GCSE(F),

Measurements can be compared using a ratio. On a drawing, the height of a door is 2cm: the actual height of the door is 200cm. The ratio for the drawing to the door is 2:200. Divide both numbers by 2 for 1:100.

Take care when translating ratios from a linear measurement to an area. A small rectangle which is 2cm long 3cm high is compared with a larger rectangle which is 6cm long and 9cm high. The ratio for the lengths and heights are 1:3. However, the area of the first rectangle is 6cm^{2}, and for the second 54 cm^{2}. For the area, this is a ratio of 6:54, or 1:9. *The area ratio increases by the square of the linear ratio*.

Similarly, a cube of 1cm by 1cm by 1cm is a small model of a cube 5cm by 5cm by 5cm. The volume of the small cube is 1 x 1 x 1 = 1cm^{3}. the volume of the larger cube is 5 x 5 x 5 = 125cm^{3}. *The volume ratio increase by the cube of the linear ratio*.

1. The distance between the two rails of a railway line is normally 1432mm. The distance between two rails of a model railway line is 16.5mm. What is the ratio of the distance between the rails as model : real?

Answer: 1 : 87

Divide both 16.5 and 1432 by 16.5 for 1:87 (rounded to the nearest integer).

2. On a drawing, a field is 50cm long and 150cm wide. The actual field is 120m by 360m wide. What is the ratio of the areas?

Answer: 1 : 57600

The area of the drawing is 0.5m by 1.5m = 0.75 m^{2}.

The area of the actual field is 120 x 360 = 43200^{2}.

The ratio of the areas are 0.75:43200; divide both numbers by 0.75 for 1:57600.

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