Every length multiplied by the same number
Here is a rectangle 2 units wide and 3 units tall. To make a bigger copy with exactly the same shape, multiply every length by the same number. Multiply by 3, and the copy is 2 × 3 = 6 units wide and 3 × 3 = 9 units tall.
The number that every length is multiplied by is called the scale factor. Here the scale factor is 3, and the copy is called an enlargement of the rectangle.
The 2 by 3 rectangle and its copy with scale factor 3, which is 6 by 9. The dashed line from the shared corner passes through the opposite corner of both rectangles.
Why the shape stays the same
In the first rectangle the width and the height are in the ratio 2 : 3. In the copy they are 6 : 9, and 6 : 9 = 2 : 3, because both numbers were multiplied by 3. The width and the height still compare in the same way, so the copy has the same shape. Only its size has changed.
Every scale factor works the same way. With scale factor 2, a rectangle 3 wide and 2 tall becomes 3 × 2 = 6 wide and 2 × 2 = 4 tall: twice as wide and twice as tall. Its sides are in the ratio 6 : 4, which is 3 : 2 again.
A rectangle 3 wide and 2 tall, and its copy with scale factor 2, which is 6 wide and 4 tall.
Going back, and scales on models
To go from the copy back to the original, divide every length by the scale factor: 6 ÷ 3 = 2 and 9 ÷ 3 = 3.
A model or a map is a copy of something real, made with one scale factor for every length. A model ship built to the scale 1 : 150 has every length of the real ship divided by 150. So to go from the model to the real ship, multiply by 150, and to go from the real ship to the model, divide by 150. Before you multiply or divide, make sure both lengths are in the same unit.
The usual mistakes
Adding the scale factor instead of multiplying. With scale factor 3, a width of 2 becomes 2 × 3 = 6, not 2 + 3 = 5. Adding 3 to every side of the 2 by 3 rectangle gives a 5 by 6 rectangle, and 5 : 6 is not the same ratio as 2 : 3, so that rectangle is a different shape.
Scaling the wrong side. The new width comes from the old width, and the new height comes from the old height. For the 2 by 3 rectangle with scale factor 3, the new width is 2 × 3 = 6. The 9 is the new height.
Worked example: A Scale Model and the Real Object
Question A model of a ship is built to the scale 1 : 150. The model is 42 cm long. (a) How long is the real ship, in meters? (b) The mast of the real ship is 27 m tall. How tall is the mast of the model, in centimeters?
1.The scale 1 : 150 means that every length on the real ship is 150 times the matching length on the model.
Model to ship, multiply by 150. Ship to model, divide by 150. 2.Multiply the length of the model by 150: 42 × 150 = 6300 cm.
The ship is 42 × 150 = 6300 cm long. 3.(a) 6300 cm = 63 m, so the real ship is 63 m long.
(a) 6300 cm = 63 m. 4.For the mast, change meters to centimeters first: 27 m = 2700 cm. Going from the ship to the model, divide by 150.
The mast of the ship is 27 m, which is 2700 cm. 5.(b) The mast of the model is 2700 ÷ 150 = 18 cm tall. Check: 18 × 150 = 2700 cm.
(b) The mast of the model is 2700 ÷ 150 = 18 cm.
Answer: (a) 63 m; (b) 18 cm
Common mistakes
- Calculating 27 ÷ 150 = 0.18 and giving 0.18 cm. The 27 is in meters, so the result is 0.18 m, which is 18 cm. Change to centimeters before dividing.
- Dividing 42 by 150 in part (a). The real ship is larger than the model, so the length of the model is multiplied by the scale factor.