Multi-step Percent Problems: Comprehensive Exercises & Solutions

Master multi-step percent problems: compound percentages, percentage changes, profit/loss, tax and discount calculations, and complex financial scenarios through these 5 detailed exercises.

Core Concepts & Formulas
\(\text{Compound Percentage} = P \times (1 + r_1) \times (1 + r_2) \times \ldots\)
Compound Percentage Formula
\(\text{Tax Total} = \text{Base Price} \times (1 + \text{Tax Rate})\)
Tax Calculation
\(\text{Discounted Price} = \text{Original Price} \times (1 - \text{Discount Rate})\)
Discount Calculation
Profit
\(\text{Profit \%} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100\%\)
Profit percentage calculation
Loss
\(\text{Loss \%} = \frac{\text{Cost Price} - \text{Selling Price}}{\text{Cost Price}} \times 100\%\)
Loss percentage calculation
Sequential
\(\text{Final} = \text{Initial} \times \text{Factor}_1 \times \text{Factor}_2\)
Multiplying change factors
Key Definitions:

Multi-step Percent Problem: A problem requiring multiple percentage calculations in sequence

Compound Percentage: Multiple percentage changes applied sequentially to the same base

Sequential Operations: Performing percentage operations in a specific order

Percentage Chain: Linking multiple percentage calculations together

Tip 1: Always identify what the percentage is being applied to (the base value).
Tip 2: Convert percentages to decimals by dividing by 100 (e.g., 15% = 0.15).
Tip 3: For sequential changes, multiply the factors, not add the percentages.
Tip 4: Keep track of each step to avoid losing track in multi-step problems.
Multi-step Exercises 1-3
1 Sequential Discounts
Exercise 1
A store offers a 20% discount followed by an additional 15% discount on an item originally priced at $150. What is the final sale price? What is the total percentage discount?
Definition:

Sequential Discounts: Multiple discounts applied one after another to the same item

Method:
  1. Apply the first discount to the original price
  2. Apply the second discount to the new price
  3. Calculate the total percentage discount
Original Price
$150
After 20% Discount
$120
Final Price
$102
Step 1: Apply first discount (20%)

Price after first discount = Original × (1 - 0.20) = $150 × 0.80 = $120

Step 2: Apply second discount (15%)

Final price = Previous price × (1 - 0.15) = $120 × 0.85 = $102

Step 3: Calculate total percentage discount

Total discount amount = $150 - $102 = $48

Total discount % = ($48/$150) × 100% = 32%

Step 4: Alternative calculation using factors

Combined factor = 0.80 × 0.85 = 0.68

Final price = $150 × 0.68 = $102

Total discount = 1 - 0.68 = 0.32 = 32%

Final price = $102, Total discount = 32%
Final answer:

The final sale price is $102, representing a total discount of 32% from the original price.

Applied rules:

• Sequential Application: Apply each percentage to the previous result

• Multiplication of Factors: Combined effect = (1 - r₁) × (1 - r₂)

• Total Discount: Cannot simply add percentages (20% + 15% ≠ 32%)

2 Tax and Tip Calculation
Exercise 2
Sarah's restaurant bill is $85 before tax. If the sales tax is 8% and she wants to leave a 15% tip on the pre-tax amount, what is her total payment?
Definition:

Tax and Tip Calculation: Applying different percentages to different base amounts in sequence

Pre-tax Amount
$85
Tax Amount
$6.80
Tip Amount
$12.75
Step 1: Calculate sales tax (on pre-tax amount)

Tax = $85 × 0.08 = $6.80

Step 2: Calculate tip (on pre-tax amount)

Tip = $85 × 0.15 = $12.75

Step 3: Calculate total payment

Total = Pre-tax amount + Tax + Tip

Total = $85 + $6.80 + $12.75 = $104.55

Step 4: Verify calculation

Check: $85 × (1 + 0.08 + 0.15) = $85 × 1.23 = $104.55 ✓

Total payment = $104.55
Final answer:

Sarah's total payment is $104.55.

