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Solving problems involving percentages
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Percent means "per hundred" or "out of 100"
Symbol: %
Key concept: 50% = 50/100 = 0.50 = 1/2
Converting:
Examples:
Type 1: Find the part What is 30% of 80?
Type 2: Find the percent 12 is what percent of 40?
Type 3: Find the whole 15 is 25% of what number?
Key equation: Part = Percent × Whole Or: is = % × of
What is 25% of 80?
Step 1: Convert the percent to a decimal: 25% = 25/100 = 0.25
Step 2: Multiply by the whole: "of" means multiply 0.25 × 80 = 20
Step 3: Alternative method - use a proportion: part/whole = percent/100 x/80 = 25/100 100x = 2000 x = 20
Answer: 20
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Question form: What is P% of W?
Equation: Part = (P/100) × W
Example 1: What is 20% of 50?
Part = 0.20 × 50 = 10
Answer: 10
Example 2: Find 35% of 200
Part = 0.35 × 200 = 70
Answer: 70
Example 3: What is 8% of 125?
Part = 0.08 × 125 = 10
Answer: 10
Example 4: Calculate 150% of 60
Part = 1.50 × 60 = 90
Answer: 90 (yes, more than 100%!)
Question form: A is what percent of B?
Equation: Percent = (Part/Whole) × 100
Example 1: 15 is what percent of 60?
Percent = (15/60) × 100 = 0.25 × 100 = 25%
Answer: 25%
Example 2: What percent of 80 is 20?
Percent = (20/80) × 100 = 0.25 × 100 = 25%
Answer: 25%
Example 3: 33 is what percent of 150?
Percent = (33/150) × 100 = 0.22 × 100 = 22%
Answer: 22%
Example 4: What percent of 25 is 30?
Percent = (30/25) × 100 = 1.2 × 100 = 120%
Answer: 120% (more than the whole!)
Question form: A is P% of what number?
Equation: Whole = Part / (P/100) = Part / Percent (as decimal)
Example 1: 20 is 25% of what number?
Whole = 20 / 0.25 = 80
Answer: 80
Example 2: 15 is 30% of what number?
Whole = 15 / 0.30 = 50
Answer: 50
Example 3: 12 is 8% of what number?
Whole = 12 / 0.08 = 150
Answer: 150
Example 4: 45 is 150% of what number?
Whole = 45 / 1.50 = 30
Answer: 30
Alternative method: Set up proportion
Form: Part/Whole = Percent/100
Example 1: What is 40% of 70?
x/70 = 40/100
Cross multiply: 100x = 2800 x = 28
Example 2: 18 is what percent of 72?
18/72 = x/100
Cross multiply: 72x = 1800 x = 25
Answer: 25%
Example 3: 24 is 60% of what number?
