Mathematics Grade 8: Percentage Profit Loss Discount Tax

Grade 8 Mathematics Study Guide: Percentage – Profit, Loss, Discount, Tax

1. The Driving Question

You see a $50 hoodie on sale for "30% off," but when you check out, the price is higher than you expected. Then your friend says, "Wait, did they add tax after the discount?" How do stores actually calculate what you pay—and how can you tell if a "deal" is really saving you money, or just tricking you into spending more?

2. The Core Idea – Built, Not Listed

Imagine you’re running a lemonade stand at the park. You buy lemons for $10, sugar for $5, and cups for $3—your cost price is $18. You sell each cup for $1.50. If you sell 20 cups, you make $30. That extra $12 ($30 - $18) is your profit. But what if it rains, and you only sell 10 cups? You make $15, which is $3 less than you spent—now you’re at a loss.

Stores do the same math, but they use percentages to compare profit or loss to the original cost. A 20% profit on your $18 lemonade means you made $3.60 extra ($18 × 0.20). Discounts work backward: a 30% discount on a $50 hoodie means you pay $15 less ($50 × 0.30 = $15), so the selling price is $35. But then the store adds 8% sales tax—$35 × 0.08 = $2.80—so you actually pay $37.80. The order matters: discount first, then tax.

Key Vocabulary: - Cost Price (CP): The amount you pay to buy or make something. Example: A baker spends $2 on flour, $1 on eggs, and $3 on chocolate to make a cake. CP = $6. - Selling Price (SP): The amount you charge customers. Example: The baker sells the cake for $12. SP = $12. - Profit: SP > CP. Example: $12 - $6 = $6 profit. College note: In business, profit can be split into gross (before expenses) and net (after all costs). - Loss: SP < CP. Example: If the baker sells the cake for $5, loss = $1. - Discount: A reduction from the original price. Example: A $40 video game is marked "25% off," so the discount is $10. - Sales Tax: A percentage added to the selling price by the government. Example: A $20 book with 7% tax costs $21.40 total.

3. Assessment Translation

How this appears on state tests (Grade 8): - Multiple Choice: Questions like: "A store buys a jacket for $40 and sells it at a 25% profit. What is the selling price?" Distractors: $30 (subtracts 25%), $50 (adds $10 instead of 25%), $45 (adds 25% of $50). - Short Answer: "A bike is on sale for 20% off its original price of $250. If sales tax is 6%, what is the total cost? Show your work." Proficient response: Calculates discount ($250 × 0.20 = $50), sale price ($250 - $50 = $200), then tax ($200 × 0.06 = $12), total = $212. Developing response: Forgets to add tax or calculates tax on the original price. - Evidence-Based Writing (rare): "Explain why a 50% discount followed by a 50% increase does not return to the original price. Use an example." Proficient response: Uses numbers (e.g., $100-$50-$75) and explains that percentages are applied to different bases.

Model Proficient Response (Short Answer): Prompt: A $120 pair of shoes is on sale for 15% off. If the sales tax is 5%, what is the total cost? Response:
1. Discount: $120 × 0.15 = $18
2. Sale price: $120 - $18 = $102
3. Tax: $102 × 0.05 = $5.10
4. Total: $102 + $5.10 = $107.10

4. Mistake Taxonomy

Mistake 1: Applying tax to the original price Question: A $80 jacket has a 10% discount and 7% sales tax. What is the total cost? Wrong Response: $80 × 0.07 = $5.60 tax-$80 - $8 + $5.60 = $77.60 Why it loses credit: Tax is applied to the discounted price, not the original. Correct Approach:
1. Discount: $80 × 0.10 = $8
2. Sale price: $80 - $8 = $72
3. Tax: $72 × 0.07 = $5.04
4. Total: $72 + $5.04 = $77.04

Mistake 2: Confusing profit percentage with markup Question: A store buys a toy for $25 and wants a 40% profit. What should the selling price be? Wrong Response: $25 + 40% = $25 + $10 = $35 Why it loses credit: 40% profit is 40% of the cost price ($25), not $10. Correct Approach:
1. Profit: $25 × 0.40 = $10
2. Selling price: $25 + $10 = $35 (Wait—this looks right, but the error is in the reasoning! The correct calculation is indeed $35, but the mistake was thinking 40% profit = $10. The real error is not recognizing that profit percentage is always relative to cost price.)

Mistake 3: Misordering discount and tax Question: A $60 shirt has a 20% discount and 8% tax. What is the total cost? Wrong Response: $60 × 0.20 = $12 discount-$60 - $12 = $48-$48 × 0.08 = $3.84 tax-$48 + $3.84 = $51.84 (then adds discount again: $51.84 - $12 = $39.84) Why it loses credit: The discount is subtracted once from the original price; the student double-subtracts it. Correct Approach:
1. Discount: $60 × 0.20 = $12
2. Sale price: $60 - $12 = $48
3. Tax: $48 × 0.08 = $3.84
4. Total: $48 + $3.84 = $51.84

5. Connection Layer

• Within Math: Percentage profit/loss-Compound interest — Both use percentages to grow or shrink a starting amount over time. A 10% profit this year and 10% next year isn’t 20% total (it’s 21% because of compounding).
• Across Subjects: Percentage discounts-Chemistry concentration — A 30% salt solution means 30g salt per 100g water, just like a 30% discount means $30 off $100. Both are ratios scaled to 100.
• Outside School: "Buy one, get one 50% off" deals-Restaurant tipping — A "50% off" deal is like a 25% discount on the total (since you’re paying full price for one item and half for the other). Similarly, a 20% tip on a $50 bill is $10, but if you split the bill with a friend, you each pay $25 + $5 tip—same math, different context.

6. The Stretch Question

"A store offers a 'buy 2, get 1 free' deal on $20 shirts. Is this a better deal than a 30% discount on all three shirts? Prove it with numbers—and explain why stores might prefer one deal over the other."

Pointer Toward the Answer: - "Buy 2, get 1 free" means you pay $40 for 3 shirts (?$13.33 per shirt). - 30% off all three: $20 × 3 = $60-$60 × 0.30 = $18 discount-$42 total (?$14 per shirt). - The "buy 2" deal is slightly better for the customer, but stores might prefer it because it encourages buying more items (what if you only wanted one shirt?). Also, "free" feels like a bigger win, even if the math is close.