Problem Solving

varun289 wrote:Lanier's construction company is offering home buyers a wide variety of home styles in a new planned community. Each home buyer must choose among 4 basic floor plans and 3 levels of bathroom fixtures (basic, intermediate, and premium). Additionally, each buyer must choose whether to purchase energy-efficient windows and whether to purchase security doors. Finally, each buyer must choose an exterior and an interior color scheme. If Lanier offers twice as many interior color schemes as exterior color schemes, and she offers the home buyer between 2500 and 3500 distinct home styles, how many exterior color schemes does Lanier offer?
A. 4
B. 5
C. 6
D. 8

Take the task of building a home and break it into stages.

Stage 1: Select a floor plan
There are 4 floor plans, so this stage can be accomplished in 4 ways.

Stage 2: Select a level of bathroom fixtures
There are 3 levels of bathroom fixtures, so this stage can be accomplished in 3 ways.

Stage 3: Select a window type (energy efficient or not)
There are 2 options, so this stage can be accomplished in 2 ways.

Stage 4: Select a door type (security or not)
There are 2 options, so this stage can be accomplished in 2 ways.

Stage 5: Select an exterior color scheme
We aren't told the the number of options, so let's let x = the # of color schemes available.
So, this stage can be accomplished in x ways.

Stage 6: Select an interior color scheme
We aren't told the the number of options, but we are told there are twice as many interior color schemes as exterior color schemes
In other words, the # of interior color schemes available = 2x
So, this stage can be accomplished in 2x ways.

By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus build the complete home) in (4)(3)(2)(2)(x)(2x) ways .

Simplify to get 96x^2 possible home styles. (we need to find the value of x)

The question tells us that there are between 2500 and 3500 distinct home styles.

In other words, 2500 < 96x^2 < 3500

Here's a quick approach to solving this for x.

Notice that 2500/100 = 25
So, 2500/96 will be a little bit bigger than 25. Let's say 2500/96 = "some # a bit bigger than 25"

When we take 2500 < 96x^2 < 3500 and divide all three parts by 96, we get:
some # a bit bigger than 25 < x^2 < ???

At this point, we can see that x cannot equal 5, which means x must equal 6.

Answer: C

Cheers,
Brent

Aside: For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775