Solved Exercises on Percent Decrease in Grade 7

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

Solution: Exercises 1 to 3
1 Basic Percent Decrease Calculation
Exercise 1
A store reduces the price of a $80 jacket by 20%. What is the new price? How much was the decrease?
Definition:

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

Method for calculating percent decrease:
  1. Convert the percentage to a decimal by dividing by 100
  2. Multiply the original amount by the decimal to find the decrease amount
  3. Subtract the decrease amount from the original amount to get the new amount
  4. Alternatively: New Amount = Original Amount × (1 - decimal)
Decrease Amount
$80 × 0.20 = $16
New Price
$80 - $16 = $64
Step 1: Convert percentage to decimal

20% = 20 ÷ 100 = 0.20

Step 2: Calculate the decrease amount

Amount of decrease = Original × Decimal

Amount of decrease = $80 × 0.20 = $16

Step 3: Calculate the new amount

New amount = Original - Decrease

New amount = $80 - $16 = $64

Decrease: $16 | New price: $64
Final answer:

The price decreased by $16, making the new price $64.

Applied rules:

• Decimal Conversion: Percent ÷ 100 = Decimal

• Decrease Formula: New Amount = Original × (1 - decimal)

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

Key Concept:

When decreasing by a percentage, you're subtracting that percentage of the original amount from the original amount. So 20% decrease means the new amount is 80% of the original.

2 Population Decline Problem
Exercise 2
A city's population was 50,000 last year. This year it decreased by 12%. Next year it's expected to decrease by another 8%. What will the population be next year?
Definition:

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

After 12% decrease
50,000 × 0.88 = 44,000
After 8% decrease
44,000 × 0.92 = 40,480
Step 1: Calculate population after first decrease

Population after 12% decrease = 50,000 × (1 - 0.12)

Population after 12% decrease = 50,000 × 0.88 = 44,000

Step 2: Calculate population after second decrease

Population after 8% decrease = 44,000 × (1 - 0.08)

Population after 8% decrease = 44,000 × 0.92 = 40,480

Step 3: Calculate overall percentage decrease

Overall decrease = (50,000 - 40,480) ÷ 50,000 × 100%

Overall decrease = 9,520 ÷ 50,000 × 100% = 19.04%

Population next year: 40,480 | Overall decrease: 19.04%
Final answer:

The population will be 40,480 next year, representing an overall decrease of 19.04% from the original population.

Applied rules:

• Sequential Decreases: Apply each percentage decrease to the current value

• Compound Effect: The second decrease is calculated on the reduced amount

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

50,000
Original
→
-12%
Decrease
→
44,000
After 1st
→
-8%
Decrease
→
40,480
Final
3 Salary Reduction Problem
Exercise 3
Sarah's salary is $60,000 per year. Due to budget cuts, she receives a 5% cut, followed by an additional 3% reduction. What is her new salary? What is the total percentage decrease from her original salary?
Definition:

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

After 5% cut
$60,000 × 0.95 = $57,000
After 3% reduction
$57,000 × 0.97 = $55,290
Total decrease
($60,000 - $55,290)/$60,000 = 7.85%
Step 1: Calculate salary after 5% cut

New salary = $60,000 × (1 - 0.05) = $60,000 × 0.95 = $57,000

Step 2: Calculate salary after additional 3% reduction

New salary = $57,000 × (1 - 0.03) = $57,000 × 0.97 = $55,290

Step 3: Calculate total percentage decrease

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

Percentage decrease = ($60,000 - $55,290) ÷ $60,000 × 100% = 7.85%

New salary: $55,290 | Total decrease: 7.85%
Final answer:

Sarah's new salary is $55,290. The total percentage decrease from her original salary is 7.85%.

Applied rules:

• Sequential Application: Apply each percentage to the current value

• Compound Effect: Each decrease reduces the base for the next decrease

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

Key Concept:

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

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

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

Original Amount: The starting value before the decrease occurs.

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

Percent decrease methodology:
  1. Identify Original Amount: Determine the starting value
  2. Convert Percent to Decimal: Divide percentage by 100
  3. Calculate Decrease Amount: Original × Decimal
  4. Find New Amount: Original - Decrease
  5. Alternative Method: Original × (1 - Decimal)
Tip 1: Remember that 100% - decrease % gives the multiplier.
Tip 2: For 20% decrease, multiply by 0.80 (not 0.20).
Tip 3: Sequential decreases compound - apply each to the current value.
Tip 4: Always verify that your new amount is less than the original.
Real-Life Applications: Sales discounts, population decline, depreciation, budget cuts, tax reductions.
Common Mistakes: Forgetting to subtract from original, applying percentages to wrong base amount.
Percent Decrease Rules:

• Multiplier Rule: Decrease by p% means multiply by (1 - p/100)

