Ratios and Proportions

According to Wikipedia, a proportion is a mathematical statement expressing the equality of two ratios, and a ratio shows how many times one number contains another.

$$a:b=c:d$$

$a$ and $d$ are called the extremes, and $b$ and $c$ are called the means. Proportions can be written as fractions:

$$\frac{a}{b}=\frac{c}{d}$$

Properties of proportions

$$\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$$

Why is this true? Consider that if $\frac{a}{b}=\frac{c}{d}$, then there is some number $x$ that we can multiply $a$ by to get $c$ (i.e: $a\cdot x=c$), which is the same number that we can multiply $b$ by to get $d$ (i.e: $b\cdot x=d$):

$$\frac{a}{b}=\frac{a}{b}\cdot \frac{x}{x}=\frac{a\cdot x}{b\cdot x}=\frac{c}{d}$$

We can multiply a ratio by $1$ and not change it. Consider multiplying the ratio by a fancy version of $1$ that equals $\frac{(1+x)}{(1+x)}$:

$$\frac{a}{b}=\frac{a}{b}\cdot \frac{(1+x)}{(1+x)}=\frac{a\cdot (1+x)}{b\cdot (1+x)}=\frac{a+a\cdot x}{b+b\cdot x}=\frac{a+c}{b+d}$$