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Intern  Joined: 16 Apr 2010
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The number of stamps that Kaye and Alberto had were in the  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 70% (02:05) correct 30% (02:21) wrong based on 1413 sessions

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The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90
Math Expert V
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Re: Pls explain how to come up to the slolution  [#permalink]

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42
kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

Given: $$\frac{K}{A}=\frac{5x}{3x}$$ (where $$x$$ is a positive integer) and $$\frac{5x-10}{3x+10}=\frac{7}{5}$$ --> $$x=30$$ --> $$K=150$$ and $$A=90$$.

Question: $$(K-10)-(A+10)=?$$ --> $$(K-10)-(A+10)=40$$.

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Re: Pls explain how to come up to the slolution  [#permalink]

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7
so before exchange ratio is 5x:3x (x basically cancels out but they might have 10:6, 15:9 etc. which reduces to 5:3) so after exchange we have 5x-10:3x+10 = 7:5 You can represent this in fractions like:
(5x-10)/(3x+10) = 7/5 and cross multiply to get
25x-50 = 21x + 70 -> 4x = 120 x = 30
NOW 30 IS NOT THE ANSWER - that is the multiplier so you need to find the number of stamps in possession so...

Kaye 7*30 =210
Alberto 5*30 = 150
210-150 = 60 = D
##### General Discussion
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Re: Pls explain how to come up to the slolution  [#permalink]

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shaselai wrote:
i think the problem here is yes you get X=30 BUT the new result is the ratio 7x:5x where the +/- 10 has already been taken into account or maybe i am wrong...?

Answer to this question is C (40), provided OA is wrong.

When we write $$\frac{K}{A}=\frac{5x}{3x}$$ it means that $$x$$ is the multiple of the initial ratio of 5/3 and not 7/5.

We found that $$x=30$$: initially K had $$5*30=150$$ stamps and now K has $$150-10=140$$ stamps. Initially A had $$3*30=90$$ stamps and now A has $$90+10=100$$ stamps. Current difference is $$140-100=40$$.

Hope it's clear.
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The number of stamps that Kaye and Alberto had were in the ratio  [#permalink]

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1
Throw x=30 back into the second equation you set up. That's the ratio after the transaction has taken place and the number of stamps is defined from that equation, not the first. Make sense?

x=30
5(30)-10=140
3(50)+10=100

140-100=40 stamps
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

1. We know the initial ratio of the number of stamps. We can represent the initial number of stamps that Kaye and Alberto had, as 5x and 3x resp
2. We can represent the final number of stamps that Kaye and Alberto had as 5x-10 and 3x + 10 resp. The final ratio is 5x-10/3x+10
3. The final ratio is also given as 7:5
4. As (2) and (3) both represent the final ratio of number of stamps and so as both are equivalent we can equate the two. 5x-10/ 3x+10 = 7/5
5. x=30
6. We now need to find out how many stamps Kaye and Alberto had after the gift.
7. Kaye had 5x-10 = 140 stamps. Alberto had 3x-10 = 100 stamps
8. So Kaye has 40 more stamps than Alberto after the gift.

Correct choice is c.
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

5+3 = 8
7+5= 12
LCM of 12 and 8 = 24
8*3 = 24. Thus (5*3) / (3*3) = 15/9
12*2 = 24. Thus (7*2) / (5*2) = 14/10
10 – 9 = 1unit
What we need is 14-10= 4
Therefore 10*4 = 40
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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2
1
Initially their counts is
5x:3x

After K gave 10 stamps to A their new ratio is
7y:5y

i.e. 5x-10=7y-----Eq 1
3x+10=5y ----- eq 2

Solving eq 1 qnd 2 we get y=20

so from 2nd ratio

7*20 - 5 *20 = 40
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Re: Pls explain how to come up to the slolution  [#permalink]

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2
Bunuel wrote:
kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

Given: $$\frac{K}{A}=\frac{5x}{3x}$$ (where $$x$$ is a positive integer) and $$\frac{5x-10}{3x+10}=\frac{7}{5}$$ --> $$x=30$$ --> $$K=150$$ and $$A=90$$.

Question: $$(K-10)-(A+10)=?$$ --> $$(K-10)-(A+10)=40$$.

Once x = 30 is calculated, we require to calculate (5x-10) - (3x+10)
= 2x-20
= 60-20
= 40 = Answer = C
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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2
so you can think about it this way. For some number (x) Kaye had 5x stamps and Alberto had 3x
then kaye gave 10 stamps and the ration became 7:5, we can turn this into an equation like so:

(5x-10)/(3x+10)=7/5
cross multiplying gives us: 21x+70=25x-50 or 120=4x x=30
but remember, we are trying to figure out how many more stamps kaye had than alberto after she gave him ten.
then she gave alberto 10, so she had 140 and he had 100
the difference is 40 stamps (C)
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Bunuel wrote:
Given: $$\frac{K}{A}=\frac{5x}{3x}$$ (where $$x$$ is a positive integer) and $$\frac{5x-10}{3x+10}=\frac{7}{5}$$ --> $$x=30$$ --> $$K=150$$ and $$A=90$$.

Hi Bunuel,
I find your method above a very elegant way to convert a 2 variable (K, A) problem into 1 variable (X).

It is clearly the same as saying: $$\frac{K}{A}=\frac{5}{3}$$ and $$\frac{K-10}{A+10}=\frac{7}{5}$$ but I find your way much clearer and straighforward (no factions, one eq, etc).

