Percent ages. Grade 7 Mathematics: Percentages - Understanding and Calculating Percentage Values Subtopic Navigator Understanding Percentages Basic Percentage Calculations Finding Percentage of a Number Finding Original Value Percentage Increase Percentage Decrease Reverse Percentage Problems Successive Percentage Changes Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Understand percentages as fractions with denominator 100 Calculate percentages of quantities using multiple methods Solve percentage increase and decrease problems Work backwards from percentages to find original values Calculate successive percentage changes Apply percentage concepts to real-world scenarios Understanding Percentages Percentages are a way of expressing fractions or proportions out of 100. The term "percent" comes from the Latin "per centum," meaning "by the hundred." Percentages are widely used in everyday life for discounts, interest rates, statistics, and many other applications where relative comparisons are important. Basic Percentage Calculations To calculate a percentage of a number, we can convert the percentage to a decimal or fraction and then multiply. The basic formula is: Percentage of a number = $frac{percentage}{100} × number$. Example 1: Multiple Percentage Calculations Calculate 35% of 480 using three different methods. Solution: Method 1: Decimal Conversion 35% = 0.35 0.35 × 480 = 168 Method 2: Fraction Conversion 35% = $frac{35}{100} = frac{7}{20}$ $frac{7}{20} × 480 = frac{7 × 480}{20} = frac{3360}{20} = 168$ Method 3: 1% Method 1% of 480 = 4.8 35% = 35 × 4.8 = 168 Example 2: Complex Percentage Problem If 15% of a number is 72, what is 40% of the same number? Solution: Let the number be x 15% of x = 72 $frac{15}{100} × x = 72$ $x = 72 × frac{100}{15} = 72 × frac{20}{3} = 24 × 20 = 480$ Now find 40% of 480: 40% of 480 = 0.4 × 480 = 192 Or: $frac{40}{100} × 480 = frac{2}{5} × 480 = 192$ Basic Percentage Problems Calculate 28% of 650 using two different methods If 12% of a number is 54, what is 75% of the same number? What is 45% of 320? If 8% of x is 24, what is 125% of x? Which is greater: 30% of 450 or 45% of 300? Finding Percentage of a Number To find what percentage one number is of another, we use the formula: Percentage = $frac{part}{whole} × 100%$. This tells us how many parts per hundred the part represents of the whole. Example 1: Complex Percentage Finding What percentage is 42 of 56? Express your answer as a mixed number percentage. Solution: Percentage = $frac{42}{56} × 100%$ Simplify fraction: $frac{42}{56} = frac{3}{4}$ $frac{3}{4} × 100% = 75%$ This can also be expressed as $75frac{0}{1}%$, but usually just 75% Example 2: Multiple Percentage Comparisons A class has 18 boys and 24 girls. What percentage of the class are boys? What percentage are girls? Express as fractions of percentages if needed. Solution: Total students = 18 + 24 = 42 Percentage boys = $frac{18}{42} × 100% = frac{3}{7} × 100% ≈ 42.857%$ As a fraction: $42frac{6}{7}%$ Percentage girls = $frac{24}{42} × 100% = frac{4}{7} × 100% ≈ 57.143%$ As a fraction: $57frac{1}{7}%$ Check: $42frac{6}{7}% + 57frac{1}{7}% = 100%$ ✓ Finding Percentage Problems What percentage is 27 of 45? If a test has 40 questions and a student gets 34 correct, what percentage did they score? What percentage is 18 of 72? Express as a mixed number percentage if necessary A recipe calls for 3 cups of flour and 1 cup of sugar. What percentage of the total is sugar? Which represents a larger percentage: 15 out of 25 or 18 out of 30? Finding Original Value Sometimes we know the percentage and the result after applying that percentage, and we need to find the original value. This requires working backwards using division instead of multiplication. Example 1: Finding Original Before Percentage After a 20% discount, a shirt costs $36. What was the original price? Solution: After 20% discount, you pay 80% of original price Let original price = x 80% of x = 36 0.8x = 36 x = $frac{36}{0.8} = 45$ Original price = $45 Check: 20% of 45 = 9, 45 - 9 = 36 ✓ Example 2: Complex Original Value Problem If 35% of a number is added to 120, the result is 218. What is the number? Solution: Let the number be x 35% of x + 120 = 218 0.35x + 120 = 218 0.35x = 218 - 120 = 98 x = $frac{98}{0.35} = 280$ The number is 280 Check: 35% of 280 = 98, 98 + 120 = 218 ✓ Finding Original Value Problems After a 15% discount, a book costs $25.50. What was the original price? If 28% of a number is 70, what is the number? When 45% is added to a number, the result is 203. What is the number? A price increased by 25% to become $150. What was the original price? If $frac{3}{5}$ of a number is 48, what is 125% of the number? Percentage Increase Percentage increase measures how much a quantity has grown relative to its original value. The formula is: Percentage increase = $frac{increase}{original} × 100%$. Example 1: Complex Percentage Increase A house's value increased from $180,000 to $225,000. What is the percentage increase? Solution: