Pre-Algebra Unit 6 Review: Percents

What Are Percents?

• Percents represent a part of a whole, where the whole is always 100
• The word "percent" comes from the Latin phrase "per centum," which means "by the hundred"
• Percents are denoted using the % symbol, which is read as "percent" (50%)
• Percents are used to express fractions or ratios with a denominator of 100
  • For example, 25% is equivalent to the fraction 25/100 or the ratio 25:100
• Percents are commonly used in everyday life to express portions of a whole (taxes, discounts, tips)
• Percents can be greater than 100% when the part exceeds the whole (150% of the original price)
• Understanding percents is essential for solving various real-world problems involving proportions and ratios

Converting Between Percents, Decimals, and Fractions

• Percents, decimals, and fractions are different ways to represent parts of a whole
• To convert a percent to a decimal, divide the percent by 100 and remove the % symbol
  • For example, 75% = 75 ÷ 100 = 0.75
• To convert a decimal to a percent, multiply the decimal by 100 and add the % symbol
  • For example, 0.4 = 0.4 × 100 = 40%
• To convert a percent to a fraction, write the percent as a fraction over 100 and simplify if possible
  • For example, 60% = 60/100 = 3/5 (simplified)
• To convert a fraction to a percent, divide the numerator by the denominator, multiply by 100, and add the % symbol
  • For example, 2/5 = (2 ÷ 5) × 100 = 40%
• When converting between percents, decimals, and fractions, it's essential to keep the place values and equivalences in mind
• Practice converting between the three forms to develop fluency and understanding

Calculating Percentages

• To calculate a percentage of a given number, convert the percent to a decimal and multiply it by the number
  • For example, to find 30% of 80, convert 30% to 0.3 and multiply: 0.3 × 80 = 24
• To find what percent one number is of another, divide the first number by the second and multiply by 100
  • For example, to find what percent 15 is of 60, divide 15 by 60 and multiply by 100: (15 ÷ 60) × 100 = 25%
• To find a number when a percentage of it is known, divide the known part by the percentage in decimal form
  • For example, if 25% of a number is 30, to find the original number, divide 30 by 0.25: 30 ÷ 0.25 = 120
• When calculating percentages, pay attention to the wording of the problem to determine the appropriate operation
• Double-check your calculations and ensure the final answer makes sense in the context of the problem

Percent Increase and Decrease

• A percent increase occurs when a value grows by a certain percentage (price increase, population growth)
  • To calculate a percent increase, find the difference between the new and original values, divide by the original value, and multiply by 100
  • For example, if a price increases from $50 to $60, the percent increase is: $((60 - 50) ÷ 50) × 100 = 20\\%$
• A percent decrease occurs when a value diminishes by a certain percentage (discount, population decline)
  • To calculate a percent decrease, find the difference between the original and new values, divide by the original value, and multiply by 100
  • For example, if a price decreases from $80 to $60, the percent decrease is: $((80 - 60) ÷ 80) × 100 = 25\\%$
• To find a new value after a percent increase or decrease, convert the percent to a decimal, multiply it by the original value, and add or subtract the result from the original value
  • For a 15% increase from $200: $200 + (0.15 × 200) = $230$
  • For a 20% decrease from $150: $150 - (0.2 × 150) = $120$

Percent of Change

• Percent of change measures the change in a value over time, expressed as a percentage of the original value
• To calculate percent of change, find the difference between the new and original values, divide by the original value, and multiply by 100
  • Percent of change = $((New Value - Original Value) ÷ Original Value) × 100$
• The percent of change can be positive (increase) or negative (decrease)
  • For example, if a population grows from 1,000 to 1,250, the percent of change is: $((1,250 - 1,000) ÷ 1,000) × 100 = 25\\%$ (increase)
  • If a stock price falls from $100 to $80, the percent of change is: $((80 - 100) ÷ 100) × 100 = -20\\%$ (decrease)
• Percent of change is useful for comparing growth or decline rates across different quantities or time periods
• When interpreting percent of change, consider the context and the original values to understand the significance of the change

Real-World Applications

• Percents are widely used in various real-world situations to express parts of a whole or to compare values
• In finance, percents are used to calculate interest rates, tax rates, and investment returns
  • For example, a savings account with a 3% annual interest rate earns 3% of the account balance each year
• In retail, percents are used to express discounts, markups, and sales tax
  • A 20% discount on a $50 item reduces the price by $10 (0.2 × $50), resulting in a sale price of $40
• In statistics, percents are used to express proportions, probabilities, and survey results
  • If 250 out of 500 surveyed people prefer a product, 50% (250 ÷ 500 × 100) of the surveyed population prefers that product
• Understanding how to apply percent concepts to real-world situations is crucial for making informed decisions and solving practical problems

Common Percent Problems

• Finding a percentage of a whole: Calculate the specified percent of a given value
  • Example: What is 30% of 80? (0.3 × 80 = 24)
• Finding a whole when a percentage is known: Determine the original value when a part and its percentage are given
  • Example: If 25% of a number is 30, what is the number? (30 ÷ 0.25 = 120)
• Percent increase or decrease: Calculate the new value after a percent increase or decrease
  • Example: If a $50 item is discounted by 15%, what is the new price? ($50 - (0.15 × $50) =$42.50)
• Percent of change: Determine the percent change between an original and new value
  • Example: If a population grows from 1,000 to 1,250, what is the percent of change? (((1,250 - 1,000) ÷ 1,000) × 100 = 25%)
• Percent error: Calculate the difference between an estimated and actual value, expressed as a percentage of the actual value
  • Example: If the estimated length is 18 cm and the actual length is 20 cm, what is the percent error? (((18 - 20) ÷ 20) × 100 = -10%)

Tips and Tricks for Working with Percents

• To quickly calculate 10% of a number, move the decimal point one place to the left (10% of 80 = 8)
• To find 1% of a number, divide the number by 100 (1% of 75 = 0.75)
• When adding or subtracting percentages, convert them to decimals or fractions with a common denominator
  • For example, to add 20% and 30%, convert to decimals (0.2 + 0.3 = 0.5) or fractions (1/5 + 3/10 = 1/2)
• To calculate a percentage of a percentage, convert both percentages to decimals and multiply
  • For example, to find 25% of 40%, convert to decimals (0.25 × 0.4 = 0.1) and multiply by 100 to get the percentage (10%)
• When solving percent problems, use the proportion method or the equation "Part = Percent × Whole" to set up and solve equations
• Always double-check your work and ensure the final answer makes sense in the context of the problem
• Practice regularly with a variety of percent problems to build confidence and mastery of the concepts