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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?

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

Algebraic soln: Note that total time taken by Holly today is 5 min less than her usual time taken Let Holly's usual speed be x and her usual time taken be t Today her speed is 1.2x and time taken is (t-5) (we should not be bothered with unit of speed since it is bound to get cancelled) Distance traveled by Holly usually as well as today is the same i.e. from her home to school Thus x*t = 1.2x*(t-5) ----> we can cancel x since speed cannot be zero -----> t = 1.2t - 6 ------> t= 30 mins Answer D

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

Draining pipe takes 4 hr to empty the pool Rain inflow = 3lt/hr Time taken to drain the water from the pool on rainy day = 6

It takes 2 hr to drain the additional water added due to rain Water accumulation in six hrs of rain = 3*6 = 18lts

It takes 2 hrs to drain 18 lts => in 4 hours 36 lts of water will be drained from the pool and the capacity of the pool is 36 liters

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?

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

Answer. C. 39

To identify the maximum number each member of the team should have contributed closer or equal to minimum number (~$10)

37 members contributing $10 each will make $370. Since no one contributes $19, 2 members share the $29 between them which makes $399 from 39 members _________________

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%

Say mar yeans $100 and spends $16 for meat and $64 for food.

Percentage spent on meat in total food expenditure is 16/64 = 25% which makes 75% not spent 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 [/b] Mary left 20 Minutes later and reached school 15 minutes later. Because of the 20% speed increase she has reached the distance 5 minutes earlier

Substituting the value in each of the answer options, Speed difference between today and usual should be 20% ~Du/Tu – Dt/Tt = 20/100 = .2 Where, Du – Distance usual; Dt – Distance today; Tu – Time Usual; Tt – Time today Lets take mary’s speed is 1 unit/ minute

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 B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

X and Y are integers X+Y = -12

A. Both x and y are negative => not true as there is a possibility for one of the number is higher negative (e.g., X = -14; Y =2) B. xy > 0 => not true; one number can be positive and one number can be negative to make the condition true C. If y < 0, then x > 0 => It is not necessary that only one number should be negative we need at least one to be negative to make the statement true; In this scenario x can be negative. D. If y > 0, then x < 0 => True as at least one number should be negative to get the negative summation E. x - y > 0 => not true in the case of x = -6 and y = -6 and so many other scenarios

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

A, B, and C, can complete a certain task in 10, 5 and x hours respectively A completes 1/10 of the total work in 1 hr B Completes 1/5 of the total work in 1 hr C completes 1/x of the total work in 1 hr A starts working alone and 2 hours later B joins. After another 2 hours joins C. A completes 2/10 of the total work in 2 hrs later B joins A A&B together can complete 1/10 + 1/5 of the total work in 1 hr = 3/10 A&B work together for 2 hour and they complete 2 * 3/10 = 6/10 of the total work

By the time C joins A&B 8/10 of the total work is complete

To complete 2/10 of the work A, B & C together take 15 minutes (1/4 hr) 2/10 = ¼( 1/10 + 1/5+ 1/x) 8/10 = 3/10 + 1/X 5/10 = 1/X X = 2 Answer = C

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

Draining pipe takes 4 hr to empty the pool Rain inflow = 3lt/hr Time taken to drain the water from the pool on rainy day = 6

It takes 2 hr to drain the additional water added due to rain Water accumulation in six hrs of rain = 3*6 = 18lts

It takes 2 hrs to drain 18 lts => in 4 hours 36 lts of water will be drained from the pool and the capacity of the pool is 36 liters

Answer:D 36 liters

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

X = ½ -X^2 = ¼ = X1 -X^3 = -1/8 = X2

-X1^2 = 1/16 -X1^3 = -1/64 (I)

-X2^2 = 1/64 (II)

Ans D. I and II only

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

Answer. C. 39

To identify the maximum number each member of the team should have contributed closer or equal to minimum number (~$10)

37 members contributing $10 each will make $370. Since no one contributes $19, 2 members share the $29 between them which makes $399 from 39 members

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%

Say mar yeans $100 and spends $16 for meat and $64 for food.

