🔢 Ratio, Proportion and Percentages

What is a Ratio?

A ratio is a comparison of two quantities of the same kind, showing how many times one quantity contains the other. It is written as \(a : b\) or \(a/b\), where \(a\) and \(b\) are the two quantities.

Example: If there are 3 boys and 5 girls in a class, the ratio of boys to girls is \(3 : 5\).

Simplifying Ratios:

To simplify a ratio, divide both terms by their greatest common factor (GCF).

Steps to Simplify a Ratio:

• Find the GCF of both numbers

• Divide both numbers by the GCF

• Write the simplified ratio in the form \(a : b\)

Equivalent Ratios:

Ratios that represent the same comparison are called equivalent ratios. They can be found by multiplying or dividing both terms by the same non-zero number.

Examples: \(3 : 5 = 6 : 10 = 9 : 15 = 12 : 20\)

Comparing Ratios:

Convert ratios to fractions and compare, or use cross multiplication.

What is Direct Proportion?

Two quantities are in direct proportion if they increase or decrease together in the same ratio. When one quantity doubles, the other doubles; when one is halved, the other is halved.

If \(x\) and \(y\) are directly proportional, then \(x/y = k\) (constant), or \(x = ky\).

What is the Unitary Method?

The unitary method is a technique for solving proportion problems by first finding the value of one unit, then multiplying to find the required value.

Steps for Unitary Method:

• Find the value of one unit (divide the given quantity by the given number)

• Multiply by the required number of units

Real-Life Examples of Direct Proportion:

• Cost of items: more items → more cost (at same price)

• Distance and time (at constant speed): more time → more distance

• Work done and number of workers (at same rate): more workers → more work…

What is a Scale Drawing?

A scale drawing is a drawing that represents a real object with all lengths reduced or enlarged proportionally. The scale is the ratio of the length in the drawing to the actual length.

Scale Notation:

• \(1 : 50\) means 1 unit in the drawing represents 50 units in real life

• \(1 cm : 5 m\) means 1 cm in drawing = 5 m in reality (units differ)

Types of Scales:

TypeExampleMeaning
Reduction scale\(1 : 100\)Drawing is smaller than real object (maps, floor plans)
Enlargement scale\(5 : 1\)Drawing is larger than real object (microscope images)

Using Scale Drawings:

• Actual length = Drawing length × Scale factor

• Drawing length = Actual length ÷ Scale factor

Real-Life Applications:

• Maps: 1 cm = 1 km, etc.

• Architectural blueprints: 1 cm = 1 m

• Model making: Model cars, airplanes

• Engineering drawings: Machine parts

Scale…