Lecture Notes: Ratios, Rates, Proportions, and Scale Drawings

Ratio: a comparison of two quantities by division or fraction. In symbols:
$\text{ratio} = \dfrac{A}{B}$
Rate: a ratio that compares two quantities with different units.
Unit rate: a rate with a denominator of one.
Proportion: an equation that states two ratios are equal.
Scale drawing: a drawing that uses a scale to represent an object's actual size.
Actual size: the real-world size of the object.

Ratios and rates are ways to compare quantities; rates specifically pair different units (e.g., miles per hour).
Unit rates express a single unit of the denominator (e.g., copies per minute).
Proportions link two ratios or fractions that are equal; solving involves cross-multiplication.
Scale drawings use a fixed scale to relate drawing units to real-world units; solving requires setting up a proportion to find an unknown real length.
The content connects to foundational ideas of division, fractions, and solving equations via cross-multiplication.
Real-world relevance: understanding recipes, speed, maps, and measurements in everyday problems.

Step 1: Write both quantities in the same units.
Step 2: Write the ratio in simplest form (divide numerator and denominator by their greatest common factor).
Step 3: Write the ratio as a pair (e.g., 15:10 or 15/10). Use the simplest form for clarity.

Step 1: Write the ratio as a fraction.
- Example: 240 copies in 8 minutes → $\frac{240}{8}$
Step 2: Divide the numerator by the denominator to find the unit rate.
- Example: $\frac{240}{8} = 30\,\text{copies per minute}$
Practical interpretation: unit rate tells you how many units per one unit of time (or per one item, etc.).

Step 1: Set up two equal fractions (or ratios). Examples: $\frac{a}{b} = \frac{c}{d}$ or $\frac{a}{b} = \frac{x}{y}$
Step 2: Use cross multiplication (the traditional method; sometimes called the butterfly method):
- If $\frac{a}{b} = \frac{c}{d}$ , then $ad = bc$ .
Step 3: Solve for the unknown by simplifying and checking that the result makes sense ("makes cents").
Note: In some problems you may also see setting up two equal fractions and then solving; cross multiplication is the standard quick method.

Example 1: Solve $\dfrac{3}{4} = \dfrac{x}{20}$
- Cross-multiply: $3\cdot 20 = 4x$ → $60 = 4x$ → $x = \dfrac{60}{4} = 15$
Example 2: Solve $\dfrac{5}{8} = \dfrac{x}{40}$
- Cross-multiply: $5\cdot 40 = 8x$ → $200 = 8x$ → $x = \dfrac{200}{8} = 25$
Quick note: Spiral-through approach is valid as long as you keep units consistent and use correct arithmetic.

Definition: A scale drawing uses a scale to represent actual size. For example, a map or blueprint uses a scale like "1 inch represents 50 miles."
Step 1: Identify the scale (e.g., 1 inch : 50 miles).
Step 2: Set up a proportion using the scale and the known measurement.
Step 3: Solve for the missing length.
Example A (map): Map scale is 1 inch to 50 miles. If two cities are 3.5 inches apart on the map, how many miles apart are they in real life?
- Proportion: $\frac{1\text{ in}}{50\text{ mi}} = \frac{3.5\text{ in}}{x\text{ mi}}$
- Solve: cross-multiply or rearrange to get $x = 50 \cdot 3.5 = 175\,\text{miles}.$
Example B (scale drawing): The map uses a scale of 1 cm to 2 m. If a wall in the drawing is 7 cm long, what is the real wall length?
- Proportion: $\frac{1\text{ cm}}{2\text{ m}} = \frac{7\text{ cm}}{x\text{ m}}$
- Solve: $x = 7 \times 2 = 14\,\text{m}.$
Important note on units: Always keep track of the units on both sides of a proportion and convert as needed to keep the equation consistent.

Example 1: A class has 15 boys and 10 girls. Write the ratio of boys to girls.
- Step 1: Write the ratio with the given quantities: $\frac{15}{10}$ or 15:10.
- Step 2: Simplify by the greatest common factor (GCF). GCF of 15 and 10 is 5.
- Simplified ratio: $\frac{15}{10} = \frac{3}{2}$ or 3:2.
Example 1 (alternative path): If there are 20 teachers and 480 students, write the ratio of teachers to students and simplify.
- Raw ratio: $\frac{20}{480}$
- GCF: 20. Simplified: $\frac{20}{480} = \frac{1}{24}$ or 1:24.
- Note on different paths: One could divide by 10 first to get $\frac{2}{48}$ and then divide by 2 to obtain $\frac{1}{24}$ . Both paths yield the same simplified ratio; note that the product of the factors used (10 and 2) multiplies to 20, the gcd.
Example 2: Unit rate from a rate problem (unit rate as a fraction): Unit rate can be found by writing the quantity as a fraction and simplifying.
- Example continuation: If 240 copies are produced in 8 minutes, the unit rate is
- Fraction form: $\frac{240}{8}$
- Quotient: $30$
- Unit rate: $30\ \text{copies per minute}$

Example: 18 miles in 3 hours.
- Write as a fraction: $\frac{18\text{ miles}}{3\text{ hours}}$
- Divide to obtain unit rate: $\frac{18}{3} = 6$
- Unit rate: $6\ \text{miles per hour}$ (often written as 6 mph).

Example 1: Solve for x in $\frac{3}{4} = \frac{x}{20}$ (see above).
- Answer: $x = 15$ .
Example 2: Solve for x in $\frac{5}{8} = \frac{x}{40}$ (see above).
- Answer: $x = 25$ .

Scale: 1 inch to 50 miles; 3.5 inches apart → real distance = $50 \times 3.5 = 175\text{ miles}.$
Scale: 1 centimeter to 2 meters; a wall is 7 centimeters on the drawing → real wall length = $7 \times 2 = 14\text{ meters}.$

Always start by aligning units in ratios and setting up consistent fractions.
For unit rates, remember the denominator should be 1 unit of the time (or other base unit). It helps to think of it as "per 1 unit."
Cross multiplication is a powerful, quick method for solving proportions; it directly uses the equality of two fractions.
When working with scale drawings, converting the drawing length to a real length relies on a direct proportion using the given scale.
If a problem seems messy, try breaking it into smaller steps: simplify the ratio first, then compute the unit rate, then apply proportions if needed.

Review the exit ticket on your own first, then compare with a peer to discuss methods and checks.
Use the steps outlined above as a checklist for similar problems in upcoming assessments.