Problem Solving

1) An amount of money was divided between some people in such a way that if there had been 4 more people, everyone would have got tk 16 less. However, if there had been 4 less people, everyone would have got tk 16 less. However, if there had been 4 less people, everyone would have got tk. 24 more. How many people were there in the group?

A) 32
B) 24
C) 20
D) 16
E) none

2) Anik, kun, and arefin, working independently, can build a wall in 10, 12, and 15 days respectively. The income from getting one wall up is tk 500. If the three work together for 16 days, how much is kun's income?

A) tk 333
B) tk 375
C) tk 667
D) tk 750
E) cannot be determined

3) Hisham noticed number at the back of his favorite mirinda bottle which consists two digits. The difference between the digits of the given number is not less than 3. If the digits of the number interchange position, the difference between the new number and the old number is 9 times the difference of the two digits. What is the number?
A) 48
B)33
C)14
D) 78
E) cannot be determined

4) Tirtho invented a board game where a square board with N number of rows with N number of squares in each row was used. Which of the following can be the possible number of squares which are not alongside the boundaries?

A) 26
B) 8
C)24
D)0
E) 82

5) Tanveer has three children. All of the children have integer ages and the product of their ages is 36. If only the sum of their ages is also known to us, figuring out their individual age is impossible unless it is also known that the oldest child has a pet dog. What are the ages of the three children?

A) 2,3,6
B) 1,2,18
C) 2,2,9
D) 3,3,4
E) Cannot be determinded

6) Nafiz had a cup filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

A) 1/3 B) 1/4 C) 1/5 D) 1/6 E) 1/7

7) Nawaz was looking for such a number that will always be a factor of a 5-digit number of the form xy0xy, where x can take values from 1 to 9 and y can take values from 0 to 9. Which of the following numbers qualifies Nawaz''s criteria?
I. 143 II. 77 III. 93

A) I only
B) II only
C) III only
D) I and II only
E) I and III only

8) Rituraj has an aquarium, and he will buy exactly 2 of 5 creatures. The available creatures are: two turtles, a goldfish, a catfish and a sucker fish. It is given that a turtle does not fight with the other, neither does it attack the sucker fish. However the turtle will eat any fish in the aquarium. None of the other creatures fights with one another. What is the probability that if Rituraj randomly chooses two creatures, both of them will live umharmed?

A) 6/25
B) 3/5
C) 5/7
D) 1/2
E) Cannot be determined

9) After receiving a call from Ehsan regarding a proxy class, Samia started towards Dhaka at a constant speed. At the same time, Ashfaq got a call from his teenage wife and started running towards chittagong from dhaka at a constant speed. After they crossed each other, it took Samia 40 minutes to reach Dhaka, and it took Ashfaq 90 minutes to reach chittagong. What is the ratio of samia's speed to ashfaq's speed?

A) 4:9
B) 9:4
C) 3:2
D) 13:8
E) cannot be determined

10) Walid and Andalib are playing a Board game. Both of their pieces start from opposite ends, and if Andalib's piece goes 3 steps, walids piece goes 2 steps. However, once the pieces cross each other , if Andalib's piece goes 3 steps, walid's piece goes 6 steps. If it takes 9 turns for Walid's piece to reach andalib's end, what is the number of places from one end to the other end of the board?

A) 24
B) 30
C) 45
D) 81
E) Cannot be determined

11) In a test consisting of 50 questions, a student scores 1 mark for a correct answer, -1/3 for a wrong answer, and -1/6 for not attempting a question. If Moutusi's final score is 32, the number of questions marked wrong by Moutusi cannot be less than:

A) 12 B) 9 C) 6 D) 3 E) Cannot be determined

12) Adib wanted to find a number 'n' which lies between 1 and 96, inclusive, for which it is true that n(n+1)(n+2) is divisible by 8. How many different numbers can Adib find?

