# Ratio, proportion, and scale.

Ratio, proportion, and scale are important concepts in mathematics and are frequently used in many different fields, including engineering, science, and finance. Here is a brief explanation of each concept:

**Ratio:** A ratio is a way of comparing two or more quantities. It is expressed as the quotient of two numbers and can be written in different ways, such as with a colon \( (3 \div 5) \), with a fraction \( \frac{3}{5} \) or as a decimal \((0.6)\). Ratios can be used to describe a wide range of relationships, such as the ratio of boys to girls in a class, or the ratio of the area of a square to the area of a circle.

**Proportion:** A proportion is a statement that two ratios are equal. In other words, if \( \frac{a}{b}=\frac{c}{d}\) , then we say that \(a\), \(b\), \(c\) and \(d\) are in proportion.

In mathematics, a proportion is a statement that two ratios are equal. A proportion can be written as \( \frac{a}{b}=\frac{c}{d}\), or as \( a \div b = c \div d \).

**The properties of proportion are:**

- Multiplying or dividing both terms of a ratio by the same non-zero number does not change the ratio or the proportionality relationship.

For example, if \( \frac{a}{b}=\frac{c}{d}\) , then \( \frac{2a}{2b}=\frac{2c}{2d}\) and \( \frac{a}{2b}=\frac{c}{2d}\) - If \( \frac{a}{b}=\frac{c}{d}\) , then \( \frac{b}{a}=\frac{d}{c}\). This is called the inverse property of proportion. It means that if we interchange the numerator and denominator of each ratio, we still have a valid proportion.
- If \(\frac{a}{b}=\frac{c}{d}\) and \(\frac{c}{d}=\frac{e}{f}\) , then \(\frac{a}{b}=\frac{e}{f}\). This is called the transitive property of proportion. It means that if two ratios are equal to a third ratio, then they are also equal to each other.
- If \( \frac{a}{b}=\frac{c}{d}\) , then \(\frac{a+b}{b}=\frac{c+d}{d}\). This is called the addition property of proportion. It means that if we add the two terms of each ratio, we still have a valid proportion.
- If \( \frac{a}{b}=\frac{c}{d}\) , then \(\frac{a-b}{b}=\frac{c-d}{d}\). This is called the subtraction property of proportion. It means that if we subtract the second term from the first term of each ratio, we still have a valid proportion.

**Scale:** Scale is a measure of the size or magnitude of something relative to a standard or reference point. In mathematics, scale is often used to represent distances or measurements in a diagram or map. For example, a map might use a scale of \( 1:1000 \), which means that 1 unit on the map represents 1000 units in the real world. Scales can also be used to represent the size of objects, such as in architectural drawings or engineering diagrams.