Applied rules:

• Correct Base Amount: Tax and tip may apply to different bases

• Additive Calculation: Sum all individual amounts

• Verification: Always check your calculation

3 Compound Interest Simulation
Exercise 3
A savings account starts with $500. It grows by 4% in the first year, then by 3% in the second year, and finally decreases by 2% in the third year. What is the final balance after 3 years?
Definition:

Compound Growth/Decay: Multiple percentage changes applied sequentially over time periods

Year 0
$500
Year 1
$520
Year 2
$535.60
Year 3
$524.89
Step 1: Calculate Year 1 balance (4% growth)

Year 1 = $500 × (1 + 0.04) = $500 × 1.04 = $520

Step 2: Calculate Year 2 balance (3% growth)

Year 2 = $520 × (1 + 0.03) = $520 × 1.03 = $535.60

Step 3: Calculate Year 3 balance (2% decrease)

Year 3 = $535.60 × (1 - 0.02) = $535.60 × 0.98 = $524.89

Step 4: Alternative calculation using combined factors

Combined factor = 1.04 × 1.03 × 0.98 = 1.049776

Final balance = $500 × 1.049776 = $524.89

Final balance = $524.89
Final answer:

The final balance after 3 years is $524.89.

Applied rules:

• Sequential Growth/Decay: Apply each percentage to the previous balance

• Factor Multiplication: For multiple changes, multiply all factors

• Direction Matters: Increases use (1 + r), decreases use (1 - r)

Multi-step Exercises 4-5
4 Markup, Discount, and Tax
Exercise 4
A retailer buys an item for $40. They mark it up by 25%, then offer a 10% discount, and finally add 6% sales tax. What is the final customer price? What is the retailer's profit percentage?
Definition:

Business Calculations: Multiple percentage operations in retail: markup, discount, tax

Cost Price
$40
Marked Price
$50
Discounted Price
$45
Final Price
$47.70
Step 1: Calculate marked price after 25% markup

Marked price = Cost × (1 + 0.25) = $40 × 1.25 = $50

Step 2: Apply 10% discount to marked price

Discounted price = Marked price × (1 - 0.10) = $50 × 0.90 = $45

Step 3: Add 6% sales tax to discounted price

Final price = Discounted price × (1 + 0.06) = $45 × 1.06 = $47.70

Step 4: Calculate retailer's profit percentage

Profit = Final price - Cost price = $47.70 - $40 = $7.70

Profit % = ($7.70/$40) × 100% = 19.25%

Final customer price = $47.70, Retailer profit = 19.25%
Final answer:

The final customer price is $47.70, and the retailer's profit percentage is 19.25%.

Applied rules:

• Sequential Operations: Apply markup, then discount, then tax in order

• Profit Calculation: Profit % = (Selling Price - Cost Price) / Cost Price × 100%

• Base Identification: Each percentage applies to a different base value

5 Population Change Over Time
Exercise 5
A town's population was 12,000. It increased by 5% in the first year, decreased by 3% in the second year, and then increased by 8% in the third year. What is the population after 3 years? What is the overall percentage change?
Definition:

Population Dynamics: Sequential percentage changes applied to population over time

Year 0
12,000
Year 1
12,600
Year 2
12,222
Year 3
13,199.76
Step 1: Calculate Year 1 population (5% increase)

Year 1 = 12,000 × (1 + 0.05) = 12,000 × 1.05 = 12,600

Step 2: Calculate Year 2 population (3% decrease)

Year 2 = 12,600 × (1 - 0.03) = 12,600 × 0.97 = 12,222

Step 3: Calculate Year 3 population (8% increase)

Year 3 = 12,222 × (1 + 0.08) = 12,222 × 1.08 = 13,199.76 ≈ 13,200

Step 4: Calculate overall percentage change

Change = 13,200 - 12,000 = 1,200

Overall % change = (1,200/12,000) × 100% = 10%

Step 5: Alternative calculation using combined factors

Combined factor = 1.05 × 0.97 × 1.08 = 1.10002

Final population = 12,000 × 1.10002 = 13,200.24 ≈ 13,200

Final population = 13,200, Overall change = +10%
Final answer:

The population after 3 years is approximately 13,200, representing an overall increase of 10%.

