you have to find the value of the following ratio:

\(ratio = \frac{P(y)}{P(x)}\)
with:
\(P(y)\): probability that the prize winner completed survey A and lived in a residence more than 75 years old at the time of the survey.
\(P(x)\): probability that the prize winner completed survey B and lived in a residence built in the previous 15 years.

We can decompose \(P(y)\) as (because of this assumption "Assume that percent of income spent on housing is proportionally distributed across residents of housing of varying ages"):
\(P(y) = \)(probability that the prize winner completed survey A)*(probability of live in a residence more than 75 years old at the time of the survey)
also, we can do the same with \(P(x)\):
\(P(x) = \)(probability that the prize winner completed survey B)*(probability of live in a residence built in the previous 15 years)

Lets start with \(P(y)\):
probability that the prize winner completed survey A (numerator: possible outcomes): \((37.5*(9\%+43\%) + 49.5*(27\%+9\%))\)
probability that the prize winner completed survey A (denominator: total outcomes): \((37.5 + 49.5)\)
probability that the prize winner completed survey A: \(\frac{(37.5*(9\%+43\%) + 49.5*(27\%+9\%))}{(37.5 + 49.5)}\)
for the second term:
probability of live in a residence more than 75 years old at the time of the survey (numerator: possible outcomes): \((100\%-85\%)*132,057,804\)
probability of live in a residence more than 75 years old at the time of the survey (denominator: total outcomes): \(132,057,804\)
probability of live in a residence more than 75 years old at the time of the survey: \(\frac{(100\%-85\%)*132,057,804}{132,057,804}\)
and then:
\(P(y) = \frac{(37.5*(9\%+43\%) + 49.5*(27\%+9\%))}{(37.5 + 49.5)}*\frac{(100\%-85\%)*132,057,804}{132,057,804}\)

and for \(P(x)\):
probability that the prize winner completed survey B (numerator: possible outcomes): \((37.5*(48\%) + 49.5*(64\%))\)
probability that the prize winner completed survey B (denominator: total outcomes): \((37.5 + 49.5)\)
probability that the prize winner completed survey B: \(\frac{(37.5*(48\%) + 49.5*(64\%))}{(37.5 + 49.5)}\)
for the second term:
probability of live in a residence built in the previous 15 years (numerator: possible outcomes): \((15\%-0\%)*132,057,804\)
probability of live in a residence built in the previous 15 years (denominator: total outcomes): \(132,057,804\)
probability of live in a residence built in the previous 15 years: \(\frac{(15\%-0\%)*132,057,804}{132,057,804}\)
therefore, \(P(x)\):
\(P(x) = \frac{(37.5*(48\%) + 49.5*(64\%))}{(37.5 + 49.5)}*\frac{(15\%-0\%)*132,057,804}{132,057,804}\)

Lets construct the ratio (a good strategy to overcome this exercise is to do not do any calculation, because we can cancel a lot of terms!):
\(ratio = \frac{P(y)}{P(x)} = \frac{37.5*(9\%+43\%) + 49.5*(27\%+9\%)}{(37.5*(48\%) + 49.5*(64\%))} = \frac{19.5 + 17.82}{18+31.68} = \frac{37.32}{49.68} = 0.75\)
(please notice that I already have simplified the expression)
Option B (2/3 = 0.66)
¿Why do I get that number?, because I picked smart numbers in order to cancel some terms (0% and 15% = 15%, and also 100%-85% = 15%), but, you can pick more accurate numbers, and you will see that the result converges to 0.66.

Top-notch explanation Sir

Below is how I tried to solve in short period of time.

Step1: See the answer choice.
A. 9/20
B. 2/3
C. 9/10
D. 13/10
E. 3/2

From this step, we can understand that we do not need accurate calculation for answer.

Step2:
Find out that we can ignore below sentences, as the both percentages with "" are pretty similar to 15%.
What is the approximate ratio of the probability that the prize winner completed survey A and "lived in a residence more than 75 years old at the time of the survey" to the probability that the prize winner completed survey B and "lived in a residence built in the previous 15 years"?

What we should focus is the ratio described below.

# of or % of people who completed survey A to # of or % of people who completed to survey B.
=
# of or % of people who completed survey A / # of or % of people who completed to survey B.

Step3
As all people in this survey answer either survey A or survey B, so we can convert equation to below.

(% of people who answered survey A) / (1- % of people who answered survey A )

Step4
# of people who pays more than 30 % of income for mortgage.
(27%+9% ) x 49.5 almost equal to 0.36 x 50 = 18 million

# of people who pays more than 30 % of income for rent.
(43%+9% ) x 37.5 almost equal to 0.5 x 38 = 19 million

18 million + 19 million = 37 million -- people answered survey A (1)
50 million + 38 million = 88 million -- people who answered either of surveys. (2)
88 million - 37 million = 51 million -- people who answered survey B. (3) = (2) - (1)

(1) < (2), so at this moment, we can eliminate D,E from answer.

37:51 = 37/51 = 0.72549..
= 0.73

Answer A) 9/20 = 0.45
Answer B) 2/3 = 0.666.. = 0.67
Answer C) 9/10 = 0.9

So most possible answer is B).
Answer : B)

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