TEAS® Math Study Guide
Ratios, Proportions & Rates
A ratio is a comparison between two quantities. It tells you how much of one thing there is compared to another.
Example:
In a classroom with 12 boys and 16 girls:
- The ratio of boys to girls is \(12:16\) → Simplify to \(3:4\)
- The ratio of girls to total students is \(16:28\) → Simplify to \(4:7\)
Tip: Always simplify ratios, just like fractions.
Ratios can be written in the following ways:
- Using a colon: \(3:2\)
- As a fraction: \(\dfrac{3}{2}\)
- Using the word "to": 3 to 2
Proportions
A proportion is a statement that two ratios are equal. It’s often used to solve for unknown values when one part of the ratio is missing.
A proportion can be represented as two equal ratios written in fraction form, like \(\dfrac{a}{b} : \dfrac{c}{d}\)
It can also be shown using a colon format: \(a : b : c : d\)
A missing part in a ratio can be solved by setting up a proportion as two equal fractions and cross-multiplying to find the unknown.
Example:
If 5 notebooks cost $20, how much would 8 notebooks cost?
Set up the proportion first with the unknown quantity.
\(5 : 20 : 8 : x\) or \(\dfrac{5}{20} : \dfrac{8}{x}\)
Cross-multiply and solve:
\(5x = 20 \times 8\)\(5x = 160\)
\(x = 32\)
Thus, 8 notebooks cost $32.
Rates
A unit rate compares a quantity to 1 unit of another quantity. It’s especially useful for comparing prices or speeds.
Example:
A pack of 6 batteries costs $9.00. To find the cost per battery (unit rate): \(\dfrac{9.00}{6} = 1.50\)
Each battery costs $1.50.
A rate of change describes how one quantity changes in relation to another. In real-world terms, it often means speed, like kilometers per hour or dollars per month.
Example:
A train travels 240 km in 3 hours.
Rate of change or speed = \(\text{Distance} ÷ \text{Time} = 240 ÷ 3 = 80 \ \text{km/h}\)
Real-World Applications
Ratios, proportions, and rates are everywhere in life. Here are a few scenarios where they are useful:
Cooking Recipes
If a recipe uses 2 cups of flour for 4 servings, how much flour is needed for 10 servings?
\(2 : 4: x : 10\) or \(\dfrac{2}{4} : \dfrac{x}{10}\)
\(4x = 2 \times 10\)\(4x = 20\)
\(x = 5\)
10 servings would require 5 cups of flour.
Maps & Scale
A map key notes that 1 cm = 5 km. If two towns are 6 cm apart on the map, how far apart are they in the real world?
\(1 : 5 : 6 : x\) or \(\dfrac{1}{5} : \dfrac{6}{x}\)
\(1x= 5 \times 6\)\(x = 30\)
The towns are 30 km apart.
Products and Pricing
When comparing two products, use unit rates to see which is cheaper.
- 1.2 kg of rice costs $4.80 → $4.00 per kg
- 1.5 kg of rice costs $6.00 → $4.00 per kg
They cost the same per kilogram.
Speed and Travel
A cyclist covers 90 km in 3 hours. How long did she take to cover 60 km?
\(90 : 3 : 60 : x\) or \(\dfrac{90}{3} : \dfrac{60}{x}\)
\(90x = 3 \times 60\)\(90x = 180\)
\(x = 2\)
She covered 60 km in 2 hours.
Key Points
- Ratios compare two values.
- Proportions solve problems when one part of a ratio is unknown.
- Unit rates simplify comparisons.
- Rates of change are useful for analyzing speed and trends.
These skills apply to money, shopping, recipes, maps, time, and other real-world situations.