Solved Exercises on Percent of a Quantity in Grade 7

Master percent of a quantity: finding percentages, percentage increases/decreases, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Percentage Calculation
Exercise 1
What is 25% of 80? What is 15% of 120?
Definition:

Percent of a Quantity: A portion of a whole amount expressed as a fraction of 100. The formula is: (Percent ÷ 100) × Quantity.

Method for calculating percent of a quantity:
  1. Convert the percentage to a decimal by dividing by 100
  2. Multiply the decimal by the quantity
  3. Round if necessary based on context
25% of 80
0.25 × 80 = 20
15% of 120
0.15 × 120 = 18
Step 1: Convert percentage to decimal

25% = 25 ÷ 100 = 0.25

15% = 15 ÷ 100 = 0.15

Step 2: Multiply decimal by quantity

25% of 80 = 0.25 × 80 = 20

15% of 120 = 0.15 × 120 = 18

Step 3: Verify the answer

Check: 20 ÷ 80 = 0.25 = 25% ✓

Check: 18 ÷ 120 = 0.15 = 15% ✓

25% of 80 = 20 | 15% of 120 = 18
Final answer:

25% of 80 is 20. 15% of 120 is 18.

Applied rules:

• Percent to Decimal: Divide percent by 100

• Multiplication Property: Decimal × Quantity = Part

• Verification: Part ÷ Whole = Decimal equivalent of percent

Key Concept:

Percent means "per hundred," so 25% literally means 25 per 100 or 25/100 = 0.25. This is why we divide by 100 to convert.

2 Percentage Increase Problem
Exercise 2
A shirt originally costs $40. Its price increases by 20%. What is the new price? If the price then decreases by 10%, what is the final price?
Definition:

Percentage Increase: Adding a percentage of the original amount to the original amount. Formula: New Amount = Original Amount × (1 + Percent/100).

First Increase
$40 × 1.20 = $48
Second Decrease
$48 × 0.90 = $43.20
Step 1: Calculate the first increase

New price after 20% increase = Original price × (1 + 0.20)

New price = $40 × 1.20 = $48

Step 2: Calculate the decrease

Final price after 10% decrease = New price × (1 - 0.10)

Final price = $48 × 0.90 = $43.20

Step 3: Verify the result

Check: $43.20 is indeed less than $40 + $8 (the 20% increase)

New price: $48 | Final price: $43.20
Final answer:

The price after a 20% increase is $48. After a subsequent 10% decrease, the final price is $43.20.

Applied rules:

• Percentage Increase: Multiply by (1 + percent/100)

• Percentage Decrease: Multiply by (1 - percent/100)

• Sequential Changes: Apply each change to the previous result

$40
Original
+20%
Increase
$48
After +20%
-10%
Decrease
$43.20
Final
3 Discount and Tax Problem
Exercise 3
A laptop costs $800. A 15% discount is applied, followed by 8% sales tax. What is the final price? What is the effective percentage change from the original price?
Definition:

Discount: A reduction in price, calculated as a percentage of the original price. Sales Tax: An additional charge added to the discounted price.

After Discount
$800 × 0.85 = $680
After Tax
$680 × 1.08 = $734.40
Effective Change
($734.40 - $800)/$800 = -8.2%
Step 1: Apply the discount

Price after discount = Original price × (1 - 0.15)

Price after discount = $800 × 0.85 = $680

Step 2: Apply sales tax

Final price = Discounted price × (1 + 0.08)

Final price = $680 × 1.08 = $734.40

Step 3: Calculate effective percentage change

Change = (Final price - Original price) ÷ Original price

Change = ($734.40 - $800) ÷ $800 = -0.082 = -8.2%

Final price: $734.40 | Effective change: -8.2%
Final answer:

The final price is $734.40. The effective percentage change from the original price is -8.2%.

Applied rules:

• Discount Application: Multiply by (1 - discount percent)

• Tax Application: Multiply by (1 + tax percent)

• Effective Change: (New - Old) ÷ Old × 100%

Key Concept:

When applying multiple percentage changes, always apply them sequentially to the previous result, not to the original amount.