Would you recommend to apply this in ALL (or most, at least) ratio problems?

My doubt is that, under time pressure, adding an extra variable (X) may not seem the best idea at first glance, as it actually is.

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The number of stamps that Kaye and Alberto had were in the  [#permalink]

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Hi

all the above methods would take close to 2.5 min as the timer above suggests. The problem is easy but time consuming. Kudos to the one who can suggest a shorter solution! thanks!

If i had to triage that in 30 secs I would kick out B, D and E because they are all the multiples of 15 = 5*3. Choosing between 20 and 40 I would pick C and hit next because the odds for correct answers are on C answers
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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10
1
kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

Another method is to use ratios:

Note that the total number of stamps is the same in both the cases.
On ratio scale, 5:3 gives 8 total stamps while 7:5 gives 12 total stamps. So total stamps must be at least LCM of 8 and 12 i.e. 24.
So in first case, Kaye had 15 stamps while Alberto had 9. After the exchange, Kaye had 14 stamps while Alberto had 10. So Kaye gave 1 stamp to Alberto. But actually, Kaye gave 10 stamps to Alberto. So actual total number of stamps is 240.
So the multiplier of 7:5 is 20 (because 12*20 = 240). Kaye had (7-5)*20 = 40 more stamps than Alberto.

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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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Bunuel wrote:
shaselai wrote:
i think the problem here is yes you get X=30 BUT the new result is the ratio 7x:5x where the +/- 10 has already been taken into account or maybe i am wrong...?

Answer to this question is C (40), provided OA is wrong.

When we write $$\frac{K}{A}=\frac{5x}{3x}$$ it means that $$x$$ is the multiple of the initial ratio of 5/3 and not 7/5.

We found that $$x=30$$: initially K had $$5*30=150$$ stamps and now K has $$150-10=140$$ stamps. Initially A had $$3*30=90$$ stamps and now A has $$90+10=100$$ stamps. Current difference is $$140-100=40$$.

Hope it's clear.

Sometimes you wonder what quant section would be like without Bunuel. The first solution to the post was ultimately wrong, but inexplicably, hapless folks like me were falling over themselves giving it kudos. Tis terrific how GMAT tests the neatness of ones reasoning.
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

K/A = 5/3
(K-10)/(A+10) = 7/5

K=5A/3
K-10=(7A+70)/5
5K=7A+120
25A/3 = 7A+120
multiply by 3
25A=21A+360
4A=360
A=90
since A = 90, then K = 150
A+10=100; K-10=140
140-100=40

C=40.
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Similar solution to what has already been posted but I find it slightly easier to understand (maybe that's because it's what I did Given
5A = 3K
7*(A+10) = 5*(K-10)

Expand and solve simultaneously

7*(A+10) = 5*(K-10).........7A + 70 = 5K - 50.........7A = 5K - 120..........35A = 25K - 600
5A = 3K .............................................................................35A = 21K

Subtract to find that 4K = 600......K = 150

Apply the ratio
..........K......... A
..........5..........3
..........150.......90

..........140......100

Difference ....40.... (C)

(Test: new ratio is 14:10 i.e 7:5)
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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40 is another game where you don't read the question carefuly. The question is asking after the exchange
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Bunuel

Hi,

(5x-10)/(3x+10)=7/5 and not (5x-10)/(3x+10)=7y/5y

Why not? Why does he directly simplify the y's there?
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plalud wrote:
Bunuel

Hi,

In his answer to the question, http://gmatclub.com/forum/the-number-of ... ml#p755395, Bunuel writes

(5x-10)/(3x+10)=7/5 and not (5x-10)/(3x+10)=7y/5y

Why not? Why does he directly simplify the y's there?

$$\frac{7y}{5y} = \frac{7}{5}$$, because y is reduced.
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Re: The number of stamps that Kaye and Alberto had were in the  [#permalink]

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Top Contributor
kilukilam wrote:
The number of stamps that Kaye and Alberto had were in the ration of 5:3 respecctively. After Kaye gave Alberto 10 of her stamps, the ration of the number of Kaye had to the number of Alberto had was 7:5. As a result of the gift, Kaye had how many more stamps than Alberto

A. 20
B. 30
C. 40
D. 60
E. 90

One option is to solve the question using TWO VARIABLES.
Let K = # of stamps K had after the exchange
Let A = # of stamps A had after the exchange
This means that K+10 = # of stamps K had before the exchange
This means that A-10 = # of stamps A had before the exchange

Note: Our goal is to find the value of K-A

The number of stamps that K and A (originally) had were in the ratio 5:3
So, (K+10)/(A-10) = 5/3
We want a prettier equation, so let's cross multiply to get 3(K+10) = 5(A-10)
Expand: 3K + 30 = 5A - 50
Rearrange: 3K - 5A = -80

After K gave A 10 of her stamps, the ratio of the number K had to the number A had was 7:5
So, K/A = 7/5
We want a prettier equation, so let's cross multiply to get 5K = 7A
Rearrange to get: 5K - 7A = 0

At this point we have two equations:
5K - 7A = 0
3K - 5A = -80

Our goal is to find the value of K - A.
IMPORTANT: We need not solve for the individual values of K and A. This is great, because something nice happens when we subtract the blue equation from the red equation.
We get: 2K - 2A = 80
Now divide both sides by 2 to get: K - A = 40

Cheers,
Brent
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