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Word Problems

by swerve » Wed Jun 09, 2021 9:45 am

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Ralph is giving out Valentine's Day cards to his friends. Each friend gets the same number of cards and no cards were leftover. If each friend gets at least one card, was the number of cards received by each friend more than one?

1) Ralph has 40 Valentine's Day cards to give out
2) If the number of friends were doubled, it would not be possible for each friend to get at least one card

The OA is B

Source: Veritas Prep

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Re: Word Problems

by Ian Stewart » Thu Jun 10, 2021 4:56 am

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swerve wrote:
Wed Jun 09, 2021 9:45 am
Ralph is giving out Valentine's Day cards to his friends. Each friend gets the same number of cards and no cards were leftover. If each friend gets at least one card, was the number of cards received by each friend more than one?

1) Ralph has 40 Valentine's Day cards to give out
2) If the number of friends were doubled, it would not be possible for each friend to get at least one card

The OA is B

Source: Veritas Prep
I don't think it's clear how the question is meant to be interpreted, except by looking at the OA. If we have N cards and f friends, from the stem we know N is a multiple of f. From Statement 2, if we have 2f friends, we no longer have enough cards to give to them, so 2f > N. If N is a multiple of f, but N is smaller than 2f, the smallest non-trivial multiple of f, then N clearly must equal f. So Statement 2 is sufficient, because everyone must be getting exactly 1 card.

But reading Statement 2, I don't know precisely what it means for it to "not be possible for each friend to get at least one card". Are we meant to also observe the restriction stated in the stem, that no cards can be left over? If not, the answer is B. But if so, the answer is E, because maybe we have 40 cards and 8 friends, and each friend gets five cards. When we double the number of friends, we can no longer distribute the cards with none left over (40 is not divisible by 16).
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