24/x = 60/100
Cross multiply: 60x = 2400 x = 40
Answer: 40
Percent Change Formula:
Percent Change = (New - Original) / Original × 100
Increase: New > Original (positive result) Decrease: New < Original (negative result)
Example 1: Percent Increase
Original price: 50 dollars New price: 60 dollars
Percent Increase = (60 - 50)/50 × 100 = 10/50 × 100 = 20%
Price increased by 20%
Example 2: Percent Decrease
Original price: 80 dollars Sale price: 64 dollars
Percent Decrease = (64 - 80)/80 × 100 = -16/80 × 100 = -20%
Price decreased by 20% (negative indicates decrease)
Example 3: Population Growth
Original population: 5,000 New population: 6,500
Percent Increase = (6,500 - 5,000)/5,000 × 100 = 1,500/5,000 × 100 = 30%
Population grew by 30%
Example 4: Test Score Improvement
First score: 70 Second score: 84
Percent Increase = (84 - 70)/70 × 100 = 14/70 × 100 = 20%
Score improved by 20%
Increase: New = Original × (1 + percent increase as decimal)
Decrease: New = Original × (1 - percent decrease as decimal)
Example 1: 15% Increase
Original: 200 Increase by 15%
New = 200 × (1 + 0.15) = 200 × 1.15 = 230
Example 2: 20% Decrease
Original: 150 Decrease by 20%
New = 150 × (1 - 0.20) = 150 × 0.80 = 120
Example 3: Price After Discount
Original price: 85 dollars 30% off discount
Sale price = 85 × (1 - 0.30) = 85 × 0.70 = 59.50 dollars
Example 4: Population After Growth
Current population: 8,000 Expected 12% growth
New population = 8,000 × (1 + 0.12) = 8,000 × 1.12 = 8,960
Total Cost = Original Price + Tax Tax = Original Price × Tax Rate
Example 1: 6% sales tax
Item cost: 50 dollars Tax = 50 × 0.06 = 3 dollars Total = 50 + 3 = 53 dollars
Shortcut: Total = 50 × 1.06 = 53 dollars
Example 2: 8.5% sales tax
Item cost: 120 dollars Total = 120 × 1.085 = 130.20 dollars
Example 3: Finding Original Price
Total with tax: 84.80 dollars Tax rate: 6%
Original price = 84.80 / 1.06 = 80 dollars
Sale Price = Original Price - Discount Discount = Original Price × Discount Rate
Example 1: 25% off
Original: 80 dollars Discount = 80 × 0.25 = 20 dollars Sale price = 80 - 20 = 60 dollars
Shortcut: Sale price = 80 × 0.75 = 60 dollars
Example 2: 40% off
Original: 150 dollars Sale price = 150 × 0.60 = 90 dollars
Example 3: Multiple Discounts
Original: 100 dollars 20% off, then additional 10% off
After first discount: 100 × 0.80 = 80 dollars After second discount: 80 × 0.90 = 72 dollars
Note: NOT the same as 30% off! (that would be 70 dollars)
Tip = Bill Amount × Tip Percentage
Example 1: Restaurant Tip
Bill: 45 dollars Tip 20%
Tip = 45 × 0.20 = 9 dollars Total = 45 + 9 = 54 dollars
Example 2: Commission
Sales: 12,000 dollars Commission rate: 5%
Commission = 12,000 × 0.05 = 600 dollars
Example 3: Finding Sales from Commission
Commission earned: 450 dollars Commission rate: 6%
Sales = 450 / 0.06 = 7,500 dollars
Formula: I = Prt
Where:
Total Amount: A = P + I = P(1 + rt)
Example 1:
Principal: 1,000 dollars Rate: 5% per year Time: 3 years
Interest = 1,000 × 0.05 × 3 = 150 dollars Total = 1,000 + 150 = 1,150 dollars
Example 2:
Borrow 2,500 dollars at 4% for 2 years
Interest = 2,500 × 0.04 × 2 = 200 dollars Total owed = 2,500 + 200 = 2,700 dollars
Example 3: Finding Rate
Principal: 800 dollars Time: 5 years Interest earned: 200 dollars
200 = 800 × r × 5 200 = 4,000r r = 0.05 = 5%
Multiply the decimals
Example 1: 20% of 50% of 200
First: 50% of 200 = 0.50 × 200 = 100 Then: 20% of 100 = 0.20 × 100 = 20
Shortcut: 0.20 × 0.50 × 200 = 20
Example 2: 10% of 30% of 500
0.10 × 0.30 × 500 = 15
Selling Price = Cost + Markup Markup = Cost × Markup Percentage
Example 1: Store markup
Cost to store: 40 dollars Markup: 60%
Markup amount = 40 × 0.60 = 24 dollars Selling price = 40 + 24 = 64 dollars
Shortcut: Selling price = 40 × 1.60 = 64 dollars
Example 2: Finding Cost
Selling price: 120 dollars Markup: 50%
Cost = 120 / 1.50 = 80 dollars
Apply percentages one at a time
Example 1: Price increases 10%, then decreases 10%
Original: 100 dollars After increase: 100 × 1.10 = 110 dollars After decrease: 110 × 0.90 = 99 dollars
NOT back to original! Lost 1 dollar overall
Example 2: Successive Growth
Population: 1,000 Grows 5% first year, 8% second year
After year 1: 1,000 × 1.05 = 1,050 After year 2: 1,050 × 1.08 = 1,134
Total growth: 13.4% (not 13%!)
