Using ratio with scale factors, map scales and scale drawings, converting between ratio forms, and combining ratios that share a part.

What this dot point is asking

AQA wants you to apply ratio to scale factors, map scales and scale drawings, convert between ratio forms (such as $1 : n$), and combine two ratios that share a common quantity into a single three-part ratio. These skills connect ratio to geometry and real-world maps and plans, and the combining-ratios technique is a reliable Higher-tier question.

Scale factors and scale drawings

A scale drawing represents a real object at a reduced (or enlarged) size, set by a scale factor. A plan with scale $1 : 50$ means every $1\,\text{cm}$ on the plan is $50\,\text{cm}$ in reality. To find a real length, multiply the drawing length by the scale; to find a drawing length, divide the real length by the scale. Keep units consistent throughout, then convert at the end.

Map scales

Map scales work the same way but the numbers are larger, so unit conversion is essential. A scale of $1 : 50\,000$ means $1\,\text{cm}$ on the map is $50\,000\,\text{cm}$ in reality, which is $500\,\text{m}$ or $0.5\,\text{km}$. Always convert from centimetres up to metres or kilometres at the end of a map calculation.

Converting between ratio forms

A ratio can be rewritten in the form $1 : n$ by dividing both parts by the first part. So $4 : 10$ becomes $1 : 2.5$, and $5 : 8$ becomes $1 : 1.6$. The $1 : n$ form makes ratios easy to compare and is the standard way map scales are written.

Combining ratios that share a part

When two ratios share a common quantity, you can link them into a single ratio. Make the shared quantity equal in both ratios (scale up to the lowest common multiple of the two values), then read off the combined three-part ratio.

For example, if $A : B = 2 : 3$ and $B : C = 4 : 5$, the shared part $B$ is $3$ and $4$, with LCM $12$. Scale the first by $4$ ($A : B = 8 : 12$) and the second by $3$ ($B : C = 12 : 15$), giving $A : B : C = 8 : 12 : 15$.

Scale factors for area and volume

A subtle Higher-tier extension is that scale factors behave differently for length, area and volume. If a shape is enlarged by a linear scale factor $k$, its areas grow by $k^2$ and its volumes by $k^3$. So doubling every length ($k = 2$) makes the area four times larger and the volume eight times larger. A model car at scale $1 : 50$ has lengths $\dfrac{1}{50}$ of the real car, but its surface area is $\dfrac{1}{2500}$ and its volume is $\dfrac{1}{125\,000}$ of the real one. Mixing these up is a frequent error: the linear scale alone does not apply to area or volume.

Reading bearings and scale together

Scale drawings often combine with bearings in navigation problems. A bearing is a three-figure angle measured clockwise from north, so $075^\circ$ means $75^\circ$ east of north. To draw a journey to scale, choose a scale (such as $1\,\text{cm}$ to $10\,\text{km}$), measure each bearing with a protractor from north, and mark each leg at the scaled length. Real distances are then read back by measuring the drawing and multiplying by the scale, tying the whole topic together with measurement and angle skills.