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Direct proportion detail analysis:
Example:
If the cost of a dress is \(₹\)300, then the price of 1 dress will be \(₹\)300. The price of the dress increases as the number of dresses increases. Proceeding the same way, we can find the cost of any number of such dresses.
Consider the above situation, when two quantities, namely the number of dresses and their prices, are related to each other. When the number of dresses increases, the price also increases in such a way that their ratio remains constant.
Let us denote the number of the dress as \(X\) and the price of the dress as \(Y\) rupees. Now observe the following table.
| Number of dress \(X\) | \(1\) | \(2\) | \(4\) | \(6\) | \(8\) | \(10\) |
| Price of the dress in \(₹Y\) | 1000 | 2000 | 4000 | 6000 | 8000 | 10000 |
From the table, we can observe that when the values of \(X\) increase, the corresponding values of \(₹Y\) also increase in such a way that the ratio in each case has the same value, which is a constant (say \(k\)).
Now let us find the ratio for each of the values from the table.
and so on.
All the ratios are equivalent, and their simplified form is .
In a general way, \(=\) \(= k\) (constant).
When \(X\) and \(Y\) are in direct proportion, we get \(= k\) or .
Important!
If any two ratios are given above, we should take them .
Their ratio will be .
[Where \(Y1\) and \(Y2\) are \(Y\) values that correspond to \(X1\) and \(X2\) values of \(X\)].
From the above table, we should take \(X1\) and \(X2\) from the values of \(X\). Similarly, \(Y1\) and \(Y2\) from \(Y\) values.
| Number of dress \(X\) | \(X1\) | \(X2\) |
| Price of the dress in \(₹ Y\) | \(Y1\) | \(Y2\) |
Unitary Method:
- This is one method 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 how 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 \(= ₹\) \(= ₹\)25.
That is the cost of \(10\) apples \(= ₹\) \(= ₹\) 250.
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