Question

To mail a package, the rate is $x$ cents for the first pound and $y$ cents for each additional pound, where $x > y$. Two packages weighing 3 pounds and 5 pounds respectively can be mailed separately or combined as one package. Which method is cheaper, and how much money is saved?
A. Combined, with a savings of $x-y$ cents.
B. Combined, with a savings of $y-x$ cents.
C. Combined, with a savings of $x$ cents.
D. Combined, with a savings of $y$ cents.
E. None of these.

Answer 1

Consider two cases. In the first case, find the cost of mailing the packages separately and add the two costs to find the total cost of mailing the packages separately. In the second case, take the sum of the weights of two packages and consider it as a single package. Now, find the cost of mailing this package. Finally, compare the total cost in both the cases and find the savings by subtracting the low cost case from the high cost case.

Complete step by step answer:
Here, in the above question, we have been provided with two methods of mailing two packages. They can be mailed separately or combined as one package. We have to determine the cheaper method. Let us consider the two cases one-by-one.
1. Case (i) : When the packages are mailed separately.
It is given that the rate is $x$ cents for the first pound and $y$ cents for each additional pound. Therefore,
Cost of mailing 3 pound package = $x+2y$
Cost of mailing 5 pound package = $x+4y$
So, the total cost of mailing the packages separately = $\left( x+2y \right)+\left( x+4y \right)=2x+6y$.
2. Case (ii) : When the packages are mailed combined.
Here, when the packages are combined as one, then the total weight of that package will be the sum of the two individual packets. Therefore,
Total weight of combined package = 3 + 5 = 8 pounds.
So, the cost of mailing the 8 pound package = $x+7y$.
This $x+7y$ can be written as $\left( x+y \right)+6y$. Now, let us compare the cost of mailing in both the cases. So, we have been provided in the question, $x > y$. Adding $x$to both sides, we get,
$\begin{aligned} & x+x>x+y \\\ & \therefore 2x > x+y \\\ \end{aligned}$
Now, adding $6y$ to both sides, we get,
$\begin{aligned} & 2x+6y> x+y+6y \\\ & \therefore 2x+6y>x+7y \\\ \end{aligned}$
Clearly, from the above relation, we can say that the cost of mailing separately is greater than the cost of mailing as a combined package. Therefore, combined mailing will be cheaper.
Now, the savings will be the difference in the costs of mailing of the two methods.
$\therefore \text{saving}=\left( 2x+6y \right)-\left( x+7y \right)=x-y$.
Hence, option A is the correct answer.

Note:
One may note that, we have found the savings by subtracting the cost of mailing combined packages from the cost of mailing the packages separately. This is because the cost of mailing separately is more than the cost of mailing combined packages and so the difference remains positive. One can see that we have used the condition, $x>y$ given in the question to start the comparison of the costs. This is an important condition because there are no other methods that can be used.