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The next set of medium/hard PS questions. I'll post OA's with detailed explanations on Friday. Please, post your solutions along with the answers.

1. The distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K. If the coordinates of K are (-3, y), what is the distance between point K and X-axis?

3. Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

4. A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

A. 9 liters B. 18 liters C. 27 liters D. 36 liters E. 45 iters

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

A. I only, B. II only, C. III only, D. I and II only E. I, II and III

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have?

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

A. 15 minutes B. 20 minutes C. 25 minutes D. 30 minutes E. 210 minutes

BONUS QUESTION: 11. Certain bowl contains 5 red marbles and 3 blue marbles only. One by one, every marble is drawn at random and without replacement. What is the probability that the seventh marble drawn is NOT blue?

1. The distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K. If the coordinates of K are (-3, y), what is the distance between point K and X-axis?

Point k is on the line x=-3, so its distance form the y-axis (x=0) is 3. Given that "The distance from the Y-axis to point K is 1/3 of the distance from the X-axis", its distance to the y-axis is 3*3=9

IMO E
_________________

It is beyond a doubt that all our knowledge that begins with experience.

2. What is the area of a region enclosed by |x/3|+|y/9|=10?

We can analize the function x/3 + y/9 = 10 first. y=90-3x This intersect the Y-axis is 90 and the X-axis in 30. The original function ( with the absolute values) has the same graph as y=90-3x, only symmetric for value of x<0 and values of y <0. So the answer is: it defines a triangle of Area=\(\frac{90*30}{2}=1350\)

IMO B
_________________

It is beyond a doubt that all our knowledge that begins with experience.

3. Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

rates : A=1/10 B=1/5 c=1/x

A*2h+(A+B)*2h+(A+B+C)*15min = 1 work.

(1/10)*2 +(1/10+1/5)*2+(1/10+1/5+1/x)*1/4=1

x=2 IMO C
_________________

It is beyond a doubt that all our knowledge that begins with experience.

4. A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

To empty X liters ==> 4 h. To empty X + 3*6 liters ==> 6 h.

\(\frac{x}{4}=\frac{x+18}{6}\)

\(x=36\)

IMO D
_________________

It is beyond a doubt that all our knowledge that begins with experience.

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

If 1/2 is in the set, then -1/2^2 and -1/3^3 are in the set. (-1/4 and -1/8 are in the set) If -1/4 is in the set, then -(-1/4)^2 and -(-1/4)^3 are in the set. -(-1/4)^3 = 1/64 so II is in the set, IMO B
_________________

It is beyond a doubt that all our knowledge that begins with experience.

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have?

To maximize the number of members we have to minimize the number of dollars donated by each one. So we use 10$ donations as long as we can, given the conditions in the question. Number of members who donated 10 dollars = 37. Tot = 370. The remaining 29 dollars are splitted between 2 members ( 15 and 14 for example) Tot members = 37+2=39

IMO C
_________________

It is beyond a doubt that all our knowledge that begins with experience.

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

Salary 100$. 64 dollars on food, 16 dollars on meat and 64-16=48 dollars on "non-meat" $on non-meat/$on food = 48/64= 0.75

IMO E
_________________

It is beyond a doubt that all our knowledge that begins with experience.

9. If x and y are integers and x + y = -12, which of the following must be true?

A. Both x and y are negative False : 1-13=-12 B. xy > 0 False: 1*(-13)<0 C. If y < 0, then x > 0 False: -6-6=-12 D. If y > 0, then x < 0 True E. x - y > 0 False: -13+1=-12 and -13-1<0
_________________

It is beyond a doubt that all our knowledge that begins with experience.

1. The distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K. If the coordinates of K are (-3, y), what is the distance between point K and X-axis?

A. 1/2 B. 1 C. 3 D. 4.5 E. 9.

The distance between Y axis and (-3,y) will be the distance between (0,y) and (-3,y)

2. What is the area of a region enclosed by |x/3|+|y/9|=10?

A. 675 B. 1350 C. 2700 D. 5400 E. 10800

|x/3|+|y/9|=10 will touch the x-axis at (30,0) and y-axis at (0,90) in Q1. Area of this region will be .5*30*90=1350. Hence the total area of the figure formed on all the 4 quadrants is 4*1350=5400

8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

s=t*v She left 20 minuters later ==>you subtract 20 minutes for her time; but arrived 15 min later ==>you add 15 to her time She increase the speed by 20% ==> you multiply her speed by 6/5 = 1.2 And finally since she walked the same distance you equal her "normal" equation : space=time*speed, to her new equation so : \((t-20+15)*\frac{6}{5}v=t*v\) divide by v \((t-5)6/5=t\) \(t=30\)

IMO D
_________________

It is beyond a doubt that all our knowledge that begins with experience.

3. Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

A. 1 B. 1.25 C. 2 D. 2.5 E. 4

Work done by : A in 2 hrs = 2/10.-------(1) A and B in 2 hrs = 2(1/10 + 1/5)--------(2) A,B and C in 15 mins = 1/4(1/10 + 1/5 + 1/x)---------(3) (1)+(2)+(3) =1

4. A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

A. 9 liters B. 18 liters C. 27 liters D. 36 liters E. 45 iters

I have actually back solved this question. Took 36 liters as capacity of the tank. Hence draining pipe rate is 36/4 = 9 l/hr.

now total out flow during rain is 9-3= 6 l/hr. multiply with 6 hrs - time taken to empty the tank- we arrive back at the original capapcity of the tank = 36 L

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

A. I only, B. II only, C. II only, D. I and II only E. I, II and III

My answer is D.

Since -1/2 is there we have -1/4 and -1/8 in the set. Since -1/4 is there we will have -(-1/4)^3= 1/64. -1/8 is there hence -(-1/8)^2 = 1/64 will be in the set.

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have?

A. 37 B. 38 C. 39 D. 40 E. 41

37 ppl each contributing 10$ = total 370 $. remaining 29$ can be split between 2 people ( 14 + 15 or any other combination meeting the criteria.

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

A. 16% B. 25% C. 32% D. 48% E. 75%

let Mary's total salary = 100. Sal spent on food = 64---(1), sal spent on meat = 16. food exp not spent on meat = 64-16=48.--(2) 2 as a % of 1 is 48/64 = 75%

8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

A. 15 minutes B. 20 minutes C. 25 minutes D. 30 minutes E. 210 minutes

D= dist of school. V= actual speed.

D/v= T time taken to reach school.

when she was 20 mins, time taken would be T+20

when she increased her speed to 6/5 ofher original. time taken is T+15.

now the difference of 5 mins in the time is due to 20% increase in speed.

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