Applied rules:

• Sequential Application: Apply each percentage change to the previous population

• Positive/Negative Changes: Increases use (1 + r), decreases use (1 - r)

• Overall Change: Compare final to initial values regardless of intermediate changes

Multi-step Percent Problem Solving Guide
\(\text{Final Value} = \text{Initial} \times (1 + r_1) \times (1 + r_2) \times \ldots \times (1 + r_n)\)
Sequential Percentage Formula
Advanced Concepts:

Base Value Identification: Each percentage calculation must be applied to the correct base value

Factor Multiplication: When applying multiple percentage changes sequentially, multiply the factors

Direction of Change: Increases use (1 + rate), decreases use (1 - rate)

Cumulative Effect: The combined effect of multiple percentages is not the sum of individual percentages

Multi-step Problem Strategy:
  1. Identify the sequence: Determine the order of percentage operations
  2. Define bases: Identify what each percentage is applied to
  3. Apply operations: Perform calculations in the correct sequence
  4. Track values: Keep track of intermediate results
  5. Verify results: Check if the final answer makes sense
Tip 1: Create a table to track values at each step in multi-step problems.
Tip 2: Remember that percentages are cumulative when applied sequentially.
Tip 3: For complex problems, use the factor multiplication method for efficiency.
Tip 4: Always verify that your final percentage change makes logical sense.
Common Mistakes: Adding percentages instead of multiplying factors, using wrong base values, forgetting direction of changes.
Key Strategies: Sequential application, factor multiplication, careful base identification, verification.
Critical Thinking Points:

• Context Awareness: Understand the real-world meaning of each percentage operation

• Order Sensitivity: Some percentage operations are order-dependent

• Verification: Always check if your answer is reasonable in the context

• Efficiency: Choose the most efficient calculation method for each situation

Multi-step Percent Problem Types

📊
Sequential Discounts
Multiply discount factors
Example: 20% then 15% = ×0.80 × 0.85 = ×0.68 (32% total)
Tax and Tip
Apply to correct base amounts
Example: Tax on pre-tax, tip on pre-tax
Growth/Decay
Use (1+r) for increases, (1-r) for decreases
Example: +5%, -3%, +8% = ×1.05 × 0.97 × 1.08
Business Calculations
Markup, discount, tax in sequence
Example: Cost → Markup → Discount → Tax

Questions & Answers

Question: Why can't I just add the percentages together when calculating sequential changes? For example, if something increases by 10% and then by 20%, shouldn't that be a 30% total increase?

Answer: This is a common misconception! You cannot simply add percentages when they're applied sequentially because each percentage operates on a different base value.

Example: Start with $100

  • First 10% increase: $100 × 1.10 = $110
  • Then 20% increase: $110 × 1.20 = $132
  • Total change: ($132 - $100) / $100 = 32%, not 30%!

If you added the percentages: 10% + 20% = 30%, giving $100 × 1.30 = $130

The 20% increase is applied to $110 (the new base), not the original $100. This is why we multiply the factors: 1.10 × 1.20 = 1.32, representing a 32% increase.

Question: In tax and tip calculations, how do I know which base amount to use for each percentage? Sometimes both seem to apply to the same thing.

Answer: The base amount for tax and tip depends on the specific situation. Generally:

Standard Restaurant Scenario:

  • Sales tax is typically calculated on the pre-tax amount (subtotal)
  • Tip is often calculated on the pre-tax amount (though some people tip on the total)

However, in some situations:

  • Tip might be calculated on the post-tax amount (total)
  • The problem statement will specify which base to use
  • Always read the problem carefully to identify the correct base amount

Example: Bill is $85, tax is 8%, tip is 15% on pre-tax amount:

  • Tax: $85 × 0.08 = $6.80
  • Tip: $85 × 0.15 = $12.75
  • Total: $85 + $6.80 + $12.75 = $104.55

Question: When calculating profit percentage, why do we divide by the cost price instead of the selling price?

Answer: Profit percentage is calculated based on the cost price because it represents the return on investment relative to what was initially spent.

Think of it this way:

  • You invest $100 (cost price) to buy an item
  • You sell it for $120 (selling price)
  • Your profit is $20
  • Your return on investment = ($20 profit / $100 investment) × 100% = 20%

If we divided by selling price: ($20 / $120) × 100% = 16.67%, which doesn't accurately reflect your return on the original investment.

The cost price is the baseline because it represents the capital you risked. This standardization allows for consistent comparison of profitability across different transactions.