SAT-ACT Math: Percentage Word Problems Discounts Markups Change

What This Is and Why It Matters

Percentage word problems involving discounts, markups, and change are crucial for understanding financial transactions and making informed decisions. These concepts are fundamental in business, retail, and everyday shopping. Mastering these problems can help you save money, make profitable decisions, and avoid financial pitfalls. On exams like the SAT-ACT, these problems are common and can significantly impact your score. Getting it wrong can lead to financial losses or incorrect business decisions. For instance, misunderstanding a discount can result in overpaying for goods or services.

Core Knowledge (What You Must Internalize)

• Discount: A reduction in the price of a good or service (Why this matters: It helps you save money).
• Markup: The amount added to the cost price of goods to cover overheads and profit (Why this matters: It helps in pricing products correctly).
• Percentage Change: The difference between the original and new values, expressed as a percentage of the original value (Why this matters: It helps in understanding trends and making comparisons).
• Key Formulas:
• Discount Formula: Discount = (Discount Rate) x (Original Price)
• Markup Formula: Markup = (Markup Rate) x (Cost Price)
• Percentage Change Formula: Percentage Change = [(New Value - Original Value) / Original Value] x 100
• Critical Distinctions:
• Discount vs. Markup: Discount reduces the price, markup increases it.
• Percentage vs. Absolute Change: Percentage change is relative, absolute change is the actual difference.
• Typical Units:
• Discounts and Markups: Usually expressed as percentages.
• Prices: Typically in currency (e.g., dollars, euros).

Step‑by‑Step Deep Dive

1. Identify the Original and New Values:
2. Action: Determine the original price and the new price.
3. Principle: Understand the context to identify these values.
4. Example: Original price of a shirt is $50, new price after discount is $40.

5. ⚠️ Pitfall: Confusing original and new values can lead to incorrect calculations.

6. Calculate the Absolute Change:

7. Action: Subtract the new value from the original value.
8. Principle: This gives the absolute difference.
9. Example: $50 - $40 = $10.

10. ⚠️ Pitfall: Misplacing the values can result in a negative change.

11. Determine the Percentage Change:

12. Action: Use the percentage change formula.
13. Principle: This converts the absolute change into a relative change.
14. Example: Percentage Change = [($10) / ($50)] x 100 = 20%.

15. ⚠️ Pitfall: Forgetting to multiply by 100 can give an incorrect percentage.

16. Apply Discounts and Markups:

17. Action: Use the discount or markup formula.
18. Principle: Adjust the price based on the rate.
19. Example: A 15% discount on a $100 item: Discount = 0.15 x $100 = $15.

20. ⚠️ Pitfall: Using the wrong rate can lead to incorrect adjustments.

21. Verify the Final Price:

22. Action: Subtract the discount or add the markup to the original price.
23. Principle: Confirm the final price after adjustments.
24. Example: Final price after a 15% discount: $100 - $15 = $85.
25. ⚠️ Pitfall: Miscalculating the final price can result in financial errors.

How Experts Think About This Topic

Experts view percentage word problems as a series of logical steps rather than complex calculations. They break down the problem into smaller parts, focusing on the relationships between the original and new values. This approach helps in quickly identifying the necessary adjustments and avoiding common pitfalls.

Common Mistakes (Even Smart People Make)

1. The mistake: Confusing discount and markup rates.
2. Why it's wrong: Leads to incorrect price adjustments.
3. How to avoid: Always verify the rate type before applying.

4. Exam trap: Questions may use similar rates to confuse.

5. The mistake: Forgetting to convert percentages to decimals.

6. Why it's wrong: Results in incorrect calculations.
7. How to avoid: Remember to divide by 100.

8. Exam trap: Questions may provide rates in percentage form.

9. The mistake: Misplacing original and new values.

10. Why it's wrong: Leads to negative or incorrect changes.
11. How to avoid: Always subtract the new value from the original.

12. Exam trap: Questions may reverse the values to trick.

13. The mistake: Not multiplying by 100 in percentage change.

14. Why it's wrong: Gives an incorrect percentage.
15. How to avoid: Always multiply by 100 after dividing.
16. Exam trap: Questions may ask for the percentage directly.

Practice with Real Scenarios

Scenario: A store offers a 20% discount on a $200 jacket. Question: What is the final price after the discount? Solution: 1. Calculate the discount: Discount = 0.20 x $200 = $40. 2. Subtract the discount from the original price: $200 - $40 = $160. Answer: $160. Why it works: The discount reduces the original price by 20%.

Scenario: A retailer marks up a product by 15% on a cost price of $50. Question: What is the selling price? Solution: 1. Calculate the markup: Markup = 0.15 x $50 = $7.50. 2. Add the markup to the cost price: $50 + $7.50 = $57.50. Answer: $57.50. Why it works: The markup increases the cost price by 15%.

Scenario: The price of a stock increases from $100 to $120. Question: What is the percentage increase? Solution: 1. Calculate the absolute change: $120 - $100 = $20. 2. Determine the percentage change: Percentage Change = [($20) / ($100)] x 100 = 20%. Answer: 20%. Why it works: The price increase is 20% of the original value.

Quick Reference Card

• Core Rule: Always verify the rate type and values before calculating.
• Key Formula: Percentage Change = [(New Value - Original Value) / Original Value] x 100.
• Critical Facts:
• Discounts reduce prices.
• Markups increase prices.
• Always convert percentages to decimals.
• Dangerous Pitfall: Confusing original and new values.
• Mnemonic: "DOC" (Discount, Original, Change).

If You're Stuck (Exam or Real Life)

• What to check first: Verify the rate type and values.
• How to reason from first principles: Break down the problem into smaller steps.
• When to use estimation: If exact values are not required, estimate to confirm reasonableness.
• Where to find the answer: Refer to the core knowledge and step-by-step deep dive sections.

Related Topics

• Interest Rates: Understanding how interest affects loans and savings.
• Profit Margins: Calculating and interpreting profit margins in business.
• Inflation: Measuring and understanding the impact of inflation on prices.