Теория:

Direct proportion detail analysis:
Example:
If the cost of a watch is \(₹\)700, then the price of 1 watch will be \(₹\)700. The price of the watch increases as the number of watches increases. Proceeding the same way we can find the cost of any number of such watches.
Consider  the above situation, when two quantities, namely the number of watches and their prices are related to each other. When the number of watches increases, the price also increases in such a way that their ratio remains constant.
 
Let us denote the number of the watch as \(X\) and the price of the watch as \(Y\) rupees. Now observe the following table,
 
Number of watch \(X\)1246810
Price of the watch in \(₹  Y\)1000200040006000800010000
From the table, we can observe that when the values of \(X\) increase the corresponding values of \(₹Y\) also increases in such way that the ratio of XY in each case has the same value which is a constant (say \(k\)).
Now let us find the ratio for each of the value from the table.
 
XY=11000=22000=44000=66000=88000=1010000 and so on.
 
All the ratios are equivalent, and its simplified form is 11000.
In general way XY \(=\) 11000 \(=  k\) (constant).
When \(X\) and \(Y\) are in direct proportion, we get XY \(=  k\) or X=kY
  
Important!
If any two ratios are given above, we should take X1,X2andY1,Y2
 
Their ratio will be X1Y1=X2Y2 
 
[Where, \(Y1\), \(Y2\) are values of \(Y\) corresponding to the values \(X1\), \(X2\) of \(X\)].
From the above table, we should take \(X1\) and \(X2\) from the values of \(X\). Similarly, \(Y1\) and \(Y2\) from the values of \(Y\).
 
That is
 
Number of watch \(X\)\(X1\)\(X2\)
Price of the watch in \(₹ Y\)\(Y1\)\(Y2\)
 
Unitary Method:
  • This is one of  the methods to find out the values.
  • First, the value of one unit will be found. It will be useful to find the value of the required number of units. 
Example:
Consider that \(4\) apples cost \(₹100\). Then what will be the cost of \(10\) apples?
 
To find this first, we have to determine the cost of one apple (price per unit).
 
Then we can use this single quantity value to find our required quantity.
 
Therefore the cost of \(4\) apples \(= ₹100\).
 
Then the cost of \(1\) apple \(= ₹\)1004 \(= ₹\)25.
 
That is the cost of \(10\) apples \(= ₹\)25 ·10 \(= ₹\) 250.