# Direct and Inversely Proportion

## Direct Proportion

Two variables are direct proportional if there is always a constant ratio between them.

(As one variable increases, the other also increases.)

**$$\frac{y}{x}=k$$**

**$$y=k\xb7x\phantom{\rule{0ex}{0ex}}\downarrow \begin{array}{ccc}a& & c\\ b& & d\end{array}\downarrow \phantom{\rule{0ex}{0ex}}a:b=c:d\phantom{\rule{0ex}{0ex}}a\xb7d=b\xb7c$$**

*(The direction of the arrows follows the growth of the variables.When committing proportionality, follow the arrows direction!)*

Example:

2 books cost 6 euros. How much do 10 books cost?*(direct proportion: more books - more money)* $$\downarrow \begin{array}{ccc}2books& & 6\u20ac\\ 10books& & x\u20ac\end{array}\downarrow \Rightarrow 2:10=6:x\Rightarrow x=30$$

## Inversely Proportion

Two variables are inversely proportional if the product of those variables is a constant.

(As one variable increases, the other decreases.)

**$$y\xb7x=k$$**

**$$y=\frac{k}{x}\phantom{\rule{0ex}{0ex}}\downarrow \begin{array}{ccc}a& & c\\ b& & d\end{array}\uparrow \phantom{\rule{0ex}{0ex}}a:b=d:c\phantom{\rule{0ex}{0ex}}a\xb7c=b\xb7d$$**

*(The direction of the arrows follows the growth of the variables.When committing proportionality, follow the arrows direction!)*

Example:

20 workers finish a job in 20 days. In how many days will 5 workers finish the same job?*(inversely proportion: more workers - less days) $$\downarrow \begin{array}{ccc}2workers& & 20days\\ 5workers& & xdays\end{array}\uparrow \Rightarrow 2:5=x:20\Rightarrow x=8$$*

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