Scale factors, diagrams and maps Worked Solutions — GCSE Maths

Write down what the scale means

Every $1$ cm measured on the map is $5$ km on the ground.

Multiply the map distance by five

Six centimetres on the map is six lots of $5$ km.

State the real distance

The real distance is $30$ km, which is far bigger than the $6$ cm on the map — as it must be.

Write down what the scale means

Every $1$ cm on the map is $4$ km in real life.

Divide the real distance by four

Going from real life to the map is a reduction, so divide.

State the map length

Check: $9 imes 4 = 36$ km, which is the real length given.

Write down what the model scale means

A scale of $1 : 100$ means every real length is $100$ times the matching length on the model.

Multiply the model height by the scale factor

Each centimetre stands for $100$ cm, so $7$ cm stands for $700$ cm.

Convert centimetres to metres

There are $100$ cm in a metre, so $700$ cm is $7$ m.

Write down what the model scale means

A scale of $1 : 24$ means every real length is $24$ times the matching length on the model.

Convert the real length into centimetres

The scale compares centimetres with centimetres, so change $6$ m into $600$ cm first.

Divide by the scale factor

The model is a reduction, so divide the real length by $24$.

Write down what the map scale means

A scale of $1 : 25\,000$ means $1$ cm on the map represents $25\,000$ cm in real life. Both lengths must be measured in the same units.

Multiply the map distance by the scale factor

Each centimetre stands for $25\,000$ cm, so $12$ cm stands for $300000$ cm.

Convert the real distance from centimetres to kilometres

$1\text{ km} = 1000\text{ m} = 100\,000\text{ cm}$, so divide by $100\,000$ to get $3$ km.