Question

Four extra-large sandwiches of exactly the same size were ordered for “m” students, where m > 4. Three of the sandwiches were evenly divided among the students. Since 4 students did not want any of the fourth sandwiches, it was evenly divided among the remaining students. If Carol ate one piece from each of the four sandwiches, the amount of sandwich that she ate would be what fraction of a whole extra-large sandwich?
A. $\dfrac{m+4}{m\left( m+4 \right)}$
B. $\dfrac{2m-4}{m\left( m-4 \right)}$
C. $\dfrac{4m-4}{m\left( m-4 \right)}$
D. $\dfrac{4m-8}{m\left( m-4 \right)}$
E. $\dfrac{4m-12}{m\left( m-4 \right)}$

Answer 1

Hint: To solve this question, we will first calculate the fraction of sandwich from a large sandwich that each student will get when 3 sandwiches are divided among m students which are given by the formula $\dfrac{\text{number of sandwiches}}{\text{number of students}}$ and then we will do the same for the remaining sandwich with the students other than the 4 students using the same formula. Then, we will add both of the calculated fractions as we have been given that Carol had a bite from all of the 4 sandwiches. Hence, we will get the required answer.

Complete step-by-step solution
Now, we have been given that there are ‘m’ number of students in which 3 out of 4 sandwiches were equally divided. Thus, 3 sandwiches have been divided into m equal pieces.
Since all the pieces are equal, the fraction of each piece from a large sandwich will be given as:
$\begin{align}
  & \dfrac{\text{number of sandwiches}}{\text{number of students}} \\
 & \Rightarrow \dfrac{3}{m} \\
\end{align}$
Now, we have also been given that 4 students didn’t want any part of the remaining sandwich. Thus, the last sandwich was divided into m-4 students equally.
Since all the pieces are equal here too, the fraction of each piece from a large sandwich will be given as:
$\begin{align}
  & \dfrac{\text{number of sandwiches}}{\text{number of students}} \\
 & \Rightarrow \dfrac{1}{m-4} \\
\end{align}$
Now, we have been given that Carol ate one piece from all of the sandwiches.
Thus, she got all the calculated pieces.
Hence, the amount of sandwich she ate as a fraction from a large sandwich will be equal to the sum of the calculated fractions.
Thus, the amount of sandwich she ate as a fraction from a large sandwich is given as:
$\dfrac{3}{m}+\dfrac{1}{m-4}$
Solving this, we get:
$\begin{align}
  & \dfrac{3}{m}+\dfrac{1}{m-4} \\
 & \Rightarrow \dfrac{3\left( m-4 \right)+1\left( m \right)}{m\left( m-4 \right)} \\
 & \Rightarrow \dfrac{3m-12+m}{m\left( m-4 \right)} \\
 & \therefore \dfrac{4m-12}{m\left( m-4 \right)} \\
\end{align}$
Thus, the required answer is $\dfrac{4m-12}{m\left( m-4 \right)}$.
Hence, the option (E) is the correct option.

Note: We have here used the formula $\dfrac{\text{number of sandwiches}}{\text{number of students}}$ for finding the required fractions. But this is a kind of general formula we can use for calculating fractions like this. For example, if we need to calculate work done in 1 second by a worker, we will calculate that by using $\dfrac{\text{amount of work done in the total time}}{\text{total time }\left( \text{in seconds} \right)}$. Similarly, we can use this in all the required situations.