Increase = $225,000 - $180,000 = $45,000 Percentage increase = $frac{45,000}{180,000} × 100%$ = $frac{1}{4} × 100% = 25%$ Alternative method using ratio: New value = $225,000 = 1.25 × $180,000 So increase is 25% Example 2: Percentage Increase Application A company's profits were $45,000 last year and increased by 20% this year. Next year, they expect a further 15% increase. What will be next year's projected profit? Solution: This year's profit = $45,000 × 1.20 = $54,000 Next year's projected profit = $54,000 × 1.15 = $62,100 Note: This is not a 35% increase from original: $45,000 × 1.35 = $60,750 ≠ $62,100 Successive percentage increases multiply, not add Percentage Increase Problems A salary increased from $42,000 to $48,300. What is the percentage increase? If a number increases from 125 to 150, what is the percentage increase? A population grows from 8,400 to 9,660. What is the percentage increase? A price increases by 15% to become $115. What was the original price? Which represents a larger percentage increase: from 80 to 100 or from 120 to 150? Percentage Decrease Percentage decrease measures how much a quantity has reduced relative to its original value. The formula is: Percentage decrease = $frac{decrease}{original} × 100%$. Example 1: Complex Percentage Decrease A car's value decreased from $25,000 to $18,750 over 3 years. What is the percentage decrease? Solution: Decrease = $25,000 - $18,750 = $6,250 Percentage decrease = $frac{6,250}{25,000} × 100%$ = $frac{1}{4} × 100% = 25%$ Alternative method using ratio: New value = $18,750 = 0.75 × $25,000 So decrease is 25% (since 100% - 75% = 25%) Example 2: Percentage Decrease Application A store reduces a $240 coat by 25%, then has an additional 20% off sale. What is the final price? Solution: First reduction: $240 × 0.75 = $180 Second reduction: $180 × 0.80 = $144 Total reduction from original: $frac{240 - 144}{240} × 100% = frac{96}{240} × 100% = 40%$ Note: This is not 25% + 20% = 45% reduction Actual: 100% - (0.75 × 0.80 × 100%) = 100% - 60% = 40% Percentage Decrease Problems A stock price fell from $84 to $63. What is the percentage decrease? If a number decreases from 200 to 150, what is the percentage decrease? A store item was $85 and is now $59.50. What is the percentage discount? A price decreases by 30% to become $105. What was the original price? Which represents a larger percentage decrease: from 100 to 75 or from 150 to 100? Reverse Percentage Problems Reverse percentage problems involve finding the original value when we know the final value and the percentage change that occurred. These require careful attention to whether the percentage was applied as an increase or decrease. Example 1: Reverse Percentage Increase After a 15% increase, a salary is $46,000. What was the original salary? Solution: After 15% increase, salary is 115% of original Let original salary = x 115% of x = $46,000 1.15x = 46,000 x = $frac{46,000}{1.15} = 40,000$ Original salary = $40,000 Check: 15% of $40,000 = $6,000, $40,000 + $6,000 = $46,000 ✓ Example 2: Complex Reverse Percentage A population decreased by 12% and then increased by 15% to become 36,708. What was the original population? Solution: Let original population = x After 12% decrease: 0.88x After 15% increase: 0.88x × 1.15 = 1.012x 1.012x = 36,708 x = $frac{36,708}{1.012} = 36,300$ Original population = 36,300 Check: 12% of 36,300 = 4,356, 36,300 - 4,356 = 31,944 15% of 31,944 = 4,791.6, 31,944 + 4,791.6 = 36,735.6 (rounding difference due to decimal places) Reverse Percentage Problems After a 20% increase, a price is $144. What was the original price? After a 30% decrease, a number is 84. What was the original number? A value increased by 25% then decreased by 20% to become 300. What was the original value? After two successive 10% increases, a salary is $48,510. What was the original salary? If 80% of a number is 64, what is 125% of that number? Successive Percentage Changes When multiple percentage changes occur in sequence, we multiply the successive multipliers. The overall percentage change is not simply the sum of individual percentage changes. Example 1: Successive Increases and Decreases A price increases by 20%, then decreases by 25%, then increases by 10%. What is the overall percentage change? Solution: Let original price = 100 (for easy calculation) After 20% increase: 100 × 1.20 = 120 After 25% decrease: 120 × 0.75 = 90 After 10% increase: 90 × 1.10 = 99 Overall change: 99 - 100 = -1 Percentage change = $frac{-1}{100} × 100% = -1%$ (1% decrease) Note: This is NOT 20% - 25% + 10% = 5% increase Example 2: Complex Successive Changes An investment loses 15% in the first year, gains 20% in the second year, and loses 10% in the third year. What is the overall percentage change? Solution: Let initial investment = 100 After 15% loss: 100 × 0.85 = 85 After 20% gain: 85 × 1.20 = 102 After 10% loss: 102 × 0.90 = 91.8 Overall change: 91.8 - 100 = -8.2 Percentage change = $frac{-8.2}{100} × 100% = -8.2%$ (8.2% decrease) Successive