Sets problem

[metallicafan]

Each of the terms in sets R, S, and T are positive integers. If 9 of the integers in R are also in S, 12 of the integers that are in S are also T, and 4 of the integers that are in R are also in T, how many unique positive integers are contained in the three sets?

(1) 1 of the integers is in R, S, and T.
(2) R has 20 terms, S has 24 terms, and T has 30 terms.

Source: www.gmathacks.com

[mathbyvemuri]

Two statements together are required to solve it.
Answer option "C"

Explanation:
R and S = 9
S and T = 12
R and T = 4
only "R" + only "S" + only "T" is required

Only "R" = R - (R and S) - (R and T) - (R and S and T)
Only "S" = S - (S and R) - (S and T) - (R and S and T)
Only "T" = T - (T and R) - (T and S) - (R and S and T)

only "R" + only "S" + only "T"
= R+S+T-2(R and S)-2(R and T)-2(S and T)-3(R and S and T)

So to get the final answer we additionally need R,S,T and (R and S and T)

statement(1) gives (R and S and T), which alone is not sufficient
statement(2) gives R, S , T values, which alone is not sufficient

But (1) and (2) together are sufficient

[hey_thr67]

Answer is C

[Stuart@KaplanGMAT]

Each of the terms in sets R, S, and T are positive integers. If 9 of the integers in R are also in S, 12 of the integers that are in S are also T, and 4 of the integers that are in R are also in T, how many unique positive integers are contained in the three sets?

(1) 1 of the integers is in R, S, and T.
(2) R has 20 terms, S has 24 terms, and T has 30 terms.

Source: www.gmathacks.com

Let's solve using the most powerful rule for data sufficiency: number of equations vs number of unknowns.

First, picture (sorry, I'm not good at computer diagrams!) a Venn diagram with 3 circles. There are 7 different sections: Only R, only S, only T, RS overlap, RT overlap, ST overlap, RST overlap. Think of each of those sections as an unknown, giving us a total of 7 variables.

Next, break down the question stem: it gives us 3 equations related to the unknowns. So, we're missing 4 equations to solve.

1) 1 equation - insufficient.
2) 3 equations - insufficient.

Together: 4 equations - just what we needed - sufficient, choose C!