Question

For 20 students, the mess bill for 12 days is $\$7000$. In how many days will the mess charges be $\$4900$ for 8 students?

Answer 1

Hint:
Here we will first find a 12 days mess bill for one student. Then we will use the unitary method to find a 1 day bill for 1 student. Using this we will find a one-day mess bill for 8 students. Then we will divide the given amount with the one-day mess bill for 8 students to find the required days.

Complete step by step solution:
It is given that the mess bill for 20 students for 12 days is equal to $\$7000$.
Therefore,
For 20 students mess bill for 12 days \[ = \$ 7000\]
Now using unitary method, we get
For 1 student mess bill for 12 days \[ = \dfrac{{7000}}{{20}} = \$ 350\]
Dividing the expression by 12, we get
$\therefore $ For 1 student mess bill for 1 day \[ = \dfrac{{350}}{{12}} = \$ \dfrac{{175}}{6}\]
Now, we have to find the bill for 8 students for 1 day
For 8 students mess bill for 1 day \[ = \dfrac{{175}}{6} \times 8 = \$ \dfrac{{700}}{3}\]
Now,
Number of days required for 8 students to have $\$4900$ bill = $\$4900$ \[ \div \] Mess bill of 8 students for 1day
\[ \Rightarrow \] Number of days required for 8 students to have $\$4900$ bill \[ = \dfrac{{4900}}{{\dfrac{{700}}{3}}}\]
\[ \Rightarrow \] Number of days required for 8 students to have $\$4900$ bill \[ = \dfrac{{4900 \times 3}}{{700}}\]
Simplifying the expression, we get
\[ \Rightarrow \] Number of days required for 8 students to have $\$4900$ bill \[ = 21\] days

So, it will take 21 days for 8 students to have a $\$4900$ mess bill.

Note:
We used the unitary method to solve the problem. The unitary method is a method where first, the per-unit quantity is calculated, and then the number of units is multiplied.
Here, we have to divide and not multiply the total amount of mess bills of 8 students by their one day mess bill. If we multiply these terms then we will get wrong answers.