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A box contains a certain number of cards, each of which is labeled by

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Afroditee
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A box contains a certain number of cards, each of which is labeled by [#permalink] New post   27 Oct 2022, 09:34
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A box contains a certain number of cards, each of which is labeled by one of the values ​​4, 8, or 10. The numbers of cards labeled by each value are directly proportional to the values ​​on the cards. What is the least possible number of cards in the box labeled by the value 10?

A. 5

B. 10

C. 11

D. 20

E. 22
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A
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Re: A box contains a certain number of cards, each of which is labeled by [#permalink] New post   27 Oct 2022, 09:51
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It's temping to pick 10, but the fact that 10 seems obviously correct is a clue indicating that we should look more carefully to see whether the only choice smaller than 10, which is 5, can somehow work.
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Re: A box contains a certain number of cards, each of which is labeled by [#permalink] New post   27 Oct 2022, 13:51
Afroditee wrote:
A box contains a certain number of cards, each of which is labeled by one of the values ​​4, 8, or 10. The numbers of cards labeled by each value are directly proportional to the values ​​on the cards. What is the least possible number of cards in the box labeled by the value 10?

A. 5

B. 10

C. 11

D. 20

E. 22


no. of cards with value 4 = 4x
no. of cards with value 8 = 8y
no. of cards with value 10 = 10z

10z will have the least integer value (as cards cant be in fractions) when z=1/2 hence the correct answer should be A.
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A box contains a certain number of cards, each of which is labeled by [#permalink] New post   Updated on: 02 Feb 2024, 12:42
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Since the numbers of cards labeled by each value are directly proportional to the values ​​on the cards, let k be the constant.

Let no. of cards with value 4 = 4k
Let no. of cards with value 8 = 8k
Let no. of cards with value 10 = 10k

HCF (4, 8, 10) = 2

So the value of k for 4k, 8k and 10k to be minimum positive integers is 1/2

Min. value of 10k = 10*0.5=5

Originally posted by JoeKan1234 on 27 Jan 2024, 06:19.
Last edited by JoeKan1234 on 02 Feb 2024, 12:42, edited 1 time in total.
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Re: A box contains a certain number of cards, each of which is labeled by [#permalink] New post   01 Feb 2024, 10:33
The question needs to be rewritten.

The nearest antecedent to "each of which" is "cards", indicating that the cards themselves are numbered and that a box contains a mixture of such cards.

The last sentence refers to a box being numbered, which is not consistent with the above.

Posted from my mobile device
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Re: A box contains a certain number of cards, each of which is labeled by [#permalink] New post   01 Feb 2024, 11:54
If we're talking about direct proportionality, it is much easier to express this question in terms of a ratio. Really what this question is saying is, "if cards a, b, and c are in a box in a ratio of 4:8:10, what is the last number of card c?" In this case, divide the proportion by 2 and you get 2:4:5.
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Re: A box contains a certain number of cards, each of which is labeled by [#permalink]
01 Feb 2024, 11:54
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