"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, $x > y$
• Given: two packages — 3 pounds and 5 pounds
• Asked: which method is cheaper (separate or combined), and by how much?

There are two ways to solve this: algebra and plugging in numbers. We'll walk through both. Pick the one that works better for you.

Method 1: Algebra

Step 1: Cost of Mailing Separately

The 3-pound package: $x$ cents for the first pound, then $y$ cents for each of the remaining 2 pounds.

$\text{Cost}_{3} = x + 2y \text{ cents}$

The 5-pound package: $x$ cents for the first pound, then $y$ cents for each of the remaining 4 pounds.

$\text{Cost}_{5} = x + 4y \text{ cents}$

Total cost of mailing separately:

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

Step 2: Cost of Mailing Combined

Combined, it's an 8-pound package: $x$ cents for the first pound, then $y$ cents for each of the remaining 7 pounds.

$\text{Cost}_{\text{combined}} = x + 7y \text{ cents}$

Step 3: Compare the Two Costs

Now compare $2x + 6y$ (separate) versus $x + 7y$ (combined).

The difference:

$(2x + 6y) - (x + 7y) = x - y \text{ cents}$

Since $x > y$, this difference is positive. That means mailing separately costs more. Combined is cheaper.

The savings: $x - y$ cents.

The answer is (A).

The Algebra Reasoning (Without Full Subtraction)

If subtracting algebraic expressions feels abstract, here's another way to see it.

$2x + 6y$ has one more $x$ and one fewer $y$ compared to $x + 7y$.

Since $x > y$, adding an extra $x$ and removing a $y$ makes the total larger. So $2x + 6y > x + 7y$.

Separate costs more. Combined is cheaper. The difference is $x - y$.

Method 2: Plugging In Numbers

Step 1: Choose Numbers

Replace the variables with simple numbers that satisfy $x > y$.

Let $x = 10$ cents and $y = 5$ cents.

Any values where $x > y$ will work. Keep the numbers small and easy to compute with.

Step 2: Compute the Cost of Mailing Separately

3-pound package: 10 cents for the first pound + 5 cents × 2 remaining pounds = $10 + 10 = 20$ cents.

5-pound package: 10 cents for the first pound + 5 cents × 4 remaining pounds = $10 + 20 = 30$ cents.

Total for separate: $20 + 30 = 50$ cents.

Step 3: Compute the Cost of Mailing Combined

8-pound package: 10 cents for the first pound + 5 cents × 7 remaining pounds = $10 + 35 = 45$ cents.

Step 4: Compare

Separate: 50 cents. Combined: 45 cents.

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

Step 5: Match to the Answer Choices

Now plug $x = 10$ and $y = 5$ into each answer choice.

(A) Combined, savings of $x - y = 10 - 5 = 5$ cents. Matches our savings of 5 cents.

(B) Combined, savings of $y - x = 5 - 10 = -5$ cents. Doesn't match.

(C) Combined, savings of $x = 10$ cents. Doesn't match.

(D) Separately, savings of $x - y = 5$ cents. Wrong method (says separate is cheaper).

(E) Separately, savings of $y = 5$ cents. Wrong method.

The answer is (A).

Which Method Should You Use?

Both methods reach the same answer. The question is which one produces fewer errors for you.

Algebra is faster if you're comfortable manipulating expressions. The entire solution is three lines of setup and one subtraction.

Plugging in numbers takes longer but converts algebra into arithmetic. You're adding and multiplying small integers instead of distributing and combining variables. For most people, that's where errors drop.

If you're unsure, try both on your next few variable-answer problems and compare your accuracy. There's no universal best method — there's the method that gets you the most correct answers with the fewest mistakes.

One practical note: if you plug in numbers, pick numbers that are easy to work with. $x = 10$ and $y = 5$ made the arithmetic trivial. You could have chosen $x = 6$ and $y = 5$ — that works too, but the math is slightly less clean. Don't overthink the choice. If your first numbers turn out to be awkward, pick new ones.

Why This Problem Matters

About 22% of test takers miss this one. What's notable is that the wrong answers are spread evenly across all four incorrect options. There's no single trap answer pulling most of the misses.

That pattern tells us something. When wrong answers cluster around one option, there's a specific trap — a common mistake that most people make. When they spread out, it usually means people are getting lost in the setup. They don't have a clear process, so they end up somewhere random.

The fix is organizational. Write what's given and asked. Choose a method. Execute it step by step. Both methods in this walkthrough do the same thing — they just organize the same information differently.

If you find yourself getting lost on variable-answer problems, plugging in numbers is usually the more reliable fix. It's harder to get lost in arithmetic than in algebra.

Want the full strategy behind this problem? Read: GMAT® Quant: The 'Use What They Give You' Principle

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