My own questions

If p, q, r are positive integers. Is p> q?

1) q> 2r

2)|p-r| > |q-r|

A cylindrical tank having a height of h units is filled with water to the maximum capacity of 120 litres. Later on identical holes are drilled at four places along the height at the heights of 0, h/4 , h/2 , 3h/4. If water leaks from the holes at 2litres / min . in how many minutes the tank would be completely empty?


A) 30 MINUTES

B) 30.25 MINUTES

C) 60 MINUTES

D) 45 MINUTES

E) 90 MINUTES

How many 5 letter arrangements of alphabets A, B,C,D,E are possible such that alphabets A and B

always come together and alphabet C is never adjacent to either of the alphabets A or B. Repeating

of alphabets is not allowed .


A) 8
B) 16
C) 24
D) 32
E) 64

lets consider AB as a one group. then we have [AB], C, D, E to be arrange.

-,-,-,- lets name these places as 1,2,3,4

Case1 AB comes at position 1 or 4

if [AB] comes at the position 1 then C can occupy either position 3 or 4 and the remaining 2 letters can arrange themselves in 2 ways. Also letters AB can arrange themselves in 2 ways. (AB and BA) Therefore total no. of ways = 1(for AB)*2 (for C)*2(for remaining letters)*2(in which A and B can arrange themselves)
=8
Also, total no. of ways will be same if [AB] comes at the position 4.

Hence total no of ways for case 1 = 8+8 =16

Case 2 AB comes at position 2 or 3

if AB comes at position 2 then C can only occupy position 4, and the remaining two letters can arrange themselves in 2 ways. Also letters AB can arrange themselves in 2 ways (AB and BA). Therefore total no. of ways =1(for AB) *1 (for C) *2(for remaining letters)*2(in which A and B can arrange themselves)
=4

Total no. of ways will be same if [AB] comes at position 3

hence total no. of ways for case 2 = 4+4=8

Total =Case1+Case 2 =16+8 =24

st1 is not sufficient as no information is provided about p

st2 it just states that the distance between p and r is greater than distance between q and r. now here two cases are possible

case1 p-------------------q---r
here q is greater than p
case 2 q----r-----------------------------p

here p is greater than q
hence statement 2 alone is not sufficient

combining 1 and 2 we have
q>2r

now for inequality |p-r| > |q-r| to hold true we must have p to be greater than q because the distance between q and r will be greater than r. if p assumes value less than p then the distance between p and r will become less than distance between q and r. which violates the condition statement 2. hence p must be greater than q. therefore answer should be C

volume =120, h=0 means hole is at the bottom of the cylinder, 3h/4 results in volume =90, h/2 results in volume =60, h/4 results in volume = 30

initially, volume =120 therefore all of these holes will work simultaneously until volume=90. since rate at which water leaks from these holes =2 therefore all of them when working will leak 8 liters /min. so, it will take ~4 min. to leak 30 liters.

Now when volume reaches 90 liters only three holes will work and together they will leak 6 liter/min and it will take them ~5 min. to leak 30 litre

volume now is 60 liters , therefore only two holes will work at 4 liter/min. Together they will leak 30 liters in ~8 min

volume left now is 30 liters and at it this instant only bottom hole will work at 2 liter/min and it will empty the cylinder in 15 min.

so therefore total time taken will be 4+5+8+15 = 32

The closest option is B.

Stmt 1 - tells about q and r only, can't relate p and q. It means A n D options are out.
Stmt 2 - Modulas implies absolute distance (dist.). Thus
Dist. between p and r > Dist. btw q and r .
It means numbers will be either in order "p , q, r" or in "r, q, p" on the number line. (note we are also given p,q,r are positive.so don't think about any number being negative)
But this also does not tell which is greater between p and q. Hence B out

Together
St. 2 already restricts to two number orders and then St.1 provides us that q>2r it means q > r always (As given p,q,r are positive)
this restricts the system to number order "r,q,p".
Sufficient . Ans C

Notice hole at 3h/4 will not be active once tank has emptied h/4 capacity water i.e. 30 liters, as water will be below that height only
If 4 holes active then total rate = 4*2 = 8 ltr/min
similarly for 3 holes = 6 ltr/min ....

T = 30/8 + 30/6 + 30/4 + 30/2 = (90+120+180+360)/24 = 750/24 = bit more than 30 min. as 750/25= 30 So Ans B

Nice questions innocous Added these to my "Tough questions list"

is the slope of line k positive?

1) x-intercept and y-intercept of line k are equal.

2) y-intercept of line k is not equal to zero.

This is how I approached it:
1) Not Sufficient: Lines with negative slope (Lines going like \) have equal intercept, however if intercepts are 0,0 then line with +ve slope (line going like /) will also have same intercepts. hence statement one alone is not sufficient.
2) Not sufficient :Does not tell anything about X intercept

Combined: Now if both intercepts are equal and Y-intercept is non zero, X-Intercept is non zero too. Hence Line must be of -Ve slope. Sufficient.

IMO C.

Bunuel, Carcass please let us know if above explaination is correct.

@aniketb

Your explanation is correct. I just don't think the GMAT likes to play with the x- and y-intercepts being zero, technically they are for a line passing through the origin, but in my experience they tend to avoid that condition, just because it borders on being ambiguous. I haven't seen them play with this in the official GMAT questions. Perhaps others can shed light on this as well.

To keep it in the GMAT spirit, I would modify the question to this:

In the xy-plane, what is the slope of line k that does not pass through the origin?

1) The x-intercept of line k is two more than the y-intercept.

2) The x-intercept of line k is twice the y-intercept.

Cheers,
Dabral