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The ratio of cupcakes to children at a particular birthday [#permalink]

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09 Dec 2012, 07:54

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The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?

I. It is a multiple of 2. II. It is a multiple of 3. III. It is a multiple of 7.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II and III

The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?

I. It is a multiple of 2. II. It is a multiple of 3. III. It is a multiple of 7.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II and III

Given that: The ratio of cupcakes to children is 104 to 7 --> \(\frac{cupcakes}{children}=\frac{104k}{7k}\);

Each child eats exactly x cupcakes --> the number of cupcakes eaten \(7kx\) and the number of cupcakes that remain uneaten is \(104k-7kx\);

The number of cupcakes that remain uneaten is less than the number of children --> \(104k-7kx<7k\) --> \(x>13\frac{6}{7}\) --> \(x=14\) (notice that x cannot be more than 14 since in this case 7kx>104k, which would mean that more cupcakes were eaten than there were).

Now, if \(x=14\), then the number of cupcakes that remain uneaten is \(104k-7k*14=6k\), thus the number of uneaten cupcakes must be a multiple of both 2 and 3.

The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?

I. It is a multiple of 2. II. It is a multiple of 3. III. It is a multiple of 7.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II and III

Assume there are 104 cupcakes and 7 children.

This line: "If the number of cupcakes that remain uneaten is less than the number of children at the birthday party,"

implies that 104 cupcakes are divided equally among 7 children till you get a remainder less than 7. 104/7 => Quotient is 14 (each child gets 14 cupcakes!) and remainder is 6. 6 is a multiple of 2 and 3.

What happens if there are 208 cupcakes and 14 children? (ratio is multiplied by 2) The quotient will still be 14 and the remainder will be 12 (the remainder will be multiplied by 2)

104a = 7a*14 + 6a

The remainder 6a will always be divisible by 2 and 3. Answer (D)
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Assuming that each kid eats the exact same amount of cupcakes just pick any number. So any multiple of 7 can be taken away from 104, i took 14 (each kid eats 2). 90 is only divisible by 2 and 3 not by seven.

One additional condition is that number of leftover cupcakes must be less than the number of kids. So 90 leftover cupcakes is not correct. But even if we ignore this condition, note that 90 is divisible by 2 and 3 but not by 7 so we can say that 7 is certainly out. But can we say that in every case, the leftover cupcakes WILL BE divisible by 2 and 3? Not necessary!

Re: The ratio of cupcakes to children at a particular birthday [#permalink]

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09 Sep 2013, 21:43

7 children eat 7x cupcakes.Let Z be the no. of cupcakes left, less than 7 7x+ z=104 7x=104-z. Now, 104-z must be more than 97 and must be divisible by 7, it can be only 98 7x=98+6 6 cupcakes left, so it is a multiplier of 2 and 3.

Re: The ratio of cupcakes to children at a particular birthday [#permalink]

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13 Feb 2014, 20:31

What is being asked is what is the remainder and its factors.

Given that 104 is not a multiple of 7 AND given that the proportionality of the ratio will be kept (unless otherwise stated), no matter which (f) factor you apply: You can express the ratio as a division with remainder (it is stated in the problem that R(f)<7(f)) to account for all the cupcacakes:

The remainder is 6(f), so it must be a multiple of 2 and 3, no matter by which (f) the R (and the entire ratio) is multiplied. You can substitute the (f) for other numbers and 6 and (f) are, obviously, the factors of the remainder.

Re: The ratio of cupcakes to children at a particular birthday [#permalink]

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10 Dec 2014, 06:38

Is it also correct to just do it like this:

Assuming that each kid eats the exact same amount of cupcakes just pick any number. So any multiple of 7 can be taken away from 104, i took 14 (each kid eats 2). 90 is only divisible by 2 and 3 not by seven.

Re: The ratio of cupcakes to children at a particular birthday [#permalink]

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20 Sep 2016, 04:47

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