Question

In a hostel, 75 students had food provision for 24 days. If 15 students leave the hostel, for how many days would the food provision last?

Answer 1

Hint: Analyze that the given data defines the direct proportion or the inverse proportion. Then find the number of students that are left after 15 students leave the hostel. Then using the inverse proportion, find the required result.

Complete step-by-step solution:
It is given in the problem that the food is available for $24$ days, when there are $75$ students in the hostel.
The goal is to find the number of days that the food provision lasts when $15$ students leave the hostel.
It can be seen that as the number of students decreases, it will take more time to finish the food, so it is the inverse proportion.
It means that a decrease in the number of students will increase the number of days to finish the food. Suppose that ${S_1}$ is the initial number of students and ${F_1}$ be the number of days needed to finish the food for $75$ students.
${S_1} = 75$ and ${F_1} = 24$
Then applying the inverse proportion, we have
${S_1} = \dfrac{k}{{{F_1}}}$
Substitute the values ${S_1} = 75$ and ${F_1} = 24$ in the equation:
$75 = \dfrac{k}{{24}}$
$ \Rightarrow k = 75 \times 24$
Now, assume that ${S_2}$ is the number of students after leaving $15$ students and ${F_2}$ is the number of days needed to finish the food when there are ${S_2}$ students.
Then the value of ${S_2}$ is given as:
${S_2} = \left( {75 - 15 = 60} \right)$
As it is also the inverse proportion so we have
${S_2} = \dfrac{k}{{{F_2}}}$
Substitute the value of${S_2}$ and$k$, then we have
$60 = \dfrac{{75 \times 24}}{{{F_2}}}$
Solve the above equation for the value of${F_2}$.
\[{F_2} = \dfrac{{75 \times 24}}{{60}}\]
\[ \Rightarrow {F_2} = 15 \times 2\]
\[ \Rightarrow {F_2} = 30\]

Therefore, it will take $30$ days to finish the food when $15$ students leave the hostel.

Note:
When two quantities vary together then these are said as the direct proportion and when two quantities vary inversely, that is one quantity decreases when other increases and vise-versa, then this relation of quantity is said as inverse proportion.
In the given problem, as the students decrease, it will take more days to finish the food, so it is the inverse proportion.