Percentage spent on meat in total food expenditure is 16/64 = 25% which makes 75% not spent meat

Answer E. 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

Mary left 20 Minutes later and reached school 15 minutes later. Because of the 20% speed increase she has reached the distance 5 minutes earlier

Substituting the value in each of the answer options, Speed difference between today and usual should be 20% ~Du/Tu – Dt/Tt = 20/100 = .2 Where, Du – Distance usual; Dt – Distance today; Tu – Time Usual; Tt – Time today Lets take mary’s speed is 1 unit/ minute

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 B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

X and Y are integers X+Y = -12

A. Both x and y are negative => not true as there is a possibility for one of the number is higher negative (e.g., X = -14; Y =2) B. xy > 0 => not true; one number can be positive and one number can be negative to make the condition true C. If y < 0, then x > 0 => It is not necessary that only one number should be negative we need at least one to be negative to make the statement true; In this scenario x can be negative. D. If y > 0, then x < 0 => True as at least one number should be negative to get the negative summation E. x - y > 0 => not true in the case of x = -6 and y = -6 and so many other scenarios

Answers D

BANG ON!!!!!! Just some issue with Q-5 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

X = ½ -X^2 = ¼ = X1 -X^3 = -1/8 = X2

-X1^2 = 1/16 -X1^3 = -1/64 (I)

-X2^2 = 1/64 (II)

Ans D. I and II only

IMO -x^2 should be \(\frac{-1}{4}\) Likewise -X2^2 will be \(\frac{-1}{64}\) And we will never get +ve values. Thus only possible soln is A

I assumed –X^2 as (-x)^2 You are right it should be –(x^2), but the answer remains same X = ½ -X^2 = -¼ = X1 -X^3 = -1/8 = X2

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

A, B, and C, can complete a certain task in 10, 5 and x hours respectively A completes 1/10 of the total work in 1 hr B Completes 1/5 of the total work in 1 hr C completes 1/x of the total work in 1 hr A starts working alone and 2 hours later B joins. After another 2 hours joins C. A completes 2/10 of the total work in 2 hrs later B joins A A&B together can complete 1/10 + 1/5 of the total work in 1 hr = 3/10 A&B work together for 2 hour and they complete 2 * 3/10 = 6/10 of the total work

By the time C joins A&B 8/10 of the total work is complete

To complete 2/10 of the work A, B & C together take 15 minutes (1/4 hr) 2/10 = ¼( 1/10 + 1/5+ 1/x) 8/10 = 3/10 + 1/X 5/10 = 1/X X = 2 Answer = C

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

Draining pipe takes 4 hr to empty the pool Rain inflow = 3lt/hr Time taken to drain the water from the pool on rainy day = 6

It takes 2 hr to drain the additional water added due to rain Water accumulation in six hrs of rain = 3*6 = 18lts

It takes 2 hrs to drain 18 lts => in 4 hours 36 lts of water will be drained from the pool and the capacity of the pool is 36 liters

Answer:D 36 liters

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

X = ½ -X^2 = ¼ = X1 -X^3 = -1/8 = X2

-X1^2 = 1/16 -X1^3 = -1/64 (I)

-X2^2 = 1/64 (II)

Ans D. I and II only

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

Answer. C. 39

To identify the maximum number each member of the team should have contributed closer or equal to minimum number (~$10)

37 members contributing $10 each will make $370. Since no one contributes $19, 2 members share the $29 between them which makes $399 from 39 members

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%

Say mar yeans $100 and spends $16 for meat and $64 for food.

Percentage spent on meat in total food expenditure is 16/64 = 25% which makes 75% not spent meat

Answer E. 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

Mary left 20 Minutes later and reached school 15 minutes later. Because of the 20% speed increase she has reached the distance 5 minutes earlier

Substituting the value in each of the answer options, Speed difference between today and usual should be 20% ~Du/Tu – Dt/Tt = 20/100 = .2 Where, Du – Distance usual; Dt – Distance today; Tu – Time Usual; Tt – Time today Lets take mary’s speed is 1 unit/ minute

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 B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

X and Y are integers X+Y = -12

A. Both x and y are negative => not true as there is a possibility for one of the number is higher negative (e.g., X = -14; Y =2) B. xy > 0 => not true; one number can be positive and one number can be negative to make the condition true C. If y < 0, then x > 0 => It is not necessary that only one number should be negative we need at least one to be negative to make the statement true; In this scenario x can be negative. D. If y > 0, then x < 0 => True as at least one number should be negative to get the negative summation E. x - y > 0 => not true in the case of x = -6 and y = -6 and so many other scenarios

Answers D

BANG ON!!!!!! Just some issue with Q-5 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

X = ½ -X^2 = ¼ = X1 -X^3 = -1/8 = X2

-X1^2 = 1/16 -X1^3 = -1/64 (I)

-X2^2 = 1/64 (II)

Ans D. I and II only

IMO -x^2 should be \(\frac{-1}{4}\) Likewise -X2^2 will be \(\frac{-1}{64}\) And we will never get +ve values. Thus only possible soln is A

I assumed –X^2 as (-x)^2 You are right it should be –(x^2), but the answer remains same X = ½ -X^2 = -¼ = X1 -X^3 = -1/8 = X2

-X1^2 = -1/16 -X1^3 = +1/64 (I)

-X2^2 = -1/64 (II)

Yup u r right....Answer will not change Another one of my legendary silly mistakes

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?

