Understanding Ratios and Proportions

Ratios

• A ratio compares two numbers with the same units.
• Ratios can be expressed in multiple formats: as "a to b" (2:3), decimal (2/3), or word form (2 to 3).

Rates

• A rate compares two numbers with different units, e.g., miles per hour or words per minute.
• A unit rate is a specific type of rate with a denominator of 1.

Unit Rates

• To find the unit rate, divide the numerator by the denominator.
• Example: From 24 miles and 3 gallons, the unit rate is 8 miles per gallon.

Proportions

• A proportion compares two ratios and is evident when two ratios are equal.
• Cross-multiplication can be used to determine if two ratios are proportional.

Checking Proportionality

• Example of proportional ratios: 1/3 = 2/6 (yes), 3/12 = 1/4 (yes).
• Example of non-proportional ratios: 3/4 = 8/10 (no).

Solving Proportions

• When solving for a missing number in proportions, cross-multiply and then divide by the remaining number.

Constant of Proportionality

• This reflects the unit rate in a graph, defined as Y/X.
• Graphs can be considered proportional if they pass through the origin.

Recognizing Proportional Graphs

• A graph is considered proportional if the line passes through (0,0) and two points from it form a proportion.
• Example: If a graph shows a line through the origin and maintains consistent ratios, it is proportional.

Word Problems Involving Ratios and Proportions

• Example: To find how many root beer cans can be bought with $10 when 6 cans cost $2.50, set up the equation $2.50/6 = $10/x and solve to find 24 cans.
• Example: David painting rooms involves setting up their rates; if 3 rooms take 7 hours, then 15 rooms would take 35 hours.

Application of Ratios in Real Contexts

• In the context of a school with 750 girls and a boy-to-girl ratio of 3:5, solve for the number of boys by using the equation (x/750 = 3/5) to find 450 boys.