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A rental car agency purchases fleet vehicles in two sizes: a

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A rental car agency purchases fleet vehicles in two sizes: a [#permalink] New post 02 Dec 2010, 13:14
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A rental car agency purchases fleet vehicles in two sizes: a full-size car costs $10,000, and a compact costs $9,000. How many compact cars does the agency own?

(1) The agency owns 7 total cars.

(2) The agency paid $66,000 for its cars.
[Reveal] Spoiler: OA
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Re: Car Dealer - Data Sufficiency [#permalink] New post 02 Dec 2010, 16:03
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Yalephd wrote:
A rental car agency purchases fleet vehicles in two sizes: a full-size car costs $10,000, and a compact costs $9,000. How many compact cars does the agency own?

(1) The agency owns 7 total cars.

(2) The agency paid $66,000 for its cars.

Don't have the OA. I think this is an easy question but I am afraid that it might be deceptively simple. I just want to double check.


This is classic C-trap question. C-trap questions are the questions which are obviously sufficient if we take statements together. When we see such questions we should become very suspicious.

Let # of full-size car be F and # of compact cars be C. Question: C=?

(1) The agency owns 7 total cars --> F+C=7. Clearly insufficient to get C.

(2) The agency paid $66,000 for its cars --> 10,000F+9,000C=66,000 --> 10F+9C=66. Here comes the trap: generally such kind of linear equations (ax+by=c) have infinitely many solutions for x and y, and we cannot get single numerical values for the variables. But since F and C represent # of cars then they must be non-negative integers and in this case 10F+9C=66 is no longer simple linear equation it's Diophantine equation (equations whose solutions must be integers only) and for such kind on equations there might be only one combination of x and y (F and C in out case) possible to satisfy it. When you encounter such kind of problems you must always check by trial and error whether it's the case.

Now, it's quite easy to check whether 10F+9C=66 has one or more solutions. 9C=66-10F so 66 minus multiple of 10 must be multiple of 9: 66 is not a multiple of 9; 56 is not; 46 is not; 36 IS A MULTIPLE OF 9 (F=3 and C=4); 26 is not; 16 is not and 6 is not. So only one combination of F and C satisfies equation 10F+9C=66, namely F=3 and C=4. Sufficient.

Answer: B.

For more on this type of questions check:
eunice-sold-several-cakes-if-each-cake-sold-for-either-109602.html
martha-bought-several-pencils-if-each-pencil-was-either-a-100204.html
a-rental-car-agency-purchases-fleet-vehicles-in-two-sizes-a-105682.html
joe-bought-only-twenty-cent-stamps-and-thirty-cent-stamps-106212.html
a-certain-fruit-stand-sold-apples-for-0-70-each-and-bananas-101966.html
joanna-bought-only-0-15-stamps-and-0-29-stamps-how-many-101743.html
at-an-amusement-park-tom-bought-a-number-of-red-tokens-and-126814.html
collections-confused-need-a-help-81062.html

Hope it helps.
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Re: Car Dealer - Data Sufficiency [#permalink] New post 02 Dec 2010, 16:46
Thank you, Bunuel. I thought it was C. Your explanation was clear as always.

Bunuel wrote:
Yalephd wrote:
A rental car agency purchases fleet vehicles in two sizes: a full-size car costs $10,000, and a compact costs $9,000. How many compact cars does the agency own?

(1) The agency owns 7 total cars.

(2) The agency paid $66,000 for its cars.

Don't have the OA. I think this is an easy question but I am afraid that it might be deceptively simple. I just want to double check.


This is classic C-trap question. C-trap questions are the questions which are obviously sufficient if we take statements together. When we see such questions we should become very suspicious.

Let # of full-size car be F and # of compact cars be C. Question: C=?

(1) The agency owns 7 total cars --> F+C=7. Clearly insufficient to get C.

(2) The agency paid $66,000 for its cars --> 10,000F+9,000C=66,000 --> 10F+9C=66. Here comes the trap: generally such kind of linear equations (ax+by=c) have infinitely many solutions for x and y, and we can not get single numerical values for the variables. But since F and C represent # of cars then they must be non-negative integers and in this case 10F+9C=66 is no longer simple linear equation it's Diophantine equation (equations whose solutions must be integers only) and for such kind on equations there might be only one combination of x and y (F and C in out case) possible to satisfy it. When you encounter such kind of problems you must always check by trial and error whether it's the case.

Now, it's quite easy to check whether 10F+9C=66 has one or more solutions. 9C=66-10F so 66 minus multiple of 10 must be multiple of 9: 66 is not multiple of 9; 56 is not; 46 is not; 36 IS MULTIPLE OF 9 (F=3 and C=4); 26 is not; 16 is not and 6 is not. So only one combination of F and C satisfies equation 10F+9C=66, namely F=3 and C=4. Sufficient.

Answer: B.

Similar problems: gmat-prep2-92785.html?hilit=linear%20type

Hope it helps.
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Re: Car Dealer - Data Sufficiency [#permalink] New post 02 Dec 2010, 18:46
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In my blog post given below, I have discussed in detail how to find integer solutions to linear equations in 2 variables (i.e. equations such as this one)
See if it helps.
http://gmatquant.blogspot.com/2010/11/integral-solutions-of-ax-by-c.html
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Re: A rental car agency purchases fleet vehicles in two sizes: a [#permalink] New post 31 May 2013, 04:43
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Re: A rental car agency purchases fleet vehicles in two sizes: a [#permalink] New post 31 May 2013, 21:16
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Yalephd wrote:
A rental car agency purchases fleet vehicles in two sizes: a full-size car costs $10,000, and a compact costs $9,000. How many compact cars does the agency own?

(1) The agency owns 7 total cars.

(2) The agency paid $66,000 for its cars.


F.S 1 clearly insufficient.

F.S 2 states that 10*f+9*c = 66 [f is the # full size cars;c of compact car]

The only way in which we could get a 6 in the units digit for the RHS is through the factor 9*c --> 9*4 = 36. Thus, the no of compact cars = 4.Sufficient.

B.
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Re: A rental car agency purchases fleet vehicles in two sizes: a [#permalink] New post 02 Jun 2013, 13:13
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Stmt 1 : f + c = 7 …insufficient since we can not determine a specific number



Stmt 2 : 10000f + 9000c = 66000

Dividing both sides by 1000

10f + 9c = 66

Since cars can only be represented by integers, we can set the following equation :

9c = 66 – 10f

C = (66 – 10f)/9

We are looking for a number that is divisible by 9.We need to plug in numbers and arrive at a multiple of 9.

C = (66 – 10*3)/9 = 36/9 = 4

Hence the number of compact cars owned is 4.

Answer : B
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Re: A rental car agency purchases fleet vehicles in two sizes: a [#permalink] New post 18 Nov 2013, 10:32
Alt.:

st. 2)



$10.000 $9000 Total
# of cars 7 0 70.000
6 1 69.000
5 2 68.000
4 3 67.000
3 4 66.000 ---> Bingo it matches
Re: A rental car agency purchases fleet vehicles in two sizes: a   [#permalink] 18 Nov 2013, 10:32
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