Proportions - Grade 8 Mathematics

Comprehensive Short Notes & Formulae

1. Understanding Proportions

Definition:

Proportion: An equation stating that two ratios are equal

General Form: \(\frac{a}{b} = \frac{c}{d}\) or \(a:b = c:d\)

Read as: "a is to b as c is to d"

Key Terms:

Means: The inner terms (b and c in \(a:b::c:d\))

Extremes: The outer terms (a and d in \(a:b::c:d\))

Property: Product of means = Product of extremes

Formula: \(b \times c = a \times d\)

Examples:

\(\frac{2}{3} = \frac{4}{6}\) is a proportion (both equal \(0.\overline{6}\))

\(3:5 = 6:10\) is a proportion (cross products: \(3 \times 10 = 5 \times 6 = 30\))

2. Solving Proportions

Cross-Multiplication Method:

Formula: If \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\)

Steps to Solve:

Step 1: Write the proportion in fraction form

Step 2: Cross-multiply (multiply diagonally)

Step 3: Solve for the unknown variable

Step 4: Check your answer by substituting back

Example 1 - Basic Proportion:

Solve: \(\frac{x}{12} = \frac{5}{6}\)

Step 1: Cross-multiply: \(x \times 6 = 12 \times 5\)

Step 2: \(6x = 60\)

Step 3: \(x = \frac{60}{6} = 10\)

Answer: \(x = 10\)

Example 2 - Variable in Different Position:

Solve: \(\frac{3}{y} = \frac{9}{15}\)

Step 1: Cross-multiply: \(3 \times 15 = y \times 9\)

Step 2: \(45 = 9y\)

Step 3: \(y = \frac{45}{9} = 5\)

Answer: \(y = 5\)

3. Solving Proportion Word Problems

Strategy for Word Problems:

Step 1: Identify what you're looking for (the unknown)

Step 2: Set up ratios with matching units (top/bottom must match)

Step 3: Write the proportion equation

Step 4: Cross-multiply and solve

Step 5: Check if your answer makes sense in context

Example - Recipe Problem:

Problem: A recipe for 4 people needs 3 cups of flour. How much flour is needed for 10 people?

Setup: \(\frac{\text{cups of flour}}{\text{people}} = \frac{3}{4} = \frac{x}{10}\)

Cross-multiply: \(4x = 3 \times 10\)

\(4x = 30\)

\(x = 7.5\)

Answer: 7.5 cups of flour needed for 10 people

Example - Speed Problem:

Problem: A car travels 120 miles in 2 hours. How far will it travel in 5 hours at the same speed?

Setup: \(\frac{\text{miles}}{\text{hours}} = \frac{120}{2} = \frac{x}{5}\)

Cross-multiply: \(2x = 120 \times 5\)

\(2x = 600\)

\(x = 300\)

Answer: The car will travel 300 miles in 5 hours

4. Estimate Population Size Using Proportions

Capture-Recapture Method:

Population Formula:

\(\frac{\text{Marked in 1st sample}}{\text{Total population}} = \frac{\text{Marked in 2nd sample}}{\text{Total in 2nd sample}}\)

Or: \(\frac{M}{N} = \frac{m}{n}\)

Where:

• M = Number marked in first sample

• N = Total population (unknown)

• m = Number of marked individuals in second sample

• n = Total size of second sample

Alternative Formula:

Direct Formula: \(N = \frac{M \times n}{m}\)

Total Population = \(\frac{(\text{Marked first}) \times (\text{Second sample size})}{\text{Marked in second sample}}\)

Example - Fish Population:

Problem: Biologists catch, mark, and release 50 fish in a lake. Later, they catch 40 fish and find 8 are marked. Estimate the total fish population.

Given: M = 50, n = 40, m = 8, N = ?

Setup: \(\frac{50}{N} = \frac{8}{40}\)

Cross-multiply: \(8N = 50 \times 40\)

\(8N = 2000\)

\(N = \frac{2000}{8} = 250\)

Answer: Estimated fish population ≈ 250 fish

Another Example - Wildlife Population:

Problem: 80 deer are tagged. In a second capture, 60 deer are caught, with 12 tagged. Estimate total population.

