Unit 1 - Lesson 1: Proportion
Master the fundamental concepts of ratio, proportion, extremes, means, and cross multiplication to solve real-world problems.
Audio Voice Explanation
Listen to an audio explanation of the core definitions and properties
1. Concept of Proportion
Definition:
A proportion is an equality between two or more ratios or rates.
Equality of Ratios
Both fractions simplify to the same value (1/2), hence they form a proportion.
Equality of Rates
The rate of 4 eggs per cup is constant across both batches.
Terms of a Proportion: Means & Extremes
If the quantities a, b, c, and d are proportional, we write:
2. The Fundamental Property: Cross Multiplication
Key Rule to Remember:
If a / b = c / d, then: a × d = b × c
Testing if Ratios Form a Proportion
Determine whether 3 / 8 and 6 / 10 represent a proportion:
- Product of extremes: 3 × 10 = 30
- Product of means: 8 × 6 = 48
- Since 30 ≠ 48, they do NOT represent a proportion.
Another Example
Determine whether 8 / 10 and 12 / 15 represent a proportion:
- Product of extremes: 8 × 15 = 120
- Product of means: 10 × 12 = 120
- Since 120 = 120, they REPRESENT a proportion!
3. Solving Proportions (Finding Unknown X)
Solving a proportion means finding the unknown value by applying cross multiplication:
Find X if: 6 / 9 = 10 / X
6X = 90
X = 90 / 6
X = 15
Find X if: X / 14 = 15 / 21
21X = 210
X = 210 / 21
X = 10
Find X if: 2 / 8 = (X + 1) / 12
8(X + 1) = 24
X + 1 = 24 / 8 = 3
X = 3 - 1 = 2
Automobile Fuel Consumption
A car consumes 3 liters of gasoline to travel 33 kilometers. How many liters of gasoline does it need to travel 121 kilometers at the same rate?
Set up proportion: X / 121 = 3 / 33
Cross multiply: 33 × X = 3 × 121 = 363
Solve: X = 363 / 33 = 11 liters
4. Interactive Proportion Lab
Test four numbers to see if they form a proportion, or solve for the unknown X
Proportion Verification: a / b = c / d
Product of Extremes: 4 × 12 = 48 | Product of Means: 8 × 6 = 48
5. Knowledge Check & Practice Quiz
5 questions to test your mastery of proportions. Choose the best answer for each question:
التعليقات والمناقشة