Data Sufficiency

A certain car rental agency rented 25 vehicles yesterday, each of which was either a compact car or a luxury car. How many compact cars did the agency rent yesterday?

(1) The daily rental rate for a luxury car was $15 higher than the rate for a compact car.

(2) The total rental rates for luxury cars was $105 higher than the total rental rates for compact cars yesterday

OA: E

P.S: It took me more than 3 mins to solve this problem. Any smarter solution available in less time ?
@Experts - Could you please share your explanation.Thanks in advance

RBBmba@2014 wrote:A certain car rental agency rented 25 vehicles yesterday, each of which was either a compact car or a luxury car. How many compact cars did the agency rent yesterday?

(1) The daily rental rate for a luxury car was $15 higher than the rate for a compact car.

(2) The total rental rates for luxury cars was $105 higher than the total rental rates for compact cars yesterday

Clearly, neither statement alone is sufficient.

Statements combined:
Let C = the number of compact cars sold.
Since a total of 25 cars are sold, the number of luxury cars sold = 25-C.

Let R = the daily rate for a compact car.
Since the daily rental rate for a luxury car is $15 higher, the daily rate for a luxury car = R+15.

Total revenue for C compact cars sold at a rate of R dollars per day = CR.
Total revenue for 25-C luxury cars sold at a rate of R+15 dollars per day = (25-C)(R+15).
Since the total revenue for the luxury cars is $105 more than the total revenue for the compact cars, we get:
(25-C)(R+15) = CR + 105.

Note that the value of C -- which represents the number of compact cars -- must be an INTEGER VALUE.
But the value of R -- the daily rate for compact cars -- does NOT have to be an integer value.
Test whether different integer values for C are possible.

Case 1: C=15
Substituting C=15 into (25-C)(R+15) = CR + 105, we get:
(25-15)(R+15) = 15R + 105
10R + 150 = 15R + 105
45 = 5R
R = 9.
Thus, it is possible that 15 compact cars are sold at at a daily rate of $9.

Case 2: C=14
Substituting C=14 into (25-C)(R+15) = CR + 105, we get:
(25-14)(R+15) = 14R + 105
11R + 165 = 14R + 105
60 = 3R.
R = 20.
Thus, it is possible that 14 compact cars are sold at at a daily rate of $20.

Since C can be different values, the two statements combined are INSUFFICIENT.

The correct answer is E.

Thanks Mitch for your reply.
I did also plug in values but it took more than 3 mins to reach to conclusion. I considered three numbers C=10,C=15 and C=20 for compact cars and got NEGATIVE values for R in case of C=10 and C=20 . So my questions are -

1. Based on the fact that as R can't take NEGATIVE values, should we conclude the OA is E as we test the above plug-in values? Or we should choose the value of C such that R yields a POSITIVE always ?

2. Don't we have any other better time saving approach ?

Look forward to your feedback.

RBBmba@2014 wrote:Thanks Mitch for your reply.
I did also plug in values but it took more than 3 mins to reach to conclusion. I considered three numbers C=10,C=15 and C=20 for compact cars and got NEGATIVE values for R in case of C=10 and C=20 . So my questions are -

1. Based on the fact that as R can't take NEGATIVE values, should we conclude the OA is E as we test the above plug-in values? Or we should choose the value of C such that R yields a POSITIVE always ?

The value of C -- the number of compact cars -- must be an INTEGER.
The value of R -- the daily rate -- must be POSITIVE.

When we combine the two statements, we must determine whether more than one integer value for C will yield a positive value for R.
The safest approach is to test cases.

If ONLY ONE value for C yields a positive value for R, then the value of C can be determined.
in this case, the correct answer is C.
If MORE THAN ONE value for C yields a positive value of R, then the value of C cannot be determined.
In this case, the correct answer is E.

2. Don't we have any other better time saving approach ?

Look forward to your feedback.

This problem seems more time-consuming than a typical official problem, so don't worry if it took you longer than 2 minutes to solve.
That said, we should not expect to be able to solve every problem in 2 minutes.
Some problems will probably take longer; others will probably take less time.