Using Direct Formula: \(N = \frac{M \times n}{m} = \frac{80 \times 60}{12} = \frac{4800}{12} = 400\)

Answer: Estimated deer population ≈ 400 deer

5. Scale Drawings - Word Problems

What is a Scale Drawing?

Scale Drawing: A drawing that is proportional to the actual object

Scale: The ratio of drawing length to actual length

Written as: 1 inch : 5 feet or 1 cm : 10 m

Scale Drawing Formula:

Proportion Formula:

\(\frac{\text{Drawing length}}{\text{Actual length}} = \frac{\text{Scale drawing}}{\text{Scale actual}}\)

Or simply: \(\frac{d_1}{a_1} = \frac{d_2}{a_2}\)

Example 1 - Finding Actual Length:

Problem: A map has a scale of 1 inch : 50 miles. Two cities are 3.5 inches apart on the map. What is the actual distance?

Setup: \(\frac{1 \text{ inch}}{50 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}\)

Cross-multiply: \(1 \times x = 50 \times 3.5\)

\(x = 175\)

Answer: Actual distance = 175 miles

Example 2 - Finding Drawing Length:

Problem: A blueprint uses a scale of 1 cm : 4 m. A room is 16 meters long. How long is it on the blueprint?

Setup: \(\frac{1 \text{ cm}}{4 \text{ m}} = \frac{x \text{ cm}}{16 \text{ m}}\)

Cross-multiply: \(4x = 1 \times 16\)

\(4x = 16\)

\(x = 4\)

Answer: Blueprint length = 4 cm

6. Scale Drawings - Scale Factor Word Problems

Understanding Scale Factor:

Scale Factor (k): The ratio by which a figure is enlarged or reduced

Formula: \(k = \frac{\text{New dimension}}{\text{Original dimension}}\)

Types of Scale Factor:

Scale Up (Enlargement): \(k > 1\)

Example: \(k = 3\) means the new figure is 3 times larger

Scale Down (Reduction): \(0 < k < 1\)

Example: \(k = \frac{1}{2}\) means the new figure is half the size

No Change: \(k = 1\)

The figures are the same size

Key Formulas:

1. Scale Factor for Enlargement:

\(k = \frac{\text{Larger dimension}}{\text{Smaller dimension}}\)

2. Scale Factor for Reduction:

\(k = \frac{\text{Smaller dimension}}{\text{Larger dimension}}\)

3. Finding New Dimension:

\(\text{New dimension} = \text{Original dimension} \times k\)

4. Finding Original Dimension:

\(\text{Original dimension} = \frac{\text{New dimension}}{k}\)

Example 1 - Finding Scale Factor:

Problem: A building is 60 meters tall. A model is 12 cm tall. What is the scale factor?

Convert to same units: 60 m = 6000 cm

Scale Factor: \(k = \frac{12}{6000} = \frac{1}{500}\)

Answer: Scale factor = \(\frac{1}{500}\) or 1:500

Example 2 - Using Scale Factor:

Problem: A photo is enlarged with a scale factor of 2.5. If the original is 4 inches wide, how wide is the enlargement?

Using Formula: New width = Original × k

New width = \(4 \times 2.5 = 10\) inches

Answer: The enlargement is 10 inches wide

Example 3 - Finding Original Dimension:

Problem: A drawing is reduced with scale factor \(\frac{1}{3}\). The reduced length is 5 cm. What was the original length?

Using Formula: \(\text{Original} = \frac{\text{New}}{k}\)

Original = \(\frac{5}{\frac{1}{3}} = 5 \times 3 = 15\) cm

Answer: The original length was 15 cm

7. Quick Reference - All Formulas

Topic	Formula	When to Use
Basic Proportion	\(\frac{a}{b} = \frac{c}{d}\)	When two ratios are equal
Cross-Multiplication	\(a \times d = b \times c\)	To solve for unknown
Population Estimate	\(N = \frac{M \times n}{m}\)	Capture-recapture method
Scale Drawing	\(\frac{d_1}{a_1} = \frac{d_2}{a_2}\)	Maps, blueprints, models
Scale Factor	\(k = \frac{\text{New}}{\text{Original}}\)	Enlargement/reduction
New Dimension	\(\text{New} = \text{Original} \times k\)	Apply scale factor