Solved Exercises on Percent Increase in Grade 7

Master percent increase: calculating percentage increases, comparing increases, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Percent Increase Calculation
Exercise 1
A store increases the price of a $60 shirt by 25%. What is the new price? How much was the increase?
Definition:

Percent Increase: The amount by which a quantity increases, expressed as a percentage of the original amount.

Method for calculating percent increase:
  1. Convert the percentage to a decimal by dividing by 100
  2. Multiply the original amount by the decimal to find the increase amount
  3. Add the increase amount to the original amount to get the new amount
  4. Alternatively: New Amount = Original Amount × (1 + decimal)
Increase Amount
$60 × 0.25 = $15
New Price
$60 + $15 = $75
Step 1: Convert percentage to decimal

25% = 25 ÷ 100 = 0.25

Step 2: Calculate the increase amount

Amount of increase = Original × Decimal

Amount of increase = $60 × 0.25 = $15

Step 3: Calculate the new amount

New amount = Original + Increase

New amount = $60 + $15 = $75

Increase: $15 | New price: $75
Final answer:

The price increased by $15, making the new price $75.

Applied rules:

• Decimal Conversion: Percent ÷ 100 = Decimal

• Increase Formula: New Amount = Original × (1 + decimal)

• Verification: (New - Original) ÷ Original × 100% = Percent increase

Key Concept:

When increasing by a percentage, you're adding that percentage of the original amount to the original amount. So 25% increase means the new amount is 125% of the original.

2 Population Growth Problem
Exercise 2
A town's population was 12,000 last year. This year it increased by 8%. Next year it's expected to increase by another 5%. What will the population be next year?
Definition:

Compound Growth: When a quantity increases by a percentage each period, and subsequent increases are calculated on the new amount.

After 8% increase
12,000 × 1.08 = 12,960
After 5% increase
12,960 × 1.05 = 13,608
Step 1: Calculate population after first increase

Population after 8% increase = 12,000 × (1 + 0.08)

Population after 8% increase = 12,000 × 1.08 = 12,960

Step 2: Calculate population after second increase

Population after 5% increase = 12,960 × (1 + 0.05)

Population after 5% increase = 12,960 × 1.05 = 13,608

Step 3: Calculate overall percentage increase

Overall increase = (13,608 - 12,000) ÷ 12,000 × 100%

Overall increase = 1,608 ÷ 12,000 × 100% = 13.4%

Population next year: 13,608 | Overall increase: 13.4%
Final answer:

The population will be 13,608 next year, representing an overall increase of 13.4% from the original population.

Applied rules:

• Sequential Increases: Apply each percentage increase to the current value

• Compound Effect: The second increase is calculated on the increased amount

• Overall Increase: Use the difference between final and initial values

12,000
Original
→
+8%
Increase
→
12,960
After 1st
→
+5%
Increase
→
13,608
Final
3 Salary Increase Problem
Exercise 3
John's salary is $45,000 per year. He receives a 4% raise, followed by a 3% bonus. What is his new total compensation? What is the total percentage increase from his original salary?
Definition:

Sequential Increases: When multiple percentage increases are applied one after another to the same base amount.

After 4% raise
$45,000 × 1.04 = $46,800
After 3% bonus
$46,800 × 1.03 = $48,204
Total increase
($48,204 - $45,000)/$45,000 = 7.12%
Step 1: Calculate salary after 4% raise

New salary = $45,000 × (1 + 0.04) = $45,000 × 1.04 = $46,800

Step 2: Calculate total compensation after 3% bonus

Total compensation = $46,800 × (1 + 0.03) = $46,800 × 1.03 = $48,204

Step 3: Calculate total percentage increase

Percentage increase = (New - Original) ÷ Original × 100%

Percentage increase = ($48,204 - $45,000) ÷ $45,000 × 100% = 7.12%

Total compensation: $48,204 | Total increase: 7.12%
Final answer:

John's new total compensation is $48,204. The total percentage increase from his original salary is 7.12%.

