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by nidhis.1408 » Wed Nov 16, 2011 3:17 pm
A car dealership sells only sports cars and luxury cars and has at least some of each type of car in stock at all times. If exactly 1/7 of the sports cars and exactly 1/2 of the luxury cars have sunroofs and there are exactly 42 cars on the lot, what is the smallest number of cars that could have sunroofs?

a. 6
b. 8
c. 11
d. 13
e. 16
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by user123321 » Wed Nov 16, 2011 4:17 pm
nidhis.1408 wrote:A car dealership sells only sports cars and luxury cars and has at least some of each type of car in stock at all times. If exactly 1/7 of the sports cars and exactly 1/2 of the luxury cars have sunroofs and there are exactly 42 cars on the lot, what is the smallest number of cars that could have sunroofs?

a. 6
b. 8
c. 11
d. 13
e. 16
let x = number of luxury cars with sunroof
y = number of sports car with sunroof
the problem says
2x + 7y = 42
and asks what is minvalue for x+y
we need to have y to be more and x to be less in order to have x+y a minimum value.
so y will be 4 & x will be 7, considering each type of car is present in the lot.
hence cars with sunroof can be 7+4 = 11

[spoiler]is it c?[/spoiler]

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by Anurag@Gurome » Wed Nov 16, 2011 7:01 pm
nidhis.1408 wrote:A car dealership sells only sports cars and luxury cars and has at least some of each type of car in stock at all times. If exactly 1/7 of the sports cars and exactly 1/2 of the luxury cars have sunroofs and there are exactly 42 cars on the lot, what is the smallest number of cars that could have sunroofs?

a. 6
b. 8
c. 11
d. 13
e. 16

Solution
Note that 1/7 < 1/2.
So, if the number of sports cars is more than the number of luxury cars by a maximum difference, we should get the smallest number of cars with sunroofs.
Also note that number of sports cars should be divisible by 7 and the number of luxury cars should be divisible by 2.
This means that 42 should be divided into two even positive integers with maximum difference such that the greater part is divisible by 7.
It can be easily deduced that 28 and 14 are the two parts.
Hence, the smallest number of cars with sunroofs = (1/7)*28 + (1/2)*14 = 4 + 7 = 11
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by GMATGuruNY » Thu Nov 17, 2011 1:50 am
nidhis.1408 wrote:A car dealership sells only sports cars and luxury cars and has at least some of each type of car in stock at all times. If exactly 1/7 of the sports cars and exactly 1/2 of the luxury cars have sunroofs and there are exactly 42 cars on the lot, what is the smallest number of cars that could have sunroofs?

a. 6
b. 8
c. 11
d. 13
e. 16
Let S = sports cars and L = luxury cars.

Sports cars with sunroofs = (1/7)S.
Luxury cars with sunroofs = (1/2)L.
A SMALLER fraction of sports cars have sunroofs.
Thus, to MINIMIZE the total number with sunroofs, we need to MAXIMIZE the value of S.
S must be a multiple of 7 (since 1/7 have sunroofs), and L must be even (since 1/2 have sunroofs).
The total number of cars is 42.

S=35, L=7.
Doesn't work, since L is not even.

S=28, L=14.
This works.
Total number with sunroofs = (1/7)28 + (1/2)14 = 4+7 = 11.

The correct answer is C.
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