Percent of a Quantity: Rules and Methods
\(\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}\)
Basic Percent Formula
Percent to Decimal
\(p\% = \frac{p}{100}\)
Converting percent to decimal
Percentage Increase
\(A \times (1 + \frac{p}{100})\)
Increasing amount A by p%
Percentage Decrease
\(A \times (1 - \frac{p}{100})\)
Decreasing amount A by p%
Key definitions:

Percent: A ratio or fraction expressed as a part of 100. The symbol % means "per hundred."

Percent of a Quantity: A portion of a whole amount calculated using percentage.

Percentage Change: The relative change in a quantity expressed as a percentage of the original amount.

Percent calculation methodology:
  1. Identify Components: Determine what is the percent, whole, and part
  2. Convert Percent: Change percentage to decimal by dividing by 100
  3. Apply Formula: Use appropriate formula based on problem type
  4. Calculate: Perform the multiplication
  5. Verify: Check that the answer makes sense in context
Tip 1: Remember that "of" means multiply in percentage problems.
Tip 2: 10% of a number = number ÷ 10 (move decimal point left once).
Tip 3: 50% = 1/2, 25% = 1/4, 75% = 3/4 (memorize these common percentages).
Tip 4: Always apply percentage changes to the current value, not the original.
Real-Life Applications: Discounts, taxes, tips, commissions, interest rates, population changes.
Common Mistakes: Forgetting to convert percent to decimal, applying changes to wrong base amount.
Percent Calculation Rules:

• Decimal Conversion: Divide percent by 100 to get decimal

• Multiplication Order: Decimal × Whole = Part

• Sequential Changes: Apply each change to the previous result

• Verification: Part ÷ Whole should equal the decimal form of the percent

Solution: Exercises 4 to 5
4 Population Percentage Problem
Exercise 4
A city has a population of 250,000 people. 40% are children, 35% are adults, and the rest are seniors. How many people are in each group? What percentage are seniors?
Definition:

Population Percentages: Dividing a total population into groups based on given percentages.

Children
250,000 × 0.40 = 100,000
Adults
250,000 × 0.35 = 87,500
Seniors
250,000 - 100,000 - 87,500 = 62,500
Step 1: Calculate number of children

Children = 40% of 250,000 = 0.40 × 250,000 = 100,000

Step 2: Calculate number of adults

Adults = 35% of 250,000 = 0.35 × 250,000 = 87,500

Step 3: Calculate number of seniors

Seniors = Total - Children - Adults

Seniors = 250,000 - 100,000 - 87,500 = 62,500

Step 4: Calculate senior percentage

Senior % = (62,500 ÷ 250,000) × 100% = 25%

Children: 100,000 | Adults: 87,500 | Seniors: 62,500 (25%)
Final answer:

There are 100,000 children (40%), 87,500 adults (35%), and 62,500 seniors (25%) in the city.

Applied rules:

• Percentage of Total: Multiply decimal by total population

• Remaining Percentage: 100% - Sum of known percentages

• Verification: All percentages should sum to 100%

40% Children
35% Adults
25% Seniors
5 Investment Growth Problem
Exercise 5
An investment of $5,000 grows by 12% in the first year, then by 8% in the second year. What is the value after two years? What is the overall percentage growth?
Definition:

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

After Year 1
$5,000 × 1.12 = $5,600
After Year 2
$5,600 × 1.08 = $6,048
Overall Growth
($6,048 - $5,000)/$5,000 = 20.96%
Step 1: Calculate value after first year

Year 1 value = $5,000 × (1 + 0.12) = $5,000 × 1.12 = $5,600

Step 2: Calculate value after second year

Year 2 value = $5,600 × (1 + 0.08) = $5,600 × 1.08 = $6,048

Step 3: Calculate overall percentage growth

Growth = (Final - Initial) ÷ Initial × 100%

Growth = ($6,048 - $5,000) ÷ $5,000 × 100% = 20.96%

Value after 2 years: $6,048 | Overall growth: 20.96%
Final answer:

The investment is worth $6,048 after two years. The overall percentage growth is 20.96%.

