Ratios and rates are powerful tools for comparing quantities and solving real-world problems. They help us understand relationships between numbers and make sense of complex information by breaking it down into simpler, more manageable parts.

From recipe measurements to unit prices, ratios and rates are everywhere in our daily lives. By mastering these concepts, you'll be better equipped to make informed decisions, whether you're shopping, cooking, or analyzing data in various fields.

Ratios and Rates

Ratios and rates as fractions

• Ratios compare two quantities by division written as $a:b$, $a$ to $b$, or $\frac{a}{b}$ (boys to girls, $3:2$, $3$ to $2$, or $\frac{3}{2}$)
• Rates compare two quantities with different units expressed as a fraction $\frac{\text{quantity 1}}{\text{quantity 2}}$ (car travels 120 miles in 3 hours, rate is $\frac{120 \text{ miles}}{3 \text{ hours}}$ or $\frac{40 \text{ miles}}{1 \text{ hour}}$)
• Ratios and rates are used for comparison between quantities

Calculation of unit rates

• Unit rates have a second quantity of 1 unit calculated by dividing the first quantity by the second quantity
  • 5 apples cost $2, unit price is $\frac{2 \text{ dollars}}{5 \text{ apples}} = \frac{0.4 \text{ dollars}}{1 \text{ apple}}$
• Unit prices represent the cost per unit of a product found by dividing the total cost by the number of units
  • 3 pounds of bananas cost $4.50, unit price is $\frac{4.50 \text{ dollars}}{3 \text{ pounds}} = \frac{1.50 \text{ dollars}}{1 \text{ pound}}$

Verbal to fractional conversions

• Identify quantities being compared in the verbal description
• Express the ratio or rate as a fraction with the first quantity as the numerator and the second quantity as the denominator
  • "For every 2 boys, there are 3 girls" written as $\frac{3 \text{ girls}}{2 \text{ boys}}$

Ratios vs rates in context

• Ratios compare quantities with the same units (recipe uses 2 cups flour for every 3 cups sugar, $\frac{2 \text{ cups flour}}{3 \text{ cups sugar}}$)
• Rates compare quantities with different units (car travels 60 miles in 2 hours, $\frac{60 \text{ miles}}{2 \text{ hours}}$)

Applications of ratio concepts

1. Identify given information and unknown quantity
2. Set up proportion using known ratio or rate
3. Solve proportion to find unknown quantity
   • Recipe calls for 2 cups flour for every 3 cups sugar, flour needed for 9 cups sugar:
     • Proportion: $\frac{2 \text{ cups flour}}{3 \text{ cups sugar}} = \frac{x \text{ cups flour}}{9 \text{ cups sugar}}$
     • Cross multiply and solve: $2 \times 9 = 3x$, $x = 6$
   • 6 cups flour needed for 9 cups sugar

Advanced ratio and rate concepts

• Scaling involves changing the size of quantities while maintaining their proportions
• Dimensional analysis uses unit conversion factors to solve problems involving different units of measurement
• Direct variation describes a relationship where one quantity changes proportionally with another, often expressed as $y = kx$ where $k$ is the constant of variation
• Percent represents a ratio expressed as a fraction of 100, useful for comparing parts to wholes