"To Mail a Package, the Rate Is x Cents for the First Pound..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

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) Separately, with a savings of $x - y$ cents

(E) Separately, with a savings of $y$ cents

Try it before reading on.

Setting Up the Problem

Write what's given and what's asked.

• Given: first pound costs $x$ cents, each additional pound costs $y$ cents, $x > y$
• Package 1: 3 pounds
• Package 2: 5 pounds
• Option: mail separately or combine into one 8-pound package
• Asked: which method is cheaper, and by how much?

There are two clean ways to solve this. We'll walk through both.

Method 1: Algebra

Step 1: Cost of mailing separately

The 3-pound package costs $x$ for the first pound plus $y$ for each of the remaining 2 pounds:

$x + 2y \text{ cents}$

The 5-pound package costs $x$ for the first pound plus $y$ for each of the remaining 4 pounds:

$x + 4y \text{ cents}$

Total cost of mailing separately:

$(x + 2y) + (x + 4y) = 2x + 6y \text{ cents}$

Step 2: Cost of mailing combined

An 8-pound package costs $x$ for the first pound plus $y$ for each of the remaining 7 pounds:

$x + 7y \text{ cents}$

Step 3: Compare the two costs

We have:

• Separately: $2x + 6y$
• Combined: $x + 7y$

The problem tells us $x > y$. Looking at the two expressions:

• $2x + 6y$ has one more $x$ and one fewer $y$ than $x + 7y$
• Since $x$ is the more expensive unit, having an extra $x$ makes the separate method cost more

So combined is cheaper.

Step 4: Calculate the savings

Subtract the combined cost from the separate cost:

$(2x + 6y) - (x + 7y)$

Line them up vertically:

  2x + 6y
-  x + 7y
  -------
   x -  y

The savings is $x - y$ cents.

The answer is (A).

Method 2: Plug In Numbers

If algebra feels uncomfortable, or if you want a built-in error check, plug in numbers instead.

Step 1: Choose numbers

The problem says $x > y$. Let's use:

• $x = 10$ cents
• $y = 5$ cents

These are easy to work with. They satisfy the constraint. That's all we need.

Step 2: Solve the problem with those numbers

3-pound package mailed separately:

First pound: 10 cents

Remaining 2 pounds: $2 \times 5 = 10$ cents

Total: $10 + 10 = 20$ cents

5-pound package mailed separately:

First pound: 10 cents

Remaining 4 pounds: $4 \times 5 = 20$ cents

Total: $10 + 20 = 30$ cents

Both mailed separately: $20 + 30 = 50$ cents

Combined 8-pound package:

First pound: 10 cents

Remaining 7 pounds: $7 \times 5 = 35$ cents

Total: $10 + 35 = 45$ cents

Combined is cheaper. Savings: $50 - 45 = 5$ cents.

Step 3: Test each answer choice

Now plug $x = 10$ and $y = 5$ into each choice and look for the one that gives 5 cents.

(A) Combined, savings of $x - y = 10 - 5 = 5$ cents ✓

(B) Combined, savings of $y - x = 5 - 10 = -5$ cents ✗

(C) Combined, savings of $x = 10$ cents ✗

(D) Separately, savings of $x - y = 5$ cents ✗ (says separately, not combined)

(E) Separately, savings of $y = 5$ cents ✗ (says separately, not combined)

The answer is (A).

Both methods get there. The plug-in method gives you a numerical answer to verify against. The algebra method is faster if you're comfortable with it. Pick the one that produces fewer errors for you.

Why This Problem Matters

About 22% of test takers miss this one. What's interesting is that the wrong answers are spread across all four incorrect choices. There's no single trap pulling everyone in one direction.

That pattern tells us something. When wrong answers are clustered around one choice, there's a specific trap — a common mistake that most people make. When they're spread out, it usually means people are getting lost in the setup. They don't have a system for organizing the information, so they make different errors depending on where they lost the thread.

The fix is a process: write what's given and what's asked. Set up expressions for each option. Compare. Subtract. Each step is simple. But skipping any step opens the door to a wrong answer.

This problem also demonstrates the value of having two methods in your toolkit. If the algebra feels uncertain, plugging in numbers provides a check. If you're confident in algebra, it's faster. The best approach is the one that gets you the right answer consistently — and that varies from person to person.

For the full strategy behind when to use each method, read: GMAT® Variables in Answer Choices: A Plug-In Numbers System That Reduces Algebra Errors

From Episode 30 of Real GMAT® Problems (The GMAT® Strategy Podcast).