Applied rules:

• Sequential Application: Apply each percentage to the current value

• Compound Effect: Each increase builds on the previous amount

• Total Calculation: Use the difference between final and original values

Key Concept:

When multiple increases are applied sequentially, the final amount is calculated by multiplying by (1 + each decimal) in sequence. The total percentage increase is not simply the sum of individual percentages.

Percent Increase: Rules and Methods
\(\text{New Amount} = \text{Original Amount} \times (1 + \frac{\text{Percent}}{100})\)
Percent Increase Formula
Percent Increase
\(A \times (1 + \frac{p}{100})\)
Increase amount A by p%
Sequential Increases
\(A \times (1 + \frac{p_1}{100}) \times (1 + \frac{p_2}{100})\)
Multiple increases applied
Total Increase
\(\frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%\)
Overall percentage change
Key definitions:

Percent Increase: The amount by which a quantity increases, expressed as a percentage of the original amount.

Original Amount: The starting value before the increase occurs.

New Amount: The value after the increase has been applied.

Percent increase methodology:
  1. Identify Original Amount: Determine the starting value
  2. Convert Percent to Decimal: Divide percentage by 100
  3. Calculate Increase Amount: Original × Decimal
  4. Find New Amount: Original + Increase
  5. Alternative Method: Original × (1 + Decimal)
Tip 1: Remember that 100% + increase % gives the multiplier.
Tip 2: For 25% increase, multiply by 1.25 (not 0.25).
Tip 3: Sequential increases compound - apply each to the current value.
Tip 4: Always verify that your new amount is greater than the original.
Real-Life Applications: Salary raises, population growth, price increases, investment returns, inflation calculations.
Common Mistakes: Forgetting to add increase to original, applying percentages to wrong base amount.
Percent Increase Rules:

• Multiplier Rule: Increase by p% means multiply by (1 + p/100)

• Sequential Application: Apply each increase to the current value, not original

• Verification: New amount should be greater than original

• Compound Effect: Multiple increases multiply rather than add percentages

Solution: Exercises 4 to 5
4 Comparison Problem
Exercise 4
Store A increases the price of a $200 item by 15%. Store B increases the price of a $300 item by 10%. Which store has a greater percentage increase? Which store has a greater dollar increase?
Definition:

Percentage vs Dollar Increase: Understanding the difference between percentage change and absolute change.

Store A
$200 × 1.15 = $230, +$30
Store B
$300 × 1.10 = $330, +$30
Step 1: Calculate Store A's increase

Price increase = $200 × 0.15 = $30

New price = $200 + $30 = $230

Percentage increase = 15%

Step 2: Calculate Store B's increase

Price increase = $300 × 0.10 = $30

New price = $300 + $30 = $330

Percentage increase = 10%

Step 3: Compare the results

Percentage increase: Store A (15%) > Store B (10%)

Dollar increase: Both stores increased by $30

Store A: Greater percentage | Same dollar increase
Final answer:

Store A has a greater percentage increase (15% vs 10%). Both stores have the same dollar increase ($30).

Applied rules:

• Percentage Comparison: Compare the percentage values directly

• Dollar Comparison: Calculate actual dollar amounts

• Relative vs Absolute: Percentage is relative, dollar is absolute

📊
Store A
15% of $200 = $30
📈
Store B
10% of $300 = $30
⚖️
Comparison
Same dollar, different %
5 Academic Performance Problem
Exercise 5
Sarah scored 75 on her first math test. Her score increased by 20% on the second test, then increased by another 10% on the third test. What was her score on the third test? What was the total percentage increase from the first to the third test?
Definition:

Sequential Performance Increases: When scores or performance metrics improve by percentages over multiple periods.

Second test
75 × 1.20 = 90
Third test
90 × 1.10 = 99
Total increase
(99 - 75)/75 × 100% = 32%
Step 1: Calculate second test score

Score after 20% increase = 75 × (1 + 0.20) = 75 × 1.20 = 90

Step 2: Calculate third test score

Score after 10% increase = 90 × (1 + 0.10) = 90 × 1.10 = 99

Step 3: Calculate total percentage increase

Total increase = (Final - Original) ÷ Original × 100%

Total increase = (99 - 75) ÷ 75 × 100% = 24 ÷ 75 × 100% = 32%

Third test score: 99 | Total increase: 32%
Final answer:

Sarah's score on the third test was 99. The total percentage increase from the first to the third test was 32%.

