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Question
How many people in a group of 50 own neither a fax machine nor a laser printer?
(1) The total number of people in the group who own a fax machine or a laser printer or both is less than 50.
(2) The total number of people in the group who own both a fax machine and a laser printer is 15.
How many people in a group of 50 own neither a fax machine nor a laser printer?
(1) The total number of people in the group who own a fax machine or a laser printer or both is less than 50.
(2) The total number of people in the group who own both a fax machine and a laser printer is 15.
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Solution
The correct option is E Statements (1) and (2) together are not sufficient.
Algebra Sets(1) This indicates that the total number who own either a fax machine or a laser printer or both is less than 50, but does not indicate how much less than 50 the total number is. Thus, the number of people in the group who own neither a fax machine nor a laser printer cannot be determined; NOT sufficient.(2) This indicates that, of the 50 people, 15 own both a fax machine and a laser printer, but does not indicate how many own one or the other. Thus, the number of people in the group who own neither a fax machine nor a laser printer cannot be determined; NOT sufficient. Taking (1) and (2) together, it is known that 15 people own both a fax machine and a laser printer and that the total number who own either a fax machine or a laser printer or both is less than 50, but the exact number who own neither still cannot be determined. For example, if 20 people own only a fax machine and 10 people own only a laser printer, then both (1) and (2) are true and the number of people who own neither a fax machine nor a laser printer is 50−20−10−15=5. However, if 10 people own only a fax machine and 10 people own only a laser printer, then both (1) and (2) are true and the number of people who own neither a fax machine nor a laser printer is 50−10−10−15=15.The correct answer is E; both statements together are still not sufficient.
Algebra Sets
(1) This indicates that the total number who own either a fax machine or a laser printer or both is less than 50, but does not indicate how much less than 50 the total number is. Thus, the number of people in the group who own neither a fax machine nor a laser printer cannot be determined; NOT sufficient.
(2) This indicates that, of the 50 people, 15 own both a fax machine and a laser printer, but does not indicate how many own one or the other. Thus, the number of people in the group who own neither a fax machine nor a laser printer cannot be determined; NOT sufficient.
Taking (1) and (2) together, it is known that 15 people own both a fax machine and a laser printer and that the total number who own either a fax machine or a laser printer or both is less than 50, but the exact number who own neither still cannot be determined. For example, if 20 people own only a fax machine and 10 people own only a laser printer, then both (1) and (2) are true and the number of people who own neither a fax machine nor a laser printer is 50−20−10−15=5. However, if 10 people own only a fax machine and 10 people own only a laser printer, then both (1) and (2) are true and the number of people who own neither a fax machine nor a laser printer is 50−10−10−15=15.
The correct answer is E; both statements together are still not sufficient.
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