Direct Proportion

Suppose the price of one piece of soap is 20 Rs.

If a person wants to buy one dozen pieces of soap, then he has to pay 240 Rs. If he wants to buy two dozen pieces of soap, he has to pay 480 Rs, and so on.
We can easily see that if the person buys more pieces he has to pay more, and he has to pay less if he buys fewer pieces.
\[\begin{array}{*{20}{c}} {{\text{No}}{\text{. of pieces of soap}}}&{}&{{\text{Total Price (Rs}}{\text{.)}}} \\ {{\text{12 Pieces}}}&{}&{{\text{240}}} \\ {{\text{24 Pieces}}}&{}&{{\text{480}}} \\ {{\text{36 Pieces}}}&{}&{{\text{720}}} \end{array}\]
That is, as pieces of soap are increased the total price also increases; conversely, if pieces of soap are decreased the total price also decreases. In this situation, we say that the number of pieces and the price are directly related.
In other words, if an increase in one quantity causes an increase in another quantity or a decrease in one quantity causes a decrease in another quantity, then we say that they are related directly (they are in direct proportion).
If $$x$$ and $$y$$ are in direct proportion, then the division of $$x$$ and $$y$$ will be constant; i.e.
\[\frac{x}{y} = c \Rightarrow x = cy\]
In the above example, we see that
\[\begin{gathered} \frac{{12}}{{240}} = \frac{1}{{20}} \\ \frac{{24}}{{480}} = \frac{1}{{20}} \\ \frac{{36}}{{720}} = \frac{1}{{20}} \\ \end{gathered} \]
Each ratio is the same.
Hence, if we are dealing with quantities which are related directly (which are in direct proportion), then we shall use the following rule:
\[\begin{array}{*{20}{c}} {{\text{No}}{\text{. of Pieces}}}&{}&{{\text{Total Cost}}} \\ {{\text{12}}}&{}&{{\text{240}}} \\ {{\text{24}}}&{}&{{\text{480}}} \end{array}\]
24 x 240 = 12 x 480
In general
\[\boxed{\begin{array}{*{20}{c}} {{\text{Quantity 1}}}&{}&{{\text{Quantity 2}}} \\ {\boxed{\begin{array}{*{20}{c}} {\text{a}} \\ {\text{d}} \end{array}}}&{}&{\boxed{\begin{array}{*{20}{c}} {\text{c}} \\ {\text{d}} \end{array}}} \\ {\frac{{\text{a}}}{{\text{b}}} = \frac{{\text{c}}}{{\text{d}}}}&{{\text{or}}}&{{\text{ad = bc}}} \end{array}}\]
Principle of Direct Proportion

If 30 dozen eggs costs 300 Rs, find the cost of 5 dozen eggs.
Let $$x$$ be the required price of 5 dozen eggs
\[\begin{array}{*{20}{c}} {{\text{Eggs (dozens)}}}&{}&{{\text{Cost (Rs}}{\text{.)}}} \\ {{\text{30}}}&{}&{{\text{300}}} \\ {\text{5}}&{}&x \end{array}\]
Since the quantities are in direct proportion, we use the above principle.
$$\frac{{30}}{5} = \frac{{300}}{x}$$
$$x$$ x 30 = 5 x 300
$$x = \frac{{5{\text{ x 300}}}}{{{\text{30}}}}$$= 50 Rs.