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 or he has to pay less if he buys less 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 total price also increased, conversely, if pieces of soap are decreased total price also decreased. In such situation, we say that pieces and price are directly related.
In other words, If increase in one quantity causes increase in other quantity or decrease in one quantity causes decrease in other quantity, then we say that they are related directly (They are direct proportion).
If x and y are in direct proportion, then 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 follow 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 dozens of eggs cost 300 Rs. Find the cost of 5-dozens of eggs.
Let x be the required price of 5 dozens eggs

\begin{array}{*{20}{c}} {{\text{Eggs  (dozens)}}}&{}&{{\text{Cost (Rs}}{\text{.)}}} \\ {{\text{30}}}&{}&{{\text{300}}} \\ {\text{5}}&{}&x \end{array}

Since quantities are in direct proportion, so 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.