Calculating and Using Map Scales

A house is 12 meters (m) long; it is represented on an architectural plan by a rectangle 48 centimeters (cm) long. What is the scale of the plan?

While walking, Peter sees that the distance that he still has to cover is represented by 5 cm on a map with a scale of $\frac{1}{25000}$. What real distance does this represent?

What calculation methods can we use to answer these questions?

I. Applying a scale

Example: Bernard wants to make a plan of his bedroom; it is rectangular and is 5 m long and 2.50 m wide.

He decides to divide the real dimensions by 20:

5 m = 500 cm and 500 cm ÷ 20 = 25 cm; 2.5 m = 250 cm and 250 ÷ 20 = 12.5 cm.

So he draws a rectangle with a length of 25 cm and a width of 12.5 cm.

This rectangle is a plan of his room to the scale of $\frac{1}{20}$.

Note: The dimensions of the plan are the real dimensions multiplied by the scale factor of $\frac{1}{20}$ ; in fact: $500\times \frac{1}{20}=500÷20=25$ and $250\times \frac{1}{20}=250÷20=12.5$;

the dimensions of the plan are proportional to the real dimensions; the scale factor is $\frac{1}{20}$.

Definition: On a map (or a plan), the dimensions are equal to the real dimensions multiplied by the same number e. The number e is called the map scale.

If D is a real distance that is represented on the map by a distance d, then

D × e = d (the distances must be expressed in the same unit).

II. Calculating a scale

Example 1: What is the scale e of the architectural plan mentioned in the introduction (12 meters represented by 48 centimeters)?

So: D = 12 m = 1,200 cm and d = 48 cm.

So: 1,200 × e = 48, or $e=\frac{48}{1200}=\frac{1}{25}$ (simplifying by 48).

The scale of the plan is equal to $\frac{1}{25}$.

Note: $\frac{1}{25}=0.04$ ; we can also say that the scale factor is equal to 0.04, but it is usual, where possible, to write the scale as a fraction with a numerator of 1 when the scale is less than 1.

Example 2: On a road map, a straight road 1 kilometer (km) long is represented by 1 cm. What is the scale of this map?

So: D = 1 km = 100,000 cm and d = 1 cm. We can call the scale of the map e.

So: 100,000 × e = 1, or $e=\frac{1}{100000}$.

The scale of the map is equal to $\frac{1}{100000}$.

Example 3: Using a microscope, you photograph a paramecium that is 0.2 millimeters (mm) long. On the photograph, the paramecium is 10 cm long. What is the scale of this photograph?

So: D = 10 cm = 100 mm and d = 0.2 mm. We can call the scale of the photograph e.

So: 0.2 × e = 100, or e = 100 ÷ 0.2 = 500.

The scale of the photograph is equal to 500.

Note: In this example, the photograph is an enlargement; this is because the scale is greater than 1.

III. Using a scale

A. Example 1: calculating a real distance

Look again at the second example given in the introduction. What is the distance that Peter must cover (the distance represented by 5 cm on a map with a scale of $\frac{1}{25000}$ )?

We apply the formula D × e = d, with $e=\frac{1}{25000}$ and d = 5 cm.

So: $D\times \frac{1}{25000}=5$, or D = 5 × 25,000 = 125,000.

Therefore D = 125,000 cm = 1.25 km.

Peter must cover 1.25 km.

B. Example 2: calculating a reduced length

On the same map, how is a path 750 m long represented?

We apply the formula D × e = d, with $e=\frac{1}{25000}$ and D = 750 m.

So: $d=750\times \frac{1}{25000}=0.03$, therefore d = 0.03 m = 3 cm.

On the map with a scale of $\frac{1}{25000}$, a path 750 m long is represented by 3 cm.