A)12 B) 24 C) 32 D) 48 E) 60

13) In 2012, Abrar took four batches (A2, 2A, 4D, and 10A) in Mentors Containing a total of 100 students. If all the students were born in 1993, what is the probability that atleast two of Abrar's students will have the same birthday>

A) (100*99)/(365*364)
B) 1/365*364
C) 1-(100!/365!)
D) 1-(365!/265!*365^100)
E)1-(100!/365^100)

14) If Dave works alone, he will take 20 more hours to complete a task than if he worked with Diana to complete the task. If diana works alone, she will take 5 more hours to complete the task, what is the ratio of the time taken by Dave to that taken by Diana if each of them worked alone to complete the task?

A) 4:1
B) 2:1
C)10:1
D) 3:1
E) 1:2

15) The number of livestock in a farm at the beginning of year 2000 was 100,000. During the year, the number was increased by p%. During the next year 2001, there was a famine and the number decreased by q%. A census at the end of year 2001 revealed that the number of livestock in the farm was 100,000. Which of the following expressions is correct?

A) P>Q
B) Q>P
C) P=Q
D) with the exception of 1 instance, p will be equal to q
E) there is no relation between p and q

3) Hisham noticed number at the back of his favorite mirinda bottle which consists two digits. The difference between the digits of the given number is not less than 3. If the digits of the number interchange position, the difference between the new number and the old number is 9 times the difference of the two digits. What is the number?
A) 48
B)33
C)14
D) 78
E) cannot be determined

The difference between the two numbers must be 3 or more. Therefore immediately rule out B and D

Consider A viz. 48. Reverse the numbers = 84

Difference between digits = 4

Difference between new number and old number = 36.

36 = 9 *4

Therefore A is the answer.

4) Tirtho invented a board game where a square board with N number of rows with N number of squares in each row was used. Which of the following can be the possible number of squares which are not alongside the boundaries?

A) 26
B) 8
C)24
D)0
E) 82

I feel there is something wrong with the options.

Total number of squares = n^2

Total number of squares alongside the boundary = 4n - 4

Total number of squares not alongside the boundary = n^2 - 4n - 4 = (n - 2) ^2

Now we have to equate this equation to the options to check which is a perfect square. But I do not see any perfect squares.

So either the options are wrong or my ideas are wrong. Any suggestions?

everything's eventual wrote:

3) Hisham noticed number at the back of his favorite mirinda bottle which consists two digits. The difference between the digits of the given number is not less than 3. If the digits of the number interchange position, the difference between the new number and the old number is 9 times the difference of the two digits. What is the number?
A) 48
B)33
C)14
D) 78
E) cannot be determined

The difference between the two numbers must be 3 or more. Therefore immediately rule out B and D

Consider A viz. 48. Reverse the numbers = 84

Difference between digits = 4

Difference between new number and old number = 36.

36 = 9 *4

Therefore A is the answer.

Hi
I think the ans for question 3 should be E

as both A & C satisfy the conditions.

For optionC 14
41-14 = 27
27/9 = 3 which is equal to (4-1)

Please let me know if I am wrong.

das.ashmita wrote: 3) Hisham noticed number at the back of his favorite mirinda bottle which consists two digits. The difference between the digits of the given number is not less than 3. If the digits of the number interchange position, the difference between the new number and the old number is 9 times the difference of the two digits. What is the number?
A) 48
B)33
C)14
D) 78
E) Cannot be determined

Hi
I think the ans for question 3 should be E

as both A & C satisfy the conditions.

For optionC 14
41-14 = 27
27/9 = 3 which is equal to (4-1)

Please let me know if I am wrong.

You are correct.

Let the two digit number be 10a + b, then the number after digit interchange is 10b + a.
The difference between the new number and the old number = 10a + b - (10b + a) = 9(a-b).
The question states that the difference between the new number and the old number is 9 times the difference of the two digits. i.e The difference is 9*(a-b) or 9*(b-a).

