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09 Oct 2023, 03:57
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Difficulty:
55%
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Question Stats:
75% (02:55) correct
25%
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Of the 45 households in a certain neighborhood, 28 subscribe to Newspaper Q, 17 subscribe to Newspaper R, 12 subscribe to Newspaper S, 7 subscribe to both Q and R, 8 subscribe to both Q and S, and 9 subscribe to both R and S. The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers. If 39 of the households subscribe to at least one of the three newspapers, how many households subscribe to only one of the newspapers?
A) 15
B) 21
C) 27
D) 33
E) 46
A) 15
B) 21
C) 27
D) 33
E) 46
24 Jan 2024, 17:15
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Of the 45 households in a certain neighborhood, 28 subscribe to Newspaper Q, 17 subscribe to Newspaper R, 12 subscribe to Newspaper S, 7 subscribe to both Q and R, 8 subscribe to both Q and S, and 9 subscribe to both R and S. The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers. If 39 of the households subscribe to at least one of the three newspapers, how many households subscribe to only one of the newspapers?
We don't need to draw a Venn diagram to answer this question. If we understand the math, we can answer the question by simply doing the math on the sheet or in our heads.
However, for the purpose of illustrating what's going on, I'll use a Venn diagram.
We can start with three circles to represent the sets of households that subscribe to the three newspapers.
Then, we work our way from the center outward, starting with the households that subscribe to all three newspapers.
The passage says, "The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers." So, we know that the number that subscribe to all three is the following:
Households That Subscribe to None = All Households - Households That Subscribe to at Least One = 45 - 39 = 6.
Then, to find the number that subscribe to two but not all three, we subtract 6 from each number of households that subscribe to two.
Q and R only = 7 - 6 = 1
Q and S only = 8 - 6 = 2
R and S only = 9 - 6 = 3
Finally, to find the number that subscribe to one newspaper only, we subtract the numbers already in each circle from the total number that subscribe to each newspaper.
Q only = 28 - (2 + 6 + 1) = 19
R only = 17 - (1 + 6 + 3) = 7
S only = 12 - (2 + 6 + 3) = 1
Then, the total number of households that subscribe to only one newspaper is Q only + R only + S only = 19 + 7 + 1 = 27
A) 15
B) 21
C) 27
D) 33
E) 46
Correct answer: C
We don't need to draw a Venn diagram to answer this question. If we understand the math, we can answer the question by simply doing the math on the sheet or in our heads.
However, for the purpose of illustrating what's going on, I'll use a Venn diagram.
We can start with three circles to represent the sets of households that subscribe to the three newspapers.
Then, we work our way from the center outward, starting with the households that subscribe to all three newspapers.
The passage says, "The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers." So, we know that the number that subscribe to all three is the following:
Households That Subscribe to None = All Households - Households That Subscribe to at Least One = 45 - 39 = 6.
Then, to find the number that subscribe to two but not all three, we subtract 6 from each number of households that subscribe to two.
Q and R only = 7 - 6 = 1
Q and S only = 8 - 6 = 2
R and S only = 9 - 6 = 3
Finally, to find the number that subscribe to one newspaper only, we subtract the numbers already in each circle from the total number that subscribe to each newspaper.
Q only = 28 - (2 + 6 + 1) = 19
R only = 17 - (1 + 6 + 3) = 7
S only = 12 - (2 + 6 + 3) = 1
Then, the total number of households that subscribe to only one newspaper is Q only + R only + S only = 19 + 7 + 1 = 27
A) 15
B) 21
C) 27
D) 33
E) 46
Correct answer: C
07 Apr 2024, 06:56
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yrozenblum
First notice that we are given that out of 45, 39 households subscribe to at least one of the three newspapers. This means 6 households subscribe to none. This also means 6 susbcribe to all three (since "The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers")
Those that subscribe to 2 or more newspapers = 7 + 8 + 9 - 2*6 = 12
Why do we subtract 6 twice? Because it is counted each time in 7, 8 and 9. We need to count it only once.
Since 39 households subscribe to at least one of the three newspapers, if we subtract from this the number that subscribe to at least 2 or more, we will get the number that subscribe to only 1 newspaper.
Those that subscribe to only 1 newspaper = 39 - 12 = 27
Answer (C)
