Understanding Ratios and Rates

Essential tools for comparing quantities and understanding relationships in everyday life

CAPS Grade 10 Mathematical Literacy

This document explores the concepts of ratios and rates as fundamental tools in Grade 10 Mathematical Literacy. These concepts are essential for comparing quantities and understanding relationships between different values in everyday life.

Key Concepts

Ratios and rates help us compare quantities and understand relationships in real-world situations.

Core Skills

Ratio (same units) Rate (different units) Unit Rate Simplifying Ratios Sharing in a Ratio Unit Pricing Speed (km/h) Consumption Rates

Ratio & Rate Challenge Game

Test your skills with real-world ratio and rate problems!

Score
0
Questions
1/6
Streak
0
Mixing juice concentrate
A recipe requires mixing concentrate and water in the ratio 1:4. How much water is needed for 2 cups of concentrate?

1. Ratios (Comparing Same Units)

What is a Ratio?

Same Units

Ratios are a way to compare two or more quantities of the same kind, such as people to people or milliliters to milliliters.

2 : 3   |   2/3   |   "2 to 3"

Fraction
a/b (e.g., 2/3)
Colon
a:b (e.g., 2:3)
Words
"a to b" (e.g., 2 to 3)

Simplifying Ratios

Divide by HCF

To simplify a ratio, divide all numbers in the ratio by their highest common factor.

8 : 12 = (8÷4) : (12÷4) = 2 : 3

8
12
2
3

Sharing in a Ratio

Step-by-Step

When sharing a quantity in a specific ratio, add the parts to find the total, then divide the quantity by total parts.

Step 1
Add the parts: 2 + 3 = 5 total parts
Step 2
Find one part: R100 ÷ 5 = R20 per part
Step 3
Multiply: 2 × R20 = R40, 3 × R20 = R60

R100 shared in ratio 2:3 → R40 : R60

Ratio Sharing Calculator

Share an Amount in a Ratio

:
First person gets:
R40.00
Second person gets:
R60.00

Simplify a Ratio

:
Simplified ratio:
3 : 2

2. Rate (Comparing Different Units)

What is a Rate?

A rate compares two quantities with different units, typically indicating "how much of one thing per unit of another."

Examples:
• Speed: 100 km per hour (km/h)
• Price: R45 per kilogram (R/kg)
• Consumption: 8 litres per 100 km

Unit Rates

Unit rates are simplified to a "per unit" basis for easy comparison.

Calculation: Divide first quantity by second
300 km in 3 hours = 300 ÷ 3 = 100 km/hour

Real-World Rate Comparison: Unit Pricing

🥣 Soap Powder - Large

2 kg for R45
R22.50 per kg
Better Deal ✓

🥣 Soap Powder - Small

500 g for R12
R24.00 per kg
More Expensive

Common Rate Contexts

Unit Pricing: Compare product sizes by calculating price per gram or per kilogram.

Consumption Rates: Measuring litres of fuel used per 100 km to determine fuel efficiency.

Tariff Rates: Cost per minute for calls or cost per kilowatt-hour (kWh) for electricity.

3. Key Difference

Ratio vs Rate

Understanding the Distinction

Ratio

Same units - compares like with like

No units in final form (e.g., 2:3)

Example: Mixing juice concentrate with water (1:4)

Rate

Different units - compares different measurements

Must be expressed with units (e.g., 100 km/hour)

Example: Speed, price per kg, fuel consumption

Ratio: 2:3 (no units)   |   Rate: 100 km/h (with units)

Real-World Examples

Mixing Juice

A recipe requires mixing concentrate and water in the ratio 2:5. If you use 500 ml of concentrate, how much water is needed?

Solution

One part: 500 ml ÷ 2 = 250 ml
Water needed: 5 × 250 ml = 1250 ml (1.25 L)

Fuel Efficiency

A car travels 450 km using 36 litres of fuel. Calculate the fuel consumption rate in km per litre.

Solution

Rate = distance ÷ fuel
450 km ÷ 36 L = 12.5 km/L

Sharing Prize Money

Three friends invest in a business in the ratio 3:2:5. They make a profit of R20,000.

Solution

Total parts: 3+2+5=10
One part: R20,000÷10=R2,000
Shares: R6,000, R4,000, R10,000

Running Speed

An athlete runs 15 km in 1.5 hours. Calculate the speed in km/h.

Solution

Speed = distance ÷ time
15 km ÷ 1.5 h = 10 km/h

Practice Problems

Simplifying Ratios

Simplify the ratio 24:36 to its simplest form.

Solution

  • HCF of 24 and 36 = 12
  • 24 ÷ 12 = 2, 36 ÷ 12 = 3
  • Simplified ratio = 2:3

Unit Price Comparison

Which is better value: 750 g cereal for R45 or 1.2 kg for R72?

Solution

  • Option 1: R45 ÷ 0.75 kg = R60/kg
  • Option 2: R72 ÷ 1.2 kg = R60/kg
  • They are the same price per kg!

Sharing in a Ratio

Share R500 between two people in the ratio 3:7.

Solution

  • Total parts: 3+7=10
  • One part: R500÷10=R50
  • First: R150, Second: R350

CAPS Curriculum Requirements

Knowledge & Understanding

  • Understand ratios as comparisons of same units
  • Understand rates as comparisons of different units
  • Simplify ratios to their simplest form
  • Distinguish between ratios and rates

Skills & Applications

  • Calculate unit rates for comparison shopping
  • Share quantities in given ratios
  • Apply rates to speed, consumption, and pricing
  • Solve real-world problems involving ratios

Competencies

  • Make informed purchasing decisions using unit pricing
  • Calculate fuel consumption and travel times
  • Interpret tariff rates on bills
  • Apply ratios in cooking and mixing contexts

Learning Resources

Ratio Games

Interactive games to practice ratio simplification and sharing

Unit Pricing Activities

Compare real product prices to find the best value

Fuel Consumption Problems

Real-world rate problems for travel efficiency

Ratio & Rate Worksheets

Printable worksheets with mixed practice problems