Scale factors in ratios help us understand how quantities change in size while maintaining the same proportions. Let's explore this concept with a simple example.
Understanding Scale Factors with Ratios
Imagine you have a small picture of a car that is 2 inches wide and 1 inch tall. This picture can be represented by the ratio of its width to its height, which is 2:1.
Scaling Up
If you want to make a larger picture of the car that keeps the same proportions, you might decide to multiply both the width and the height by the same number, called the scale factor. Let's use a scale factor of 3.
- Original width: 2 inches
- Original height: 1 inch
- Scale factor: 3
New dimensions:
- New width = 2×3=62×3=6 inches
- New height = 1×3=31×3=3 inches
The new ratio of width to height is still 2:1 because 66 and 33 are in the same proportion as 22 and 11.
Scaling Down
Similarly, you can make a smaller picture by using a scale factor less than 1. Suppose the scale factor is 1221?.
- Original width: 2 inches
- Original height: 1 inch
- Scale factor: 1221?
New dimensions:
- New width = 2×12=12×21?=1 inch
- New height = 1×12=121×21?=21? inch
The new ratio of width to height is still 2:1 because 11 and 1221? are in the same proportion as 22 and 11.
Summary
- Scale Factor: A number you multiply both parts of a ratio by to increase or decrease the quantities while keeping the same proportion.
- Scaling Up: Multiplying by a scale factor greater than 1 to make everything bigger.
- Scaling Down: Multiplying by a scale factor less than 1 to make everything smaller.
In both cases, the ratio remains the same because the relationship between the quantities doesn't change.