Ratios and proportions
What is Ratios and proportions?
A comparison of the size of two quantities of the same kind.
Key formula / rule: Ratio
Key points
- Define and differentiate between ratios and proportions.
- Calculate and simplify ratios.
- Set up and solve proportions to find unknown quantities.
- Identify and solve problems involving direct and inverse proportions.
Common exam trap
Inverting the ratio (e.g., writing girls:boys instead of boys:girls).
Definitions
- Term
Ratio
- Meaning
A comparison of the size of two quantities of the same kind.
- Term
Proportion
- Meaning
A statement that two ratios are equal.
- Term
Terms of a Ratio
- Meaning
The two quantities being compared (antecedent and consequent).
- Term
Means and Extremes
- Meaning
In a proportion a/b = c/d, b and c are the means, and a and d are the extremes.
- Term
Direct Proportion
- Meaning
Two quantities are in direct proportion if they increase or decrease together in the same ratio.
- Term
Inverse Proportion
- Meaning
Two quantities are in inverse proportion if an increase in one causes a decrease in the other, and vice versa, such that their product remains constant.
Learning objectives
Define and differentiate between ratios and proportions.
Calculate and simplify ratios.
Set up and solve proportions to find unknown quantities.
Identify and solve problems involving direct and inverse proportions.
Apply ratio and proportion concepts to real-world scenarios.
Formulae
- Name
Ratio
- Note
Compares two quantities.
- Expression
a:b or a/b
- Name
Proportion
- Note
States equality of two ratios.
- Expression
a/b = c/d
- Name
Cross-Multiplication
- Note
Used to solve for an unknown in a proportion.
- Expression
If a/b = c/d, then ad = bc
- Name
Direct Proportion
- Note
As x increases, y increases proportionally.
- Expression
y = kx (where k is a constant)
- Name
Inverse Proportion
- Note
As x increases, y decreases proportionally.
- Expression
y = k/x (where k is a constant)
Prerequisites
Basic arithmetic operations (addition, subtraction, multiplication, division).
Understanding of fractions and their simplification.
Basic algebraic manipulation.
Common mistakes
Inverting the ratio (e.g., writing girls:boys instead of boys:girls).
Incorrectly setting up the proportion, especially with inverse relationships.
Calculation errors when simplifying ratios or solving for unknowns.
Confusing direct and inverse proportion scenarios.
Keywords
Ratio
Proportion
Direct Proportion
Inverse Proportion
Comparison
Equality
Scaling
Rate
Unitary Method
Practice preview
Simplify the ratio 45:75.…
easy
Divide Rs. 720 between A and B in the ratio 4:5. How much does B receive?…
easy
If A:B = 2:3 and B:C = 4:5, what is A:B:C?…
medium