Applied rules:

• Sequential Application: Apply each percentage to the current score

• Compound Growth: Each increase builds on the previous score

• Total Calculation: Use original and final values for total percentage

Comprehensive Summary: Percent Increase
\(\text{New Amount} = \text{Original Amount} \times (1 + \frac{\text{Percent}}{100}), \quad \text{Percent Increase} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%\)
Core Percent Increase Formulas
Core Definitions:

Percent Increase: The amount by which a quantity increases, expressed as a percentage of the original amount.

Original Amount: The starting value before the increase occurs.

New Amount: The value after the increase has been applied.

Sequential Increases: When multiple percentage increases are applied one after another.

Percent Increase Problem-Solving Steps:
  1. Identify Original Value: Determine the starting amount
  2. Convert Percent to Decimal: Divide percentage by 100
  3. Calculate Multiplier: Add 1 to the decimal (1 + decimal)
  4. Find New Amount: Multiply original by the multiplier
  5. Verify Result: Ensure new amount is greater than original
  6. Calculate Total Increase: If needed, find overall percentage change
Quick Tip: For a 25% increase, multiply by 1.25 (original + 25% of original).
Memory Aid: 100% + increase % = multiplier (20% increase = 1.20).
Strategy: For sequential increases, apply each to the current value.
Verification: Always check that your new amount is greater than the original.
Real-Life Applications: Salary raises, tax increases, price hikes, population growth, investment returns.
Common Scenarios: Inflation, rent increases, tuition fees, commodity prices, stock gains.
Key Rules and Properties:

• Multiplier Rule: Increase by p% means multiply by (1 + p/100)

• Sequential Application: Apply each increase to the current value

• Compound Effect: Multiple increases multiply rather than add percentages

• Verification: New amount must be greater than original amount

• Total Increase: (New - Original) ÷ Original × 100% = Total percent increase

📈

Questions & Answers

Question: I'm confused about why we multiply by (1 + decimal) instead of just the decimal. Why do we add 1?

Answer: Great question! The "1" represents the original amount, and the decimal represents the increase.

Here's why:

  • When you increase by 20%, you're keeping 100% of the original AND adding 20%
  • 100% + 20% = 120% = 1.20
  • So 1 represents the original (100%), and 0.20 represents the increase (20%)

Example: $100 increased by 20%

  • Method 1: $100 × 0.20 = $20 (just the increase), then $100 + $20 = $120
  • Method 2: $100 × 1.20 = $120 (original + increase in one step)

Method 2 is more efficient and combines both steps into one calculation!

Question: When do I add percentages together versus multiplying the multipliers? I see both approaches.

Answer: This depends on whether you're finding the total percentage increase or applying sequential increases:

Adding percentages:

  • Only when finding the total percentage increase from original to final value
  • Example: If total went up 15% then 10%, total increase is NOT 25% (it's compound)

Multiplying multipliers:

  • When applying sequential increases to the same base
  • Example: $100 × 1.15 × 1.10 = $126.50
  • Final result: (126.50 - 100)/100 × 100% = 26.5% total increase

Note: 15% + 10% = 25%, but actual total increase is 26.5% due to compounding!

Always multiply multipliers for sequential increases, then calculate the total percentage if needed.

Question: I sometimes get confused about whether my answer is correct. How can I verify my percent increase calculations?

Answer: Here are several verification methods:

Method 1: Reverse calculation

  • Take your new amount and decrease it by the same percentage
  • Should return to the original amount
  • Example: $120 decreased by 20% → $120 × 0.80 = $96 (close to $100, accounting for rounding)

Method 2: Calculate the difference

  • Find the actual increase amount
  • Divide by original amount and multiply by 100
  • Should equal your original percentage

Method 3: Reasonableness check

  • New amount should definitely be greater than original
  • Magnitude should make sense (20% of $100 is $20, not $2)

Always use at least one verification method to ensure accuracy!