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**Hint:**We have given that the stock of grain in a government warehouse lasts $30$ days for $4000$ people. Assume that one person consumes $x$ food in one day and simplify for $30$ days and $4000$ people. After that, let the number of days for $6000$ people be $y$ days and equate both of them.

**Complete step-by-step answer:**Now let us assume that one person consumes $x$food in one day.

So, $1$ person $1$ day $\to $$x$

Now, for $4000$ peoples $1$ day$\to $$4000x$

Also, for $4000$ peoples $30$ day$\to $$30\times 4000x$

Hence, total food $=30\times 4000x$ ………… (1)

So, let the number of days for $6000$ people be $y$days.

Now, for $6000$ peoples $1$day$\to $$6000x$

Also, for $6000$ peoples $y$day$\to $$6000xy$

Total food $=6000xy$ ………….. (2)

Now, equating (1) and (2), we get,

$\Rightarrow$ $30\times 4000x=6000xy$

Simplifying we get,

$\Rightarrow$ $\dfrac{30\times 4}{6}=y$

Again, simplifying we get,

$\Rightarrow$ $y=20$days

**Therefore, the stock of grain in a government warehouse lasts $20$ days for $6000$ people.**

**Additional information:**

Direct Variation is said to be the relationship between two variables in which one is a constant multiple of the other. For example, when one variable changes the other, then they are said to be in proportion. If $b$ is directly proportional to $a$ the equation is of the form $b=ka$ (where $k$ is a constant). Two variables are said to be in direct variation when the variables are related in such a way that the ratio of their values always remains the same. Direct variation means when one quantity changes, the other quantity also changes in direct proportion. Inverse variation is exactly opposite to this.

**Note:**The quantities are said to be in direct proportion if an increase in the quantity A leads to an increase in quantity B and vice versa, provided their respective ratios are the same. Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value.

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