Overview
This lecture focuses on solving proportions as practice for learning the foundational process of solving equations, emphasizing the importance of isolating variables.
Understanding Proportions as Equations
- A proportion is an equation showing two equal ratios (fractions).
- Solving a proportion involves finding the unknown variable that makes both ratios equal.
- Proportions can represent real-world relationships, such as comparing groups (e.g., boys to girls).
Steps to Solve Proportions
- Always aim to isolate the variable (get it alone on one side of the equation).
- If the variable is in the denominator, multiply both sides by that variable to move it to the numerator.
- Use the property that equal items on the top and bottom of a fraction (like x/x) cancel each other out (become 1).
- Perform the opposite operation to "undo" multiplication or division to isolate the variable.
Example Problems & Solutions
- For 1/3 = 15/x, solving gives x = 45.
- For w/24 = 4/6, multiply both sides by 24, yielding w = 16.
- For 4/10 = h/25, multiplying by 25 and simplifying gives h = 10.
- For 6/y = 15/20, first multiply both sides by y, then isolate y to get y = 8.
- For k/5 = 12/15, multiply both sides by 5, then k = 4.
- For 5/2 = 15/c, multiply both sides by c, then proceed to get c = 6.
Key Terms & Definitions
- Proportion — An equation stating that two ratios (fractions) are equal.
- Variable — A letter representing an unknown number.
- Isolate (a variable) — To get the variable alone on one side of the equation.
- Cancel — When a term appears in both numerator and denominator, it simplifies to 1.
Action Items / Next Steps
- Practice solving all example problems independently.
- Continue to the next lesson for further practice with solving equations and proportions.