example 6. A salesperson earns 5% commission on everything she sells. In a month when she sells $2,400 of goods, she earns 0.05 × 2,400 = $120 on top of her salary.

Worked example 7 (working backwards). After a 20% price increase, a bike costs $96. What did it cost before? The $96 is 120% of the original, so the original was 96 ÷ 1.2 = $80. Reverse problems divide by the multiplier — never subtract 20% of the new price (that gives 96 − 19.20 = $76.80, a classic wrong answer).

Quick practice

• 1. 30% off a $90 game: what do you pay?
• 2. Tax at 9% on a $300 purchase: how much tax?
• 3. A 15% tip on a $24 bill.
• 4. A town of 800 people grows by 10%. New population?

Answers. 1. You pay 70%: 0.70 × 90 = $63. 2. 0.09 × 300 = $27. 3. 10% = $2.40, 5% = $1.20, total $3.60. 4. 800 + 80 = 880 people.

Investigation: the member price

Trainers listed at $85. Today only: 30% off. Members: a further 15% off. Then 8% tax at the till. What does a member actually pay? Step 1: 30% off gives 0.70 × 85 = $59.50. Step 2: the member discount acts on $59.50, not on $85: 0.85 × 59.50 = $50.575, about $50.58. Step 3: tax multiplies by 1.08: 50.575 × 1.08 = $54.62 (to the nearest cent). Now the two questions shops hope you never ask. What percent of the list price did the member pay before tax? The combined multiplier is 0.70 × 0.85 = 0.595, so 59.5% paid — a 40.5% saving, nowhere near the "30% + 15% = 45% off" the banner implies. And what did the tax really cost? $54.62 − $50.58 = $4.04 on the discounted price. A non-member today pays 59.50 × 1.08 = $64.26 at the till. Same shoes, three prices ($91.80 at full price with tax, $64.26 today, $54.62 for members) — and every gap between them is explained by one idea: percents act on the current amount, in sequence.

Challenge corner

A jacket falls in price from $64 to $48. The decrease is $16, and the percent decrease is 16 ÷ 64 × 100 = 25% — computed against the original $64, always. Now the trap that catches adults daily: a TV drops 15% from $400 to $340 in a sale, then rises 15% when the sale ends: 340 × 1.15 = $391 — not $400! The rise acted on the smaller amount, so equal percents down and up do not cancel. Retailers know this asymmetry; now you do too. The general principle deserves emphasis: a percent is always of something, and when the something changes mid-problem, the same percent buys a different amount of change. Naming the base of every percent — "15% of 340" — is the habit that makes these problems safe.