Proportions

Unit Review Sheet

Facts and Definitions

Lesson 1: Proportional Relationships

• A proportion is an equation that shows two equal ratios or fractions.
• A relationship is proportional if the ratio between two quantities stays the same when increased or decreased.
• To solve a proportion, you solve for $x$ in one of the fractions by using the eyeball method, multiplication/division method, or cross-multiplying.
• To check if two ratios are proportional, simplify both fractions to see if they are equal.

Lesson 2: Unit Rates

• A unit rate is a ratio that tells us "how much for one" of something.
• To find a unit rate, decide what you need per one unit (e.g., miles per hour). Then divide the total amount by the number of units.
• To compare unit rates, divide the total by the number of units for each option, then compare the results to see which is better.
• A complex fraction is just a fraction where the numerator, denominator, or both are also fractions.
• To use Keep-Change-Flip (KCF), keep the first fraction, change ÷ to ×, flip the second fraction, then multiply and simplify.

Lesson 3: Constant Rate

• The independent variable (x-value) is the one you choose or control.
• The dependent variable (y-value) is the result that changes based on the independent variable.
• A constant is a number that stays the same.
• The formula for determining the constant of proportionality is $k=\frac{y}{x}$.
• If every pair of numbers in a table or chart gives you the same k, the relationship is proportional.
• The graph of a proportional relationship will always be a line that begins or passes through the origin (0, 0).
• If the graph of a line represents a proportional relationship, then the ratio y/x of any ordered pair on the line will be the constant of proportionality, k.
• A proportional relationship shown as a line on a graph can be written as an equation: $y=kx$.
• Direct variation means that two values change at the same steady rate. It is expressed as an equation in the form $y=kx$

Lesson 4: Graphing Proportions

• In a proportional relationship when one quantity changes, the other changes at a constant rate.
• All proportional relationships can be expressed with an equation of the form: y = kx.
• If each pair of values in a table simplifies to the same ratio, the relationship is proportional.
• If the graph forms a straight line through (0, 0), the relationship is proportional.

Lesson 5: Proportional Relationship Equations

• When working with cost or money, y = kx is expressed as t = pn. (total cost = price per item x number of items).

Lesson 6: Taxes, Tips, and Commissions

• There are three main types of taxes: Sales Tax – Added to purchases at checkout; Property Tax – Paid on homes, land, or cars; and Income Tax – Deducted from earnings.
• Sales Tax = Price × Tax Rate
  Final Price = Original Price + Sales Tax
• Property Tax = Property Value × Tax Rate
• Income Tax = Income × Tax Rate
• Tip Amount = Original Bill × Tip Percentage
  Total Amount = Original Bill + Tip Amount
• Commission = Total Sales × Commission Percentage
  Total Earnings = Base Salary + Commission

Lesson 7: Markups and Discounts

• Discount = Original Price * Discount Percentage
• Discounted Price = Original Price - Discount
• Markup = Cost Price × Markup Percentage
• Selling Price = Cost Price + Markup
• For multiple discounts, apply each discount one at a time to the new price before calculating the next discount.
• Percent Change = ((New Value - Original Value) ÷ Original Value) × 100

Lesson 8: Simple Interest and Percent Error

• Simple Interest = Principal × Interest Rate (in decimal form) × Time (in years)
  Balance = Principal + Interest
• Percent error is a way to measure how accurate an estimate or measurement is compared to the actual value.
  Percent Error = (∣Actual Value - Estimated Value∣ / Actual Value) × 100.

Lesson 9: Unit 2 Test

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Final Project: Lemonade Stand

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