📊 Ratios, Rates, Proportions — Reference

Essential concepts and definitions for working with ratios, rates, and proportions in MATHCOUNTS.

Ratios

Basic Concepts

Definition: A ratio compares two quantities and can be written as $a:b$, $\frac{a}{b}$, or $a$ to $b$.

Types of ratios:

• Part-to-part: Compares parts of a whole (e.g., 3:4 boys to girls)
• Part-to-whole: Compares part to total (e.g., 3:7 boys to total students)
• Whole-to-part: Compares total to part (e.g., 7:3 total to boys)

Ratio Properties

Order matters: $a:b \neq b:a$ (unless $a = b$) Equivalent ratios: $a:b = ka:kb$ for any $k \neq 0$ Simplest form: Divide by GCF of $a$ and $b$

Examples:

• $6:8 = 3:4$ (simplified)
• $15:20 = 3:4$ (equivalent to above)

Ratio Operations

Addition: $a:b + c:d = (ad + bc):bd$ Subtraction: $a:b - c:d = (ad - bc):bd$ Multiplication: $(a:b) \times (c:d) = ac:bd$ Division: $(a:b) \div (c:d) = ad:bc$

Rates

Basic Concepts

Definition: A rate compares two different quantities with different units.

Common rate types:

• Speed: Distance per unit time (mph, km/h)
• Price: Cost per unit (dollars per pound)
• Work: Amount per unit time (pages per hour)
• Flow: Volume per unit time (gallons per minute)

Rate Calculations

Formula: Rate = $\frac{\text{Quantity 1}}{\text{Quantity 2}}$ Unit rate: Rate with denominator of 1

Examples:

• Speed: 60 miles in 2 hours = 30 mph
• Price: $15 for 3 pounds = $5 per pound
• Work: 24 pages in 3 hours = 8 pages per hour

Proportions

Basic Concepts

Definition: A proportion states that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$

Cross products: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$ Means and extremes: In $a:b = c:d$, $a$ and $d$ are extremes, $b$ and $c$ are means

Solving Proportions

Method 1: Cross multiply

• $\frac{a}{b} = \frac{c}{d}$ becomes $ad = bc$

Method 2: Use equivalent ratios

• $\frac{a}{b} = \frac{c}{d}$ becomes $\frac{a}{c} = \frac{b}{d}$

Method 3: Scale factor

• If $\frac{a}{b} = \frac{c}{d}$, then $c = \frac{ad}{b}$

Direct and Inverse Proportions

Direct Proportion

Definition: As one quantity increases, the other increases proportionally Formula: $y = kx$ where $k$ is the constant of proportionality Graph: Straight line through origin

Examples:

• Distance = Speed × Time
• Cost = Price × Quantity
• Work = Rate × Time

Inverse Proportion

Definition: As one quantity increases, the other decreases proportionally Formula: $y = \frac{k}{x}$ where $k$ is the constant of proportionality Graph: Hyperbola

Examples:

• Time = Distance ÷ Speed
• Pressure = Force ÷ Area
• Time = Work ÷ Rate

Scale and Similarity

Scale Factor

Definition: The ratio of corresponding lengths in similar figures Formula: Scale factor = $\frac{\text{New length}}{\text{Original length}}$

Applications:

• Maps and blueprints
• Model building
• Photography
• Similar triangles

Similar Figures

Definition: Figures with same shape but different size Properties:

• Corresponding angles are equal
• Corresponding sides are proportional
• Ratio of areas = (scale factor)²
• Ratio of volumes = (scale factor)³

Unit Conversions

Length Conversions

Metric Conversions

Time Conversions

Common Applications

Mixture Problems

Template: Use weighted averages Formula: $\frac{a_1 \times p_1 + a_2 \times p_2 + … + a_n \times p_n}{a_1 + a_2 + … + a_n}$

Example: Mix 2 liters of 20% solution with 3 liters of 80% solution

• $\frac{2 \times 0.20 + 3 \times 0.80}{2 + 3} = \frac{0.4 + 2.4}{5} = 0.56 = 56%$

Work Problems

Template: Use combined rates Formula: $\frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2} + … + \frac{1}{t_n}$

Example: Pipe A fills tank in 3 hours, Pipe B in 6 hours. Together: $\frac{1}{t} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2}$, so $t = 2$ hours

Distance Problems

Template: Use $d = rt$ formula Variations:

• Same direction: $d_1 - d_2 = r_1 t - r_2 t = (r_1 - r_2)t$
• Opposite direction: $d_1 + d_2 = r_1 t + r_2 t = (r_1 + r_2)t$

Problem-Solving Strategies

Setting Up Equations

1. Identify variables and what they represent
2. Write ratios in fraction form
3. Set up proportion using cross products
4. Solve for unknown variable
5. Check answer for reasonableness

Common Patterns

Part-whole: $\frac{\text{part}}{\text{whole}} = \frac{\text{part}}{\text{whole}}$ Rate-time-distance: $\frac{d_1}{t_1} = \frac{d_2}{t_2}$ Work-rate: $\frac{w_1}{t_1} = \frac{w_2}{t_2}$ Price-quantity: $\frac{p_1}{q_1} = \frac{p_2}{q_2}$

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