A. 7/8 B. 3/4 C. 2/3 D. 5/8 E. 3/8

There are 8 marbles in total. The first thing we have to do is to calculate the number of outcomes of the first 6 drawn marbles. 1. R R R B B B (20 outcomes) (3 red 3 blue marbles can be arranged in 20 different ways) 2. R R R R B B (16 outcomes) (4 red 2 blue marbles can be arranged in 16 different ways) 3. R R R R R B (6 outcomes) (5 red 1 blue marbles can be arranged in 6 different ways) We need to find the probability of the seventh marble is RED. case 1. two remaining marbles are red (seventh always red) case 2. two remaining marbles are either blue or red (50% red) case 3. two remaining marbles are blue (always blue) So, the probability that the seventh marble drawn red is ((20+(16/2)+0))/(20+16+6)=28/42=2/3

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.

Coordinate of K is (-3,y). ==> Distance of K from Y-axis is 3. ==> Distance of K from X-axis is 9.

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

Area is enclosed by following four lines: 3x + y = 90 Passing through (0, 90) and (30, 0) 3x - y = 90 Passing through (0, -90) and (30, 0) - 3x + y = 90 Passing through (0, 90) and (-30, 0) - 3x - y = 90 Passing through (0, -90) and (-30, 0) This makes rhombus area with diagonals of length 60 and 180. Area of the region = d1*d2/2 = 60*180/2 = 5400

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

Time spent on the work is 4.25 hrs by A, 2.25 hrs by B, and .25 hr by C. Let’s say the work as w. A does w in 10 hrs ==> In 4.25 hrs, A does 0.425w B does w in 5 hrs ==> In 2.25 hrs, B does 0.45w C does w in x hrs ==> In 0.25 hrs, C does 0.25w/x ==> 0.425w + 0.45 w + 0.25w/x = w ==> 0.25w/x = w - 0.425w - 0.45 w = 0.125w ==> x = 2

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

Rain inflow in 6 hrs = 18 lt. ==> Draining pipe took extra 2 hrs to drain out extra 18 lt. water ==> Draining pipe can clear 36 lt. in 4 hrs. ==> Capacity of the pool is 36 liters.

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

If x is in the set, then both -x^2 and -x^3 are also in the set ==> There cannot be any other +ve number derived from the presence of 1/2 (as both -x^2 and -x^3 are –ve) ==> Option II and option III are not must

1/2 is in the set ==> -1/4 and -1/8 are in the set ==> -1/16, -1/64, and -1/512 are in the set ==> Option I is correct

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

Greatest number of members can be obtained by contributing least amount by most of the members. So, maximum number of members = upper round ($399) / 10 – 1 = 400/10 – 1 = 40 – 1 = 39

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%

Assuming salary as $100, $64 is spend on food and $16 spent on meat. So, 16/64 = 1/4 = 25% of the food expenditure was spent on meat. This means 75% of the salary spent on food was 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

Assuming usual time for Holly is t min. By increasing speed by 20%, today Holly took t/1.2 min. As found from the given data, today Holly has spend 5 mins less on commuting. So, t – 5 = t/1.2 ==> 1.2t – 6 = t ==> 0.2t = 6 ==> t = 30 min

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 B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

A. Not mandatory. X,y can have the values as -6,-6 or -20,8. Condition is true for first case and false for second case. B. Not mandatory. If x,y is -6,-6, xy >0. If x,y is -20,8, xy < 0. Condition is true for first case and false for second case. C. Not mandatory. X,y can be -6,-6 or 8,-20. Condition is true for second case and false for first case. D. As the sum of two numbers is –ve, if one is +ve, other one must be –ve. This is correct condition. E. Not mandatory. If x,y is -20,8, x-y < 0. If x,y is 8,-20, x-y > 0. Condition is true for second case and false for first case.