ziyuenlau wrote:

A street vendor sells only hot dogs and hamburgers, and at the beginning of the day has a ratio of two hot dogs for every one hamburger. At the end of the day in which he did not add any new items or sell any hamburgers, and only sold some of his hot dogs, his new ratio is one hot dog for every two hamburgers. Which of the following cannot represent the number of hot dogs he sold?

(A) 2

(B) 3

(C) 6

(D) 9

(E) 24

Here's an approach that uses three variables.

Let D = # of hotdogs the vendor STARTED with

Let H = # of hamburgers the vendor STARTED with

Let x = # of hotdogs sold

At the beginning of the day has a ratio of two hot dogs for every one hamburger.So, D/H = 2/1

Cross multiply to get:

D = 2H At the end of the day in which he did not add any new items or sell any hamburgers, and only sold some of his hot dogs, his new ratio is one hot dog for every two hamburgers. So, (D - x)/H = 1/2

Cross multiply to get:

2(D - x) = HSo, we have the following system:

D = 2H2(D - x) = HTake

2(D - x) = H and replace D with

2HWe get: 2(2H - x) = H

Expand: 4H - 2x = H

Solve for H to get: H = 2x/3

Since H must be an INTEGER, we can see that x must be divisible by 3.

Answer choice A is the only answer that is NOT divisible by 3.

-----ALTERNATE APPROACH-------------------------------------------

Once we get to the point where we have 4H - 2x = H, we can also solve for x to get:

x = 3H/2At that point, we can check each answer choice to see what happens when x = that certain amount.

For example. let's check answer choice E first

It tells us that x = 24

Plug x = 24 into our equation:

24 = 3H/2Multiply both sides by 2 to get: 48 = 3H

Solve for H to get: H = 16

No problem. When x = 24, we get a nice integer value for the number of Hamburgers sold.

Now let's check answer choice A

It tells us that x = 2

Plug x = 2 into our equation:

2 = 3H/2Multiply both sides by 2 to get: 4 = 3H

Solve for H to get: H = 4/3

Problem.

When x = 2, we get a NON-integer value for the number of Hamburgers sold.

So, it is impossible to sell 2 hotdogs.

Cheers,

Brent

_________________

Brent Hanneson – Founder of gmatprepnow.com