Ratio and Proportion

A ratio $a : b$ compares two quantities. Read $3 : 5$ as 'three to five'.

Simplify a ratio just like a fraction — divide both parts by their HCF.

$24 : 18 = (24 \div 6) : (18 \div 6) = 4 : 3.$

Equivalent ratios are the same comparison written differently: $1 : 2 = 2 : 4 = 5 : 10 = 100 : 200.$

A ratio can also be written as $a : b : c$ for three parts.

Sharing in a given ratio. The 'parts' method:

1. Add the ratio numbers: this is the total number of parts.
2. Find the size of ONE part: total ÷ number of parts.
3. Multiply to find each share.

Worked example. Share $60 in the ratio $2 : 3$.

• Total parts: $2 + 3 = 5$.
• One part: 60 \div 5 = \12$.
• Share 1: 2 \times 12 = \24$. Share 2:$3 \times 12 = $36$.
• Check: $24 + 36 = 60$. ✓

Worked example (3 parts). Share $120 in the ratio $1 : 3 : 4$.

• Total parts: $1 + 3 + 4 = 8$.
• One part: 120 \div 8 = \15$.
• Shares: \15, $45, $60$.

Two quantities are directly proportional if their ratio is CONSTANT — i.e. as one doubles, the other doubles.

$y \propto x \quad \iff \quad y = kx$

where $k$ is the constant of proportionality (or constant of variation).

Worked example. If $y$ is directly proportional to $x$ and $y = 12$ when $x = 4$, find $y$ when $x = 9$.

• Find $k$: $y = kx \Rightarrow 12 = 4k \Rightarrow k = 3$.
• The relation is $y = 3x$.
• When $x = 9$: $y = 3 \times 9 = 27$.

Other examples of direct proportion:

• Cost of fruit at a fixed price per kg: $\text{cost} \propto \text{weight}$.
• Distance at constant speed: $\text{distance} \propto \text{time}$.
• Circumference of a circle: $C = 2\pi r$ — $C \propto r$ with $k = 2\pi$.

A graph of direct proportion is a STRAIGHT LINE through the ORIGIN. If a graph doesn't pass through the origin, it's NOT a direct proportion.

Two quantities are inversely proportional if their PRODUCT is constant:

$y \propto \frac{1}{x} \quad \iff \quad y = \frac{k}{x} \quad \iff \quad xy = k.$

As $x$ increases, $y$ DECREASES (and vice versa).

Worked example. If $y$ is inversely proportional to $x$ and $y = 8$ when $x = 3$, find $y$ when $x = 12$.

• Find $k$: $xy = k \Rightarrow 3 \times 8 = 24$. So $k = 24$.
• The relation is $y = \dfrac{24}{x}$.
• When $x = 12$: $y = \dfrac{24}{12} = 2$.

(Check: doubled $x$ from 6 to 12 → halved $y$ from 4 to 2.)

Other examples of inverse proportion:

• Speed and time for a fixed journey: faster speed → less time. $st = d$.
• Number of workers and time for a fixed job: more workers → less time.
• Pressure and volume at constant temperature (Boyle's law): $PV = k$.

The graph of $y = \dfrac{k}{x}$ is a HYPERBOLA — a smooth curve that approaches both axes but never touches them.

For Extended Level, you may also see proportion to a POWER:

• $y \propto x^2$: $y = kx^2$ (e.g. area of a circle vs radius).
• $y \propto \dfrac{1}{x^2}$: $y = \dfrac{k}{x^2}$ (inverse-square law in physics).