GCSE · Maths · AQA · Spec 8300

Scale factors, scale diagrams and maps

A map squeezes 100 km of countryside into 4 cm of paper — and one number lets you stretch it straight back out.

Maths · Map scales

Every 1 cm on this map is 25 km in real life

Map length: 4/10. Map length 4 cm. Real distance 100 km

Map length4 cmReal distance100 km

Drag the map length along the ruler. Each extra centimetre adds another 25 km, so the real distance is always the map length × 25 km — at 4 cm, that's 4 × 25 = 100 km. Before you drag to 9 cm, predict the real distance. Then check.

Exam line: Map length → real length: multiply by the scale. At 1 cm to 25 km, 4 cm stands for 4 × 25 = 100 km.

Watch out: Take the scale away and the real-distance number has nothing to go on. A map with no scale can't tell you how far apart anything is.

Maths · Enlargement

Make DEF an enlargement of ABC, scale factor 2

AB 4.0 cm. AC 3.0 cm. BC 5.0 cm. DE 8.0 cm. DF 4.5 cm. EF 9.2 cm. Relationship: Enlargement, scale factor 2: every length of DEF = 2 × the matching length of ABC

AB4.0 cmAC3.0 cmBC5.0 cmDE8.0 cmDF4.5 cmEF9.2 cmdrag F up or down

Enlargement, scale factor 2: every length of DEF = 2 × the matching length of ABC

The same multiplier idea works on shapes. DE is already 2 × AB (4 cm → 8 cm). Drag F up or down. First try DF = 7 cm — adding 4, just like the base grew by 4. Then try DF = 6 cm — doubling 3. Watch EF: which one makes it exactly 2 × BC = 10 cm?

Exam line: In an enlargement, every length is multiplied by the scale factor. A whole-number scale factor bigger than 1 makes the image bigger than the original.

Watch out: Enlarging means multiplying every length, not adding to it. Here, adding 4 to both short sides knocks the triangle out of shape.

Scales without units

What does 1 : 25 actually mean?

A plan of a garden shed has its scale printed in the corner as 1 : 25 — just two numbers, no units.

Which is closest to what you think 1 : 25 tells you?

1 cm on the plan stands for 25 km in real life.1 cm on the plan stands for 25 cm in real life.You can't use it — a scale needs units to mean anything.

How sure are you?

I'm sureI think soI'm guessing

1 cm on the plan stands for 25 km in real life.

This is a common misconception

This is the classic trap. A scale with no units uses the SAME unit on both sides — you don't get to choose a different unit for the second number. So 1 : 25 means 1 cm to 25 cm. If you want the answer in metres or kilometres, that's a conversion you do afterwards, yourself.

1 cm on the plan stands for 25 cm in real life.

This matches the evidence

Spot on. No units written means the same unit on both sides, so every length on the plan is the real length divided by 25. It's exactly how 1 : 100 means 1 cm stands for 100 cm, which is 1 m.

You can't use it — a scale needs units to mean anything.

This is a common misconception

It's actually a complete scale. Because both sides share one unit, it works in whatever unit you measure in: 1 cm to 25 cm, or 1 mm to 25 mm. The ratio hands you the multiplier, 25, and that's all you need.

A scale written as a ratio has the same unit on both sides. 1 : 25 means 1 cm to 25 cm — never 1 cm to 25 km.

Exam line: No units means the same unit on both sides: 1 : 100 means 1 cm stands for 100 cm, which is 1 m.

Converting scales, step by step

Problem

(a) Write the scale 1 cm to 25 km as a ratio without units. (b) On a map with scale 1 : 50,000, two villages are 9 cm apart. How far apart are they in real life, in km?

Your turn — finding a scale

Work backwards to the scale

On a map, a road that is 6,000 m long is drawn 10 cm long. Find the scale of the map as a ratio without units.

The map length is 10 cm and the real length is 6,000 m.
missing step

Which line is step 2?

Find what 1 cm stands for: 6,000 m ÷ 10 = 600 m, so the scale is 1 cm to 600 m.Multiply: 6,000 m × 10 = 60,000 m, so the scale is 1 cm to 60,000 m.Divide the map length by the real length: 10 ÷ 6,000, so the scale is 1 cm to 0.0017 m.

Exam line: To find a scale: real length ÷ map length tells you what 1 cm stands for. Then make the units match and write it as 1 : n.

Spot the slip

Where does this scale drawing go wrong?

A rectangular room is 300 cm wide and 800 cm long. Make a scale drawing of it using the scale 1 : 50. How long should each side of the rectangle be drawn?

