Cheatsheet: Rates And Percentages

1. Understanding Rates

1.1 Definition of Rate

1.2 Common Examples of Rates

• Speed: miles per hour (mph), kilometers per hour (km/h)
• Price: dollars per pound, cents per ounce
• Wage: dollars per hour
• Density: people per square mile
• Fuel efficiency: miles per gallon

1.3 Writing Rates

2. Calculating Unit Rates

2.1 Finding Unit Rate

2.2 Steps to Calculate Unit Rate

1. Identify the two quantities being compared
2. Write the rate as a fraction
3. Divide the numerator by the denominator
4. Write the answer with the appropriate unit

2.3 Example Calculations

3. Comparing Rates

3.1 Using Unit Rates to Compare

• Convert each rate to a unit rate
• Compare the unit rates to find the better deal
• Lower unit rate means better value for price comparisons
• Higher unit rate means better value for distance/speed comparisons

3.2 Comparison Example

4. Introduction to Percentages

4.1 Definition and Meaning

4.2 Understanding Percent

• 50% means 50 out of 100
• 25% means 25 out of 100 or 1/4
• 100% means the whole amount
• Percentages can be greater than 100%

5. Converting Between Forms

5.1 Percent to Decimal

5.2 Decimal to Percent

5.3 Percent to Fraction

5.4 Fraction to Percent

5.5 Common Conversions to Memorize

6. Finding Percent of a Number

6.1 Methods to Calculate

6.2 Example Problems

6.3 Mental Math Shortcuts

• 10% of a number: move decimal point one place left
• 50% of a number: divide by 2
• 25% of a number: divide by 4
• 1% of a number: move decimal point two places left
• 5% of a number: find 10%, then divide by 2

7. Finding What Percent One Number Is of Another

7.1 Formula and Method

7.2 Steps

1. Identify the part and the whole
2. Divide the part by the whole
3. Multiply the result by 100
4. Add the percent sign

7.3 Examples

8. Finding the Whole When Percent and Part Are Known

8.1 Formula and Method

8.2 Steps

1. Convert the percent to a decimal
2. Divide the part by the decimal
3. The result is the whole

8.3 Examples

9. Percent Increase and Decrease

9.1 Percent Increase

9.1.1 Example

• Price increases from $40 to $50
• Amount of increase = $50 - $40 = $10
• Percent increase = ($10 ÷ $40) × 100 = 25%

9.2 Percent Decrease

9.2.1 Example

• Price decreases from $80 to $60
• Amount of decrease = $80 - $60 = $20
• Percent decrease = ($20 ÷ $80) × 100 = 25%

9.3 Key Points

• Always use the original amount as the denominator
• Find the amount of change first (new - original)
• Amount of change can be positive (increase) or negative (decrease)

10. Applications of Percentages

10.1 Sales Tax

10.1.1 Example

• Item costs $50, sales tax is 8%
• Tax amount = $50 × 0.08 = $4
• Total cost = $50 + $4 = $54 or $50 × 1.08 = $54

10.2 Discounts

10.2.1 Example

• Item costs $80, 25% discount
• Discount amount = $80 × 0.25 = $20
• Sale price = $80 - $20 = $60 or $80 × 0.75 = $60

10.3 Tips and Gratuity

10.4 Simple Interest

10.4.1 Example

• Principal = $1000, rate = 5% per year, time = 2 years
• Interest = $1000 × 0.05 × 2 = $100

11. Word Problems with Rates and Percentages

11.1 Problem-Solving Steps

1. Read the problem carefully
2. Identify what is being asked
3. Identify the given information
4. Determine the appropriate formula or method
5. Solve and check if the answer makes sense

11.2 Key Words to Identify

11.3 Common Problem Types

• Finding unit rates for comparison shopping
• Calculating discounts and sale prices
• Computing tax and total cost
• Determining percent increase or decrease
• Finding tips and total restaurant bills
• Calculating distance, time, or speed using rates

12. Proportions and Rates

12.1 Setting Up Proportions

12.2 Solving Proportions with Rates

1. Set up the proportion with matching units
2. Cross multiply: a × d = b × c
3. Solve for the unknown variable
4. Check that units make sense

12.3 Example

• If 3 pounds cost $12, how much do 5 pounds cost?
• Set up: 3/12 = 5/x
• Cross multiply: 3x = 60
• Solve: x = 20 dollars

13. Quick Reference Formulas