Scale factors, diagrams and maps Worked Solutions — GCSE Maths

Write down what the scale means

Every $1\,\mathrm{cm}$ measured on the map is $5\,\mathrm{km}$ on the ground.

Multiply the map distance by five

Six centimetres on the map is six lots of $5\,\mathrm{km}$.

State the real distance

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

Write down what the scale means

Every $1\,\mathrm{cm}$ on the map is $4\,\mathrm{km}$ 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: $9 \times 4 = 36\,\mathrm{km}$, 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 $100\,\mathrm{cm}$, so $7\,\mathrm{cm}$ stands for $700\,\mathrm{cm}$.

Convert centimetres to metres

There are $100\,\mathrm{cm}$ in a metre, so $700\,\mathrm{cm}$ is $7\,\mathrm{m}$.

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 $6\,\mathrm{m}$ into $600\,\mathrm{cm}$ 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 $1\,\mathrm{cm}$ on the map represents $25\,000\,\mathrm{cm}$ in real life. Both lengths must be measured in the same units.

Multiply the map distance by the scale factor

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

Convert the real distance from centimetres to kilometres

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