chacha0212 wrote:A housing subdivision contains only two types of homes: ranch-style homes and townhomes. There are twice as many townhomes as ranch-style homes. There are 3 times as many townhomes with pools than without pools. What is the probability that a home selected at random from the subdivision will be a townhome with a pool?

A. 1/6
B. 1/5
C. 1/4
D. 1/3
E. 1/2

This is an EITHER/OR group problem.
Every house is EITHER townhome OR ranch.
Every house EITHER has a pool arts OR does not.
For an EITHER/OR group problem, use a GROUP GRID (also known as a double-matrix) to organize the data.

Let R = ranch, T = townhome, P= pool, NP = no pool.
In the grids below, the entries in any given row or column must add up to the TOTAL of that row or column.

There are 3 times as many townhomes with pools than without pools.
Let the number of townhomes without a pool = 1, implying that the number of townhouses with a pool = 3 and that the total number of townhomes = 1+3 = 4.
The following grid is yielded:


There are twice as many townhomes as ranch-style homes.
Since there are 4 townhomes, the number of ranches = 2.
The following grid is yielded:


What is the probability that a home selected at random from the subdivision will be a townhome with a pool?
According to the resulting grid:
(townhomes with a pool)/total homes) = 3/6 = 1/2.

The correct answer is E.

I'd rate this as a 500-600 level problem.

chacha0212 wrote:A housing subdivision contains only two types of homes: ranch-style homes and townhomes. There are twice as many townhomes as ranch-style hoes. There are 3 times as many townhomes with pools than without pools. What is the probability that a home selected at random from the subdivision will be a townhome with a pool?

A. 1/6
B. 1/5
C. 1/4
D. 1/3
E. 1/2

Thank you in advance for explaining. Also what do you think the level of this question is?

Another way to solve:

Probability of a Subdivision to be townhome = 2/3
Probability of a Subdivision of a townhome have pool = 3/4

The probability of a randomly selected subdivision to be Townhome with pool = (2/3)x(3/4) = 1/2

Answer: Option E

I picked a smart number and solved this question

Let the no of ranch houses be 30
So, the no of town homes will be 60
Total houses = 90

From the 60 town houses, 45 will have a pool & 15 won't

The probability then becomes : 45/90 = 1/2

Hope that helps !!!!

It's a smart thing to solve such a question by picking a number but the Logic of choosing smart number should be clear.

1) The number should be a multiple of all the denominators as using numbers is beneficial to avoid calculation of fractions

2) The chosen numbers should be as small as possible so that we can avoid cumbersome calculation.

I would prefer to choose a number like 12 as base number if given a choice however 60 is not too bad either,