4.1 Ratios
1. Concept of Ratio
A ratio is used to compare two or more quantities of the same kind measured in the same units.
- Notation: The ratio of a to b is written as a : b.
- Three Quantities: The ratio of a to b to c is written as a : b : c.
- Unitless Property: Ratios do not have units. If quantities are in different units, convert them to the same unit first before writing the ratio.
- Example: Compare 500 g to 2 kg. Convert 2 kg to 2000 g first → 500 : 2000 = 1 : 4.
2. Equivalent Ratios
Ratios that express the same relationship are called equivalent ratios. They are obtained by multiplying or dividing each term of the ratio by the same non-zero number.
- Example: 2 : 3 = (2 × 3) : (3 × 3) = 6 : 9.
- Simplest Form: A ratio is in its simplest form when its terms are whole numbers with no common factor other than 1.
4.2 Rates
1. Concept of Rate
A rate is a special ratio that compares two quantities of different kinds with different units.
- Format: Quantity / Time or Quantity / Distance (expressed with 'per' or '/').
- Examples:
- Speed: km/h or m/s
- Density: g/cm³
- Price per unit mass: RM/kg
2. Conversion of Units for Rates
Rates can be converted from one set of units to another by converting the units of the numerator and denominator independently.
- Example: Convert 72 km/h to m/s:
(72 km / 1 hour) = (72 × 1000 m) / (1 × 3600 seconds) = 72000 / 3600 = 20 m/s
4.3 Proportions
1. Concept of Proportion
A proportion is a statement or equation showing that two ratios or two rates are equal.
- Form: a/b = c/d or a : b = c : d.
2. Methods for Solving Proportions
- Unitary Method: Find the value corresponding to one unit first, then calculate the required value.
- Cross-Multiplication Method: If a/b = c/d, then a × d = b × c.
- Proportion Method: Compare ratios directly: a : b = c : d ⇒ a/b = c/d.
4.4 Percentage, Ratio and Proportion
Percentages represent fractions out of 100 and can be expressed as ratios or used in proportion calculations.
- Converting Percentage to Ratio: 35% = 35 / 100 = 7 : 20.
- Determining Percentage from Ratio: If a part-to-whole ratio is 3 : 5, the percentage is (3 / 5) × 100% = 60%.
4.5 Relationship Between Ratios, Rates and Proportions with Fractions, Decimals and Percentages
Real-world problem solving often requires combining ratios and proportions to find unknown values or compare different situations effectively.
- Combined Ratios: If a : b = 2 : 3 and b : c = 4 : 5, equalise the value of b (LCM of 3 and 4 is 12):
- a : b = 8 : 12
- b : c = 12 : 15
- Therefore, a : b : c = 8 : 12 : 15.