Additional Rates

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The GMAT loves rates. Combined rates, average rates, simultaneous rates; you name it, the GMAT loves to test it. There's one more type of rate question that the GMAT will throw you, and its one that doesn't show up in many test-prep guides. I first wrote about it in October of 2007 when I highlighted it as a topic the GMAT had recently begun focusing on. Now that the 12th edition of the Official Guide has been released, it's clear that this question type is more and more common.

"Additional rates"

For lack of a better term, I'm going to call this type of item an "additional rate" question. Here are a couple of examples:

A hotel charges $90 for the first occupant of a room and $x for each additional occupant. If the total charge for a room is $180, how many occupants are in the room?

Wanda's weekly compensation is based on her sales. If she is paid 20 percent of her first $1,000 in weekly sales and 25 percent of all additional sales, what is her compensation in a week in which she makes $6,000 in sales?

Neither of these problems look particularly difficult, and they don't have to be. But it's easy to make a simple mistake.

The pattern

In both of the examples above, you'll find the word "additional." That is, there's one rate for some of the occupants or sales and another rate for the remaining occupants or sales. Consider translating the first sentence of the first example into algebra. Call the number of occupants p.

A hotel charges $90 for the first occupant of a room and $x for each additional occupant.

Clearly the result will have to include $90. The number of additional occupants is (p-1), so the algebraic expression is:

90 + x(p-1)

The pattern applies to the other question, as well. Instead of $90, the initial amount of compensation is 20% of $1,000, or $200. And instead of (p-1)--that is, the number of occupants minus the one occupant who was charged $90--we'll use (s - 1,000) for the amount of sales beyond the first 1,000:

200 + 0.25(s - 1,000)
200 + 0.25(6,000 - 1,000)

Almost every question of this nature can be dealt with using the same template:

  1. Start with the initial amount--in these cases, $90 or $200.
  2. Figure out the number of people, or dollars, or whatever, that is subject to the "additional rate." In the first example, that was (p - 1); in the second, it was (s - 1,000). Remember to subtract something to account for the word "additional."
  3. Multiply the result by the additional rate. In the first case, that was x (x dollars per additional occupant); in the second, it was 0.25 (25 percent commission on additional sales).

As with every type of GMAT math question, these come in all sorts of guises and difficulty levels, but the overall structure stays the same.



About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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