A student's answer — which line goes wrong?

Scale 1 : 50 means 1 cm on the drawing stands for 50 cm in the room.
Width: 300 cm in the room is drawn as 300 ÷ 50 = 6 cm.
Length: 800 cm in the room is drawn as 800 × 50 = 40,000 cm.
So the drawing is a rectangle 6 cm by 40,000 cm, with 'Scale 1 : 50' written beside it.

Exam line: Real → drawing: divide by the scale. Drawing → real: multiply. Then write the scale beside the finished drawing.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A scale is a multiplier: every real length is the map or drawing length multiplied by one fixed number.

What you need to know

A scale factor is a multiplier: in an enlargement, every length of the shape is multiplied by it.
On a map or scale drawing, drawn lengths and real lengths are linked by one fixed multiplier — the scale.
Map → real: multiply by the scale. Real → drawing: divide by the scale.
A scale written without units, like 1 : 100, uses the same unit on both sides: 1 cm stands for 100 cm.
To find a scale, divide the real length by the map length, then make both sides the same unit.
A map with no scale can't be used to measure or compare real distances.

The big picture

A scale is a multiplier. In an enlargement every length is multiplied by the scale factor, and maps and scale drawings work the same way: map lengths and real lengths are locked together by one fixed number. Map to real, multiply by the scale; real to drawing, divide; to find a scale, divide the real length by the map length. A unit-free scale like 1 : 50,000 has the same unit on both sides, so convert units before giving a real distance. No scale, no measurement.

Key points

1Enlarging multiplies every length by the same number — adding the same amount to each side is not the same thing.

2A whole-number scale factor bigger than 1 makes the image bigger than the original.

31 km = 1000 m = 100,000 cm, so converting a unit-free scale usually means big numbers.

4Match the units before you finish: the scale ratio in centimetres, real distances in whichever of cm, m or km the question asks for.

5Choose a scale that makes the drawing easy to draw accurately, and always write the scale beside it.

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Worked example

Problem

A scale drawing of a garden uses the scale 1 : 200. (a) A path on the drawing is 7.5 cm long. How long is the real path, in metres? (b) A shed in the garden is 3 m wide. How wide should it be drawn?

⚠ Watch out

Reading a unit-free scale with mixed units — treating 1 : 25 as 1 cm to 25 km. Both sides of the ratio share one unit, so 1 : 25 means 1 cm to 25 cm; change the units yourself before you state a real distance.

🧠

Memory hook

Heading OUT into the real world, things get bigger — multiply. Coming back IN to the paper, things get smaller — divide. And no units on the ratio? Same unit both sides.

✓

Check yourself

On a map with scale 1 : 300,000, two towns are 5 cm apart. How far apart are they in real life, in km? (Answer: 5 × 300,000 = 1,500,000 cm, which is 15 km.)

Flashcards

(14)

What is a scale factor?

A multiplier. In an enlargement, every length of the shape is multiplied by it.

A shape is enlarged by a whole-number scale factor bigger than 1. How does the image compare with the original?

It is bigger: every length of the image is the original length × the scale factor.

Is adding the same amount to every side the same as enlarging?

No. Enlarging multiplies every length by the same number; adding can change the shape.

You measure a length on a map. How do you get the real length?

Multiply the map length by the scale.

Why can't you measure real distances on a map with no scale?

There's no multiplier linking map lengths to real lengths, so the map length could stand for any distance.

What does the scale 1 : 100 mean?

1 cm stands for 100 cm (which is 1 m) — the same unit on both sides.

Does 1 : 25 mean 1 cm to 25 km?

No. With no units, both sides share one unit: 1 cm to 25 cm.

How many centimetres are in 1 km?

100,000 cm (1 km = 1000 m, and 1 m = 100 cm).

How do you write a scale like 1 cm to 25 km without units?

Convert the real length into cm: 25 km = 2,500,000 cm, so the ratio is 1 : 2,500,000.

How do you find the scale from a map length and the real length it stands for?

Divide real length by map length to get what 1 cm stands for, then convert to the same unit and write 1 : n.

How do you work out a length for a scale drawing?

Divide the real length by the scale (in the same units).

A table is 180 cm long. How long is it on a 1 : 20 scale drawing?

180 ÷ 20 = 9 cm.

What must always go beside a finished scale drawing?

The scale.

What makes a good choice of scale for a drawing?

One that makes the lengths easy to draw accurately.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What does 1 : 25 actually mean?

Work backwards to the scale

Where does this scale drawing go wrong?

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