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

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Difficulty:   55% (hard)

Question Stats: 65% (02:11) correct 35% (02:48) wrong based on 312 sessions

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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

Originally posted by mun23 on 09 Dec 2012, 08:54.
Last edited by Bunuel on 09 Dec 2012, 09:19, edited 1 time in total.
Edited the question.
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Re: The ratio of cupcakes to children at a particular birthday  [#permalink]

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mun23 wrote:
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.

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

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

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

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

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1
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:

$$104(f)=7x(f) + R(f)$$
$$104(f)=7(14)(f) + 6(f)$$

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

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mun23 wrote:
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.
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Re: The ratio of cupcakes to children at a particular birthday  [#permalink]

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0<=104y-7xy<7y
0<=104-7x<7
13 6/7<x<=14 6/7
x=14

Uneaten:
104-7(14)=6 (mult. of 3 and 2, so D)
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Re: The ratio of cupcakes to children at a particular birthday  [#permalink]

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

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JoostGrijsen wrote:
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.

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!

Check my solution above which assumes values but uses logic at the end to ensure that it will hold in every case and hence must be true: the-ratio-of-cupcakes-to-children-at-a-particular-birthday-143833.html#p1332088
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Re: The ratio of cupcakes to children at a particular birthday  [#permalink]

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VeritasPrepKarishma wrote:
JoostGrijsen wrote:
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.

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!

Check my solution above which assumes values but uses logic at the end to ensure that it will hold in every case and hence must be true: the-ratio-of-cupcakes-to-children-at-a-particular-birthday-143833.html#p1332088

I still don't understand why it can't also be a multiple of 7? And hence why the answer can't be E instead of D. Please explain?
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Re: The ratio of cupcakes to children at a particular birthday  [#permalink]

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kooks123 wrote:
VeritasPrepKarishma wrote:
JoostGrijsen wrote:
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.

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!

Check my solution above which assumes values but uses logic at the end to ensure that it will hold in every case and hence must be true: the-ratio-of-cupcakes-to-children-at-a-particular-birthday-143833.html#p1332088

I still don't understand why it can't also be a multiple of 7? And hence why the answer can't be E instead of D. Please explain?

Check this first: the-ratio-of-cupcakes-to-children-at-a-particular-birthday-143833.html#p1152610

The question asks what MUST be true about the number of uneaten cupcakes? Not COULD be true. Thus, while the number of uneaten cupcakes COULD be a multiple of 7 (for example if k=7), it's not necessary to be (for example if k=1).

Hope it's clear.
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The ratio of cupcakes to children at a particular birthday  [#permalink]

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mun23 wrote:
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

Another approach is to use numbers and reasoning.

The less number to create the ratio

# Cupcakes (C)=104
# Children or kids (K) =7

C/k =104/7

After each kid has eaten x cupcakes the number left was less than # of kids. It means leftover (uneaten) is less than 7. It is between 1-6 cupcake.

Based on above we can eliminate choices C & E. 1 to 6 can't be multiple of 7

Now we need to work backward to find the number eaten by kids and the number should divide into 7. With sense of numbers 105 divides into 7. The next number divides into 7 is 98. We do not need to continue as uneaten cupcakes not huge.

Let uneaten cupcakes = 6
104 - 6 =98 ...........Hence the whole kids ate 98 cupcakes then each kid must have eaten x= 14 cupcakes

104- 5= 99.............x can't divide into 7

No need to continue till 1 as there is no number divides into 7 between 98 & 105

Hence uneaten cupcakes (6) should multiple of 2 & 3

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

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mun23 wrote:
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

let x=14 cupcakes
7*14=98 cupcakes eaten
104-98=6 cupcakes uneaten
I and II only
D
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Re: The ratio of cupcakes to children at a particular birthday  [#permalink]

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cupcakes/children = 104/7
given that
the number of cupcakes that remain uneaten is less than the number of children at the birthday party
max cupcakes that can be eaten has to be multiple of 7 <104 ; i.e 98 7*14
so left cupcakes = 6
which is multiple of 2 & 3
IMO D

mun23 wrote:
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 Re: The ratio of cupcakes to children at a particular birthday   [#permalink] 17 Oct 2019, 11:57
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