# The population of a city increases at a rate proportional to the population at that time if the population of the city increases from \[20\]lakhs, the population after \[15\] years.

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**Hint:**In order to find the population after \[15\] years, we must apply the method of proportion between the time period and the population as time period is directly proportional to population. Upon solving the obtained equation, we get the required answer.

**Complete step by step answer:**

Now let us be brief about proportion. Now let us briefly discuss proportions. Proportion is nothing but saying that two ratios are equal. Two ratios can be written in proportion in the following ways- \[\dfrac{a}{b}=\dfrac{c}{d}\] or \[a:b=c:d\]. From the second way of notation, the values on the extreme end are called as extremes and the inner ones as means. Proportions are of two types: direct proportions and indirect or inverse proportions. In the direct proportion, there would be direct relation between the quantities. In the case of indirect proportion, there exists indirect relation between the quantities.

Now let us find the population after \[15\]years.

We know that, \[\text{time}\] \[\alpha\] \[\text{ population}\]

We can express this relation as

\[\dfrac{{{P}_{1}}}{{{T}_{1}}}=\dfrac{{{P}_{2}}}{{{T}_{2}}}\]

Let us consider the present year as \[1\].

\[\begin{align}

& {{P}_{1}}{{T}_{2}}={{P}_{2}}{{T}_{1}} \\

& \Rightarrow 2000000\times 15={{P}_{2}}\times 1 \\

& \Rightarrow 30000000 \\

\end{align}\]

\[\therefore \] The population after \[15\] years is \[30000000\].

**Note:**We must always assign the variable to the value to be found. We can use ratios and proportions in our daily life. We can apply a ratio for adding the quantity of milk to water or water to milk. We can apply proportions for finding the height of the buildings and trees and many more.

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