4.
1 Ratio and proportion
INTRODUCTION
❖ Ratios and proportions are fundamental mathematical concepts used to compare quantities and
understand relationships between them. Ratios express the relationship between two quantities by
division, while proportions equate two ratios, showing they are the same in value.
KEY HIGHLIGHT
𝑦
• Direct Proportionality Formula: (𝑦 = 𝑘𝑥) 𝑜𝑟 𝑘 = .
𝑥
𝑘
• Inverse Proportionality Formula: 𝑦 = 𝑥, k = y × x
• Cross Multiplication for Proportions: a.d = b.c
Definition and Forms of Ratios
❖ Definition: A ratio is a way to compare two or more quantities of the same kind and units by
division.
❖ Notation: The ratio of quantity (𝑎) to quantity (𝑏) is denoted as (𝑎 ∶ 𝑏),
❖ Forms of Representation:
• (𝑎 ∶ 𝑏) — read as "a to b"
𝑎
• 𝑏 — read as "a over b"
• (𝑎 ÷ 𝑏) — "a divided by b" (provided 𝑏 ≠ 0 )
8
❖ Example: The ratio of 8 to 4 can be written as (8 ∶ 4), 4, or 8 ÷ 4
❖ Simplified Form: Ratios are often simplified, so (𝑎) and (𝑏) become natural numbers.
❖ Unit Consistency: Ratios can only be applied to quantities with the same units and are unitless.
Differentiating Ratio and Proportion
❖ Proportion Definition: A proportion is an equation that equates two ratios, stating they are
equal.
Example of Proportion:
❖ (6 ∶ 10) and (48 ∶ 80) form a proportion as both simplify to (3 ∶ 5).
❖ However, (1 ∶ 3) and (5 ∶ 10) do not form a proportion, as they simplify to (1 ∶
3) and (1 ∶ 2), respectively.
Proportion Components:
• In a proportion (𝑎 ∶ 𝑏 = 𝑐 ∶ 𝑑):
• Extremes: (𝑎) and (𝑑)
• Means: (𝑏) and (𝑐)
• Product of Means and Extremes: (𝑎 ∶ 𝑏 = 𝑐 ∶ 𝑑) holds if (𝑎. 𝑑 = 𝑏. 𝑐 ).
Simplification and Unitless Nature of Ratios
❖ Simplification: Ratios (𝑎 ∶ 𝑏) are expressed in their simplest form by dividing (𝑎) and (𝑏) by
their greatest common divisor (GCD).
4.1 Ratio and proportion
❖ Unitless Property: Ratios are dimensionless quantities, meaning they are independent of
units.
Direct and Inverse Proportionality
A. Direct Proportionality
• Definition: Two variables, (𝑦) and (𝑥), are directly proportional (written as (𝑦 ∝ 𝑥)) if y
= kx for some constant (𝑘) (constant of proportionality).
• Relationship: As (𝑥) increases, (𝑦) increases proportionally; as (𝑥) decreases, (𝑦)
decreases.
𝑦
• Constant Ratio: The ratio 𝑥 remains constant.
• Example: If a car’s speed is directly proportional to distance traveled, doubling the speed
doubles the distance in the same time.
B. Inverse Proportionality
• Definition: Two variables, (𝑦) and (𝑥), are inversely proportional (written as
• Relationship: As (𝑥) increases, (𝑦) decreases; as (𝑥) decreases, (𝑦) increases.
• Constant Product: The product (𝑦 × 𝑥) remains constant.
• Example: For a constant amount of work, the time taken is inversely proportional to the
number of workers.
Solving Proportions using Cross Multiplication
❖ Cross Multiplication Method: To determine if two ratios (𝑎 ∶ 𝑏) and (𝑐 ∶ 𝑑) form a
proportion, use cross multiplication:
a.d=b.c
❖ Example: For the ratios (4 ∶ 6) and (2 ∶ 3), check if they form a proportion:
❖ Cross-multiplying gives (4 . 3 = 6 . 2), which simplifies to (12 = 12); hence, they are in
proportion.
Summary
• Understanding ratios and proportions is essential for comparing quantities and
establishing relationships between variables. Ratios allow for comparative analysis,
while proportions and proportionality explain how changes in one variable affect
another. Mastering these concepts is crucial for practical problem-solving in science,
economics, and daily life.
• Ratios are widely used in real-life applications such as cooking, mixing
chemicals, and design to maintain consistency in proportions.