Confusing part and whole "25% of 80" → 80 is the whole, not the part!
Wrong formula for percent change Use (New - Original)/Original, not (Original - New)/New
Adding percents incorrectly 20% off then 10% off ≠ 30% off!
Forgetting to convert percent to decimal 30% = 0.30 (not 30 in calculations!)
Percent of percent errors 50% of 50% = 25% (not 100%!)
Using whole instead of original After 20% increase: use original as base for percent change
10%: Move decimal one place left 50 → 5
1%: Move decimal two places left 50 → 0.5
5%: Half of 10% 10% of 50 = 5, so 5% = 2.5
25%: Divide by 4 25% of 80 = 80/4 = 20
50%: Divide by 2 50% of 60 = 30
Double: Same as 200% 200% of 15 = 30
Shopping: Calculate discounts, sales tax, total cost
Finance: Interest, investments, loans
Statistics: Survey results, data analysis
Science: Percent error, concentration, growth rates
Business: Profit margins, commission, markup
Everyday: Tips, grades (percent correct), batting averages
Part = Percent × Whole (is = % × of)
Percent = (Part/Whole) × 100
Whole = Part / Percent (as decimal)
Percent Change = (New - Original) / Original × 100
After Increase: New = Original × (1 + r)
After Decrease: New = Original × (1 - r)
Simple Interest: I = Prt
Percent problems are everywhere in daily life. Master them and you'll be a more informed consumer, investor, and decision-maker!
What is 15% of 80?
Convert the percent to a decimal and multiply:
Answer: 12
A shirt originally costs $40. It is on sale for 30% off. What is the sale price?
Step 1: Find the discount amount: 30% of 12
Step 2: Subtract from original price: 12 = $28
Alternative method (finding what you pay): If it's 30% off, you pay 70% 70% of 28
Answer: $28
18 is what percent of 72?
Set up the equation: part = percent × whole
Solve for :
Convert to percent:
Answer: 25%
18 is what percent of 60?
Step 1: Set up the percent equation: part = percent × whole 18 = x × 60
Step 2: Solve for x: x = 18/60 x = 0.3
Step 3: Convert to percent: 0.3 = 30/100 = 30%
Alternative - use a proportion: part/whole = percent/100 18/60 = x/100 60x = 1800 x = 30
Answer: 30%
A population increased from 500 to 650. What is the percent increase?
Step 1: Find the amount of change: Change = New - Old Change = 650 - 500 = 150
Step 2: Use the percent change formula: Percent change = (amount of change/original amount) × 100%
Step 3: Substitute and calculate: Percent increase = (150/500) × 100% = 0.3 × 100% = 30%
Step 4: Check: 30% of 500 = 150 500 + 150 = 650 ✓
Answer: 30% increase
A shirt originally costs $40. After a sale, it costs $32. What is the percent decrease?
Use the percent change formula:
The negative indicates a decrease.
Answer: 20% decrease
After a 20% discount, a laptop costs $640. What was the original price?
Step 1: Understand what we know: After 20% off, the price is $640 If 20% is taken off, we're paying 80% of the original
Step 2: Set up the equation: Let x = original price 80% of x = $640 0.80x = 640
Step 3: Solve for x: x = 640/0.80 x = 800
Step 4: Check: 20% off of 160 Sale price = 800 - 160 = $640 ✓
Alternative check: 80% of 640 ✓
Answer: $800