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

• Verification: New amount should be less than original

• Compound Effect: Multiple decreases multiply rather than add percentages

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

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

Store A
$200 × 0.85 = $170, -$30
Store B
$300 × 0.90 = $270, -$30
Step 1: Calculate Store A's decrease

Price decrease = $200 × 0.15 = $30

New price = $200 - $30 = $170

Percentage decrease = 15%

Step 2: Calculate Store B's decrease

Price decrease = $300 × 0.10 = $30

New price = $300 - $30 = $270

Percentage decrease = 10%

Step 3: Compare the results

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

Dollar decrease: Both stores decreased by $30

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

Store A has a greater percentage decrease (15% vs 10%). Both stores have the same dollar decrease ($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
Tom scored 90 on his first math test. His score decreased by 10% on the second test, then decreased by another 5% on the third test. What was his score on the third test? What was the total percentage decrease from the first to the third test?
Definition:

Sequential Performance Decreases: When scores or performance metrics decrease by percentages over multiple periods.

Second test
90 × 0.90 = 81
Third test
81 × 0.95 = 76.95
Total decrease
(90 - 76.95)/90 × 100% = 14.5%
Step 1: Calculate second test score

Score after 10% decrease = 90 × (1 - 0.10) = 90 × 0.90 = 81

Step 2: Calculate third test score

Score after 5% decrease = 81 × (1 - 0.05) = 81 × 0.95 = 76.95

Step 3: Calculate total percentage decrease

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

Total decrease = (90 - 76.95) ÷ 90 × 100% = 13.05 ÷ 90 × 100% = 14.5%

Third test score: 76.95 | Total decrease: 14.5%
Final answer:

Tom's score on the third test was 76.95. The total percentage decrease from the first to the third test was 14.5%.

Applied rules:

• Sequential Application: Apply each percentage to the current score

• Compound Reduction: Each decrease builds on the previous score

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

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

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

Original Amount: The starting value before the decrease occurs.

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

Sequential Decreases: When multiple percentage decreases are applied one after another.

Percent Decrease Problem-Solving Steps:
  1. Identify Original Value: Determine the starting amount
  2. Convert Percent to Decimal: Divide percentage by 100
  3. Calculate Multiplier: Subtract decimal from 1 (1 - decimal)
  4. Find New Amount: Multiply original by the multiplier
  5. Verify Result: Ensure new amount is less than original
  6. Calculate Total Decrease: If needed, find overall percentage change
Quick Tip: For a 20% decrease, multiply by 0.80 (original - 20% of original).
Memory Aid: 100% - decrease % = multiplier (20% decrease = 0.80).
Strategy: For sequential decreases, apply each to the current value.
Verification: Always check that your new amount is less than the original.
Real-Life Applications: Sales discounts, depreciation, population decline, budget cuts, tax reductions.
Common Scenarios: Store sales, asset depreciation, enrollment drops, revenue decreases.
Key Rules and Properties:

• Multiplier Rule: Decrease by p% means multiply by (1 - p/100)

• Sequential Application: Apply each decrease to the current value

• Compound Effect: Multiple decreases multiply rather than add percentages

• Verification: New amount must be less than original amount

• Total Decrease: (Original - New) ÷ Original × 100% = Total percent decrease

📉

Questions & Answers

Question: I'm confused about why we multiply by (1 - decimal) instead of just subtracting the decimal. Why do we subtract from 1?

Answer: Great question! The "1" represents the original amount (100%), and we subtract the decimal (the decrease percentage).

Here's why:

  • When you decrease by 20%, you're keeping 100% - 20% = 80% of the original
  • 80% = 0.80
  • So 1 represents the original (100%), and 0.20 represents the decrease (20%)
  • Therefore: 1 - 0.20 = 0.80 (what remains after decrease)

Example: $100 decreased by 20%

  • Method 1: $100 × 0.20 = $20 (just the decrease), then $100 - $20 = $80
  • Method 2: $100 × 0.80 = $80 (remaining amount in one step)

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

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

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

Subtracting percentages:

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

Multiplying multipliers:

  • When applying sequential decreases to the same base
  • Example: $100 × 0.85 × 0.90 = $76.50
  • Final result: (100 - 76.50)/100 × 100% = 23.5% total decrease

Note: 15% + 10% = 25%, but actual total decrease is 23.5% due to compounding!

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

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

Answer: Here are several verification methods:

Method 1: Reverse calculation

  • Take your new amount and increase it by the same percentage
  • Should return close to the original amount
  • Example: $80 increased by 20% → $80 × 1.20 = $96 (close to $100, accounting for rounding)

Method 2: Calculate the difference

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

Method 3: Reasonableness check

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

Always use at least one verification method to ensure accuracy!