Advanced Topics
In a nutshell: Use ratios, proportions, and rates to compare and solve problems involving different quantities.
## Comparing and Relating Quantities
Ratios, proportions, and rates help you make sense of how things relate to each other, from recipes to travel.
### Ratios
Show the relationship between two quantities. If there are 2 cats for every 3 dogs, the ratio is 2:3.
### Proportions
A proportion is when two ratios are equal, like \( \frac{2}{3} = \frac{4}{6} \).
### Rates
A rate compares things with different units, like miles per hour or dollars per pound.
### Solving Proportions
You can solve for unknowns using cross-multiplication: If \( \frac{a}{b} = \frac{c}{d} \), then \( ad = bc \).
### Everyday Uses
- Cooking (recipes)
- Comparing prices
- Figuring out speed or cost per minute
\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc
Examples
- If 3 pencils cost $1.50, the rate is $0.50 per pencil.
- A recipe calls for 2 cups of flour for every 3 eggs. For 6 eggs, you need 4 cups of flour.
Key terms
- Ratio
- A comparison of two quantities by division.
- Rate
- A ratio comparing two quantities with different units.
- Proportion
- An equation stating that two ratios are equal.