Advanced Percentage Applications

Grade 10 · mathematics · 100 practice problems · read aloud

🔊 Listen to this explanation

📊 Advanced Percentage Applications

Advanced percentage problems go beyond basic "find X% of Y" calculations. You'll encounter successive percentage changes, reverse percentage calculations, and percentage point differences. These are essential for understanding real-world scenarios like compound interest, inflation, and statistical changes.

🔢 Step-by-Step Guide

  1. Identify the type: Is it successive change, reverse calculation, or percentage points?
  2. For successive changes: Multiply by (1 ± percentage) for each step, DON'T just add percentages
  3. For reverse percentages: Divide by (1 ± percentage) to find the original value
  4. For percentage points: Subtract percentages directly (no multiplication)
  5. Check your answer: Does it make sense in context?

📝 Visual Examples

Example 1: Successive Changes

A $200 phone gets a 20% discount, then 10% tax. Final price?

Step 1: $200 × 0.80 = $160 (after discount)

Step 2: $160 × 1.10 = $176 (after tax)

Note: NOT 200 × (1 - 0.20 + 0.10)!

Example 2: Reverse Percentage

After a 15% discount, a book costs $25.50. Original price?

Solution: $25.50 ÷ 0.85 = $30

Check: $30 × 0.85 = $25.50 ✓

⚠️ Common Mistakes

Adding successive percentages: 20% off then 10% tax ≠ 10% net change

Confusing % points with %: Going from 8% to 10% is a 2 percentage point increase, but a 25% relative increase

Reverse percentage errors: Thinking "15% off = multiply by 0.15" instead of 0.85

💡 Tips & Tricks

Decimal conversion: Always convert percentages to decimals for calculations (25% = 0.25)

Successive change formula: Original × (1 ± p₁) × (1 ± p₂) × ...

Reverse percentage shortcut: "Divide by what's left" - if 20% off, divide by 0.80

Estimate first: Rough calculations help catch calculator errors

🎯 Practice Suggestions

  • Create your own successive percentage scenarios (sales, investments, population changes)
  • Practice identifying when to use reverse percentages
  • Work with real data from news articles about inflation or interest rates
  • Mix different types of percentage problems in practice sessions
  • Time yourself to build fluency with these multi-step calculations

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 A = 5000(1 + 0.06/4)^(4×3) = ?

Hint: This involves compound interest calculation. Remember to divide the annual rate by the compounding frequency and multiply the time by the same frequency.

Show the answer

Answer: 5978.09

  1. Identify the values: P = 5000, r = 0.06, n = 4, t = 3
  2. Calculate r/n = 0.06/4 = 0.015
  3. Calculate n×t = 4×3 = 12
  4. Calculate (1 + r/n) = (1 + 0.015) = 1.015
  5. Calculate (1 + r/n)^(n×t) = 1.015^12 = 1.195618
  6. Calculate A = 5000 × 1.195618 = 5978.09

The answer is 5978.09.

2 Mere bought a dress for $120 with a 30% discount and paid 8% sales tax. What was the final price?

Hint: First calculate the discounted price, then apply the sales tax to find the total amount paid.

Show the answer

Answer: 90.72

  1. Calculate the discount amount: 30% of $120 = 0.30 × 120 = $36
  2. Calculate the discounted price: $120 - $36 = $84
  3. Calculate the sales tax: 8% of $84 = 0.08 × 84 = $6.72
  4. Add tax to discounted price: $84 + $6.72 = $90.72 The final price Mere paid was $90.72.

3 Mere bought a dress for $120 with a 30% discount and then paid 8% sales tax. What was the final price?

Hint: First calculate the discount amount, then the discounted price, then add the sales tax to find the final amount.

Show the answer

Answer: 90.72

  1. Calculate the discount amount Discount = 30% of $120 = 0.30 × 120 = $36
  2. Calculate the discounted price Discounted price = Original price - Discount = $120 - $36 = $84
  3. Calculate the sales tax Sales tax = 8% of $84 = 0.08 × 84 = $6.72
  4. Calculate the final price Final price = Discounted price + Sales tax = $84 + $6.72 = $90.72 The final price Mere paid was $90.72.

4 A laptop originally priced at $1200 is on sale for 25% off. If the sales tax is 8%, what is the final price?

Hint: First calculate the discount amount, then subtract it from the original price to find the sale price. Finally, calculate the tax on the sale price and add it to get the total cost.

Show the answer

Answer: 972

  1. Calculate the discount amount: 25% of $1200 = 0.25 × 1200 = $300
  2. Calculate the sale price: $1200 - $300 = $900
  3. Calculate the sales tax: 8% of $900 = 0.08 × 900 = $72
  4. Calculate the final price: $900 + $72 = $972

The answer is 972.

5 A car's value depreciates by 20% annually. If the original value was $25,000, what is its value after 3 years?

Hint: Consider how repeated percentage decreases work over multiple time periods. Think about multiplying by a factor that represents the remaining value each year.

Show the answer

Answer: 12800

  1. Calculate the depreciation factor: 100% - 20% = 80% = 0.8
  2. Apply this factor for 3 years: Value after 1 year = $25,000 × 0.8 = $20,000
  3. Value after 2 years = $20,000 × 0.8 = $16,000
  4. Value after 3 years = $16,000 × 0.8 = $12,800
  5. Alternatively, use the compound formula: $25,000 × (0.8)^3 = $25,000 × 0.512 = $12,800 The final value is $12,800.

6 A laptop originally priced at $950 is discounted by 20% and then has 7% sales tax added. What is the final price?

Hint: First calculate the discount amount, then the discounted price, then add the sales tax to find the final price.

Show the answer

Answer: 813.20

  1. Calculate the discount amount 20% of $950 = 0.20 × 950 = $190
  2. Calculate the price after discount $950 - $190 = $760
  3. Calculate the sales tax amount 7% of $760 = 0.07 × 760 = $53.20
  4. Calculate the final price $760 + $53.20 = $813.20 The final price is $813.20.
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