Percentage Problems A price increases by 10% then decreases by 10%. What is the overall percentage change? An investment gains 15%, then loses 20%, then gains 25%. What is the overall percentage change? A salary increases by 12% then by another 8%. What is the overall percentage increase? Which results in a higher final value: two successive 10% increases or one 20% increase? A number decreases by 30% then increases by 40%. What is the overall percentage change? Real-World Applications Percentage calculations are essential in everyday life for understanding discounts, interest rates, tax calculations, statistical data, and many financial decisions. Example 1: Sales Tax and Discount A computer is priced at $850. There's a 15% discount, and then 8% sales tax is added. What is the final price? Solution: After 15% discount: $850 × 0.85 = $722.50 Add 8% tax: $722.50 × 1.08 = $780.30 Final price = $780.30 Alternative: Combined multiplier = 0.85 × 1.08 = 0.918 $850 × 0.918 = $780.30 ✓ Example 2: Population Growth A town's population was 24,000. It grew by 5% in the first year and 8% in the second year. What is the population after two years? What was the average annual growth rate? Solution: After first year: 24,000 × 1.05 = 25,200 After second year: 25,200 × 1.08 = 27,216 Population after 2 years = 27,216 Average annual growth rate: Overall growth factor = $frac{27,216}{24,000} = 1.134$ Average annual rate = $(1.134)^{1/2} - 1 ≈ 1.0649 - 1 = 0.0649 = 6.49%$ (Not simply $frac{5% + 8%}{2} = 6.5%$) Real-World Application Problems A TV is priced at $600 with 25% off. If sales tax is 7.5%, what is the final price? An investment of $5,000 grows by 6% in the first year and 8% in the second year. What is it worth after 2 years? A shirt costs $45 after a 25% discount. What was the original price? If a restaurant bill is $85 and you want to leave a 18% tip, how much should you pay total? A car depreciates 15% in the first year and 10% in the second year. If it was originally $28,000, what is its value after 2 years? Cumulative Exercises Calculate 28% of 450 What percentage is 36 of 48? After a 15% increase, a salary is $46,000. What was the original salary? A price decreases by 20% to become $96. What was the original price? If 40% of a number is 64, what is 125% of that number? A population grows from 25,000 to 28,750. What is the percentage increase? An item costs $120 after a 20% discount. What was the original price? Which is larger: 35% of 280 or 28% of 350? A value increases by 10% then decreases by 10%. What is the overall percentage change? If $frac{3}{4}$ of students are present and there are 18 absent, how many students are there total? Show/Hide Answers Problem: Calculate 28% of 450 Answer: 28% = 0.28 0.28 × 450 = 126 Or: $frac{28}{100} × 450 = frac{28 × 450}{100} = frac{12600}{100} = 126$ Problem: What percentage is 36 of 48? Answer: $frac{36}{48} × 100% = 0.75 × 100% = 75%$ Problem: After a 15% increase, a salary is $46,000. What was the original salary? Answer: 115% of original = $46,000 Original = $frac{46,000}{1.15} = 40,000$ Original salary = $40,000 Problem: A price decreases by 20% to become $96. What was the original price? Answer: 80% of original = $96 Original = $frac{96}{0.8} = 120$ Original price = $120 Problem: If 40% of a number is 64, what is 125% of that number? Answer: Let the number be x 0.4x = 64, so x = $frac{64}{0.4} = 160$ 125% of 160 = 1.25 × 160 = 200 Problem: A population grows from 25,000 to 28,750. What is the percentage increase? Answer: Increase = 28,750 - 25,000 = 3,750 Percentage increase = $frac{3,750}{25,000} × 100% = 0.15 × 100% = 15%$ Problem: An item costs $120 after a 20% discount. What was the original price? Answer: 80% of original = $120 Original = $frac{120}{0.8} = 150$ Original price = $150 Problem: Which is larger: 35% of 280 or 28% of 350? Answer: 35% of 280 = 0.35 × 280 = 98 28% of 350 = 0.28 × 350 = 98 They are equal Problem: A value increases by 10% then decreases by 10%. What is the overall percentage change? Answer: Let original = 100 After 10% increase: 110 After 10% decrease: 110 × 0.9 = 99 Overall change: 99 - 100 = -1 Percentage change = -1% Problem: If $frac{3}{4}$ of students are present and there are 18 absent, how many students are there total? Answer: $frac{3}{4}$ present means $frac{1}{4}$ absent $frac{1}{4}$ of total = 18 Total = 18 × 4 = 72 students Conclusion/Recap Percentage calculations are fundamental mathematical skills with wide-ranging applications in daily life, business, finance, and statistics. Mastery of percentage concepts enables informed decision-making regarding discounts, investments, population changes, and data interpretation. Understanding that successive percentage changes multiply rather than add, and being able to work backwards from percentages to find original values, are particularly important skills for mathematical literacy and practical problem-solving. Clip It! Share your ANSWER in the Chat. 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