Applied rules:

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

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

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

Comprehensive Summary: Percent of a Quantity
\(\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}, \quad \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\)
Core Percent Formulas
Core Definitions:

Percent: A ratio or fraction expressed as a part of 100. The symbol % means "per hundred" or divided by 100.

Percent of a Quantity: A portion of a whole amount calculated by multiplying the decimal equivalent of the percent by the quantity.

Percentage Change: The relative change in a quantity expressed as a percentage of the original amount.

Percent Problem-Solving Steps:
  1. Identify Components: Determine what is given (percent, whole, or part)
  2. Convert Percent: Change percentage to decimal by dividing by 100
  3. Select Formula: Use appropriate formula based on what you need to find
  4. Substitute Values: Plug known values into the formula
  5. Calculate: Perform the arithmetic operations
  6. Verify: Check that your answer makes sense in context
Quick Tip: "Of" means multiply in percentage problems (25% of 80 means 0.25 × 80).
Memory Aid: 10% = move decimal left once, 50% = half, 25% = quarter.
Strategy: For increases, multiply by (1 + decimal). For decreases, multiply by (1 - decimal).
Verification: Always check that percentages in a group sum to 100% when appropriate.
Real-Life Applications: Sales discounts, tax calculations, tip amounts, interest rates, grade calculations.
Common Scenarios: Shopping (discounts), dining (tips), banking (interest), statistics (survey results).
Key Rules and Properties:

• Decimal Conversion: Percent ÷ 100 = Decimal equivalent

• Multiplication Rule: Decimal × Whole = Part

• Sequential Changes: Apply each percentage change to the current value

• Percentage Increase: Multiply by (1 + percent/100)

• Percentage Decrease: Multiply by (1 - percent/100)

• Verification: Part ÷ Whole should equal the decimal form of the percent

25%
Children
35%
Adults
25%
Seniors
15%
Others

Questions & Answers

Question: I get confused about when to multiply by the decimal and when to divide. How do I know which operation to use?

Answer: Great question! The key is to identify what you're trying to find:

Finding a percent of a quantity (Part):

  • Formula: Part = Percent × Whole
  • Example: What is 20% of 50? → 0.20 × 50 = 10
  • Operation: MULTIPLY

Finding what percent one number is of another:

  • Formula: Percent = (Part ÷ Whole) × 100
  • Example: What percent is 15 of 60? → (15 ÷ 60) × 100 = 25%
  • Operation: DIVIDE then multiply by 100

Finding the whole when given a part and percent:

  • Formula: Whole = Part ÷ Percent
  • Example: 20 is 10% of what number? → 20 ÷ 0.10 = 200
  • Operation: DIVIDE

The most common case is finding a percent of a quantity, which uses multiplication!

Question: When do I multiply by (1 + percent) versus just the percent? I see both in examples.

Answer: This depends on what the problem is asking:

When to multiply by the percent only:

  • Finding the PART of a whole (e.g., 20% of $100)
  • Calculating just the AMOUNT of increase/decrease
  • Formula: Decimal × Whole

When to multiply by (1 + percent):

  • Finding the NEW total after an increase
  • Formula: Whole × (1 + Decimal)
  • Example: $100 increased by 20% = $100 × 1.20 = $120

When to multiply by (1 - percent):

  • Finding the NEW total after a decrease
  • Formula: Whole × (1 - Decimal)
  • Example: $100 decreased by 20% = $100 × 0.80 = $80

The (1 + percent) method gives you the final total, while just the percent gives you the change amount.

Question: I sometimes get decimal answers when calculating percentages. Is this normal? What if I get 0.45 instead of a whole number?

Answer: Yes, decimal answers are completely normal in percentage calculations! Many real-world percentage problems result in decimal answers.

Examples of normal decimal results:

  • 15% of 30 = 4.5
  • 23% of 200 = 46 (whole number in this case)
  • 7% of 150 = 10.5

About getting 0.45: This could mean:

  • 0.45 of the original unit (like 0.45 pounds)
  • 45% if it's a decimal representing a percentage

When dealing with decimals:

  • Round appropriately based on context (money to nearest cent)
  • Keep more precision during calculations, round final answer
  • Check that your answer makes sense in the real-world context

Decimals often occur because real-world measurements rarely divide evenly!