Scale Drawings - figuring out actual lengths using millimetres

Question

3 A scale drawing has a scale of 1:50000. What is the drawing length of these actual lengths?

Express your answer in millimetres.

a 4 km

b 1250 m

c 5000 cm

d 6.5 km

e 20000 mm

f 2125 m

Answer 1

The ratio of the drawing length to the actual length is 1:50,000.

We can think about this several ways: every 1 km (or m, cm, etc.) of length in the drawing, corresponds to 50,000 km (or m, cm, etc.) in real life, or that the actual length is 50,000 longer than the drawing length.

Let d=drawing length and a=actual length

Then we can write this relationship as an equation:

d x 50,000 = a

To solve for d (drawing length) in terms of a (actual length) divide both sides by 50,000

(d x 50,000) / 50,000 = a / 50,000

d = a / 50,000.

So our drawing length is our actual length divided by 50,000.

However, before we divide we will start by converting our lengths into mm.

1 cm = 10 mm

and since there are 100 cm per m, we multiply 10 mm (mm per cm) by 100 (cm per m) to get...

1 m = 1,000 mm

and since there are 1,000 m per km, we multiply 1,000 mm (mm per m) by 1,000 (m per km) to get...

1 km = 1,000,000 mm

a. 4 km = 4 * 1,000,000 = 4,000,000 mm

4,000,000 mm / 50,000 = 80 mm

(So for an actual length of 4 km the scale length is 80 mm.)

b. 1,250 m = 1,250 * 1,000 = 1,250,000 mm

1,250,000 mm / 50,000 = 25 mm

c. 5,000 cm = 5,000 * 10 = 50,000 mm

50,000 mm / 50,000 = 1 mm

d. 6.5 km = 6.5 * 1,000,000 = 6,500,000 mm

6,500,000 mm / 50,000 = 130 mm

e. 20,000 mm / 50,000 = .4 mm

f. 2,125 m = 2,125 *1,000 = 2,125,000 mm

2,125,000 mm / 50,000 = 42.5 mm