So, we can say that for any two digit number, the difference between the new number and the old number is 9 times the difference of the two digits. So the answer is E

A) 48, new number = 84. 84-48 = 36 = 9*(8-4). Answer choice A satisfies the condition.
B) 33, new number = 33. 33-33 = 0 = 9*(3-3). Answer choice B satisfies the condition but 3-3 is not greater than equal to 3. So, Answer choice B is not the answer
C) 14, new number = 41. 41-14 = 27 = 9*(4-1). Answer choice C satisfies the condition.
D) 78, new number = 87. 87-78 = 9 = 9(8-7). Answer choice D satisfies the condition but 8-7 is not greater than equal to 3. So, Answer choice D is not the answer.
E) Is the answer because we have two answer choices which seem correct.

p.s: The correct answer choice would be A if option C is 13 instead of 14.

4) Tirtho invented a board game where a square board with N number of rows with N number of squares in each row was used. Which of the following can be the possible number of squares which are not alongside the boundaries?

A) 26
B) 8
C)24
D)0
E) 82

I tried doing it manually,
n squares
4 5
5 14
6 26

hence A

I feel there is something wrong with the options.

Total number of squares = n^2

Total number of squares alongside the boundary = 4n - 4

Total number of squares not alongside the boundary = n^2 - 4n - 4 = (n - 2) ^2

Now we have to equate this equation to the options to check which is a perfect square. But I do not see any perfect squares.

So either the options are wrong or my ideas are wrong. Any suggestions?

Hi EE

The ans need not be perfect square.
You need to consider squares which can be formed by combining the smaller squares as well.

example for N=4. refer the attachment.

everything's eventual wrote:

4) Tirtho invented a board game where a square board with N number of rows with N number of squares in each row was used. Which of the following can be the possible number of squares which are not alongside the boundaries?

A) 26
B) 8
C)24
D)0
E) 82

I feel there is something wrong with the options.

Total number of squares = n^2

Total number of squares alongside the boundary = 4n - 4

Total number of squares not alongside the boundary = n^2 - 4n - 4 = (n - 2) ^2

Now we have to equate this equation to the options to check which is a perfect square. But I do not see any perfect squares.

So either the options are wrong or my ideas are wrong. Any suggestions?

If n = 2 then (n - 2) ^2 = 0.

6) Nafiz had a cup filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

A) 1/3 B) 1/4 C) 1/5 D) 1/6 E) 1/7

Let us say that the amount of liquid with 3 parts water and 5 parts syrup after drawing some of it be x.
Let us say that the amount of water mixed is y.

% of syrup in the new mixture = 1/2 = Total syrup / Total liquid = ((5/8)*x + (0)*y)/(x+y)
1/2 = (5/8)*x /[x+y]
x = 4y
x:y = 1:4
So out of 5 parts, one part is removed and one part of water is added.

The answer is [spoiler]1/5[/spoiler]

arifaisal wrote: Anik, kun, and arefin, working independently, can build a wall in 10, 12, and 15 days respectively. The income from getting one wall up is tk 500. If the three work together for 16 days, how much is kun's income?

A) tk 333
B) tk 375
C) tk 667
D) tk 750
E) cannot be determined

Let one wall = the LCM of 10, 12 and 15 = 60 units.

Anik's rate = w/t = 60/10 = 6 units per day.
Kun's rate = w/t = 60/12 = 5 units per day.
Arefin's rate = w/t = 60/15 = 4 units per day.

Combined rate for all 3 = 6+5+4 = 15 units per day.
Since each wall = 60 units, the time for all 3 to complete one wall = w/r = 60/15 = 4 days.
Since each wall takes 4 days, the number of walls completed in 16 days = 4.
Since the payment for each wall = 500, the total payment for 4 walls = 4*500 = 2000.

When all 3 work together, of the 15 units produced each day, Kun produces 5.
Thus, the fraction produced by Kun = 5/15 = 1/3.
Since Kun produces 1/3 of the work, he receives 1/3 of the total payment:
(1/3)(2000) = 666.7.

The correct answer is C.

arifaisal wrote: After receiving a call from Ehsan regarding a proxy class, Samia started towards Dhaka at a constant speed. At the same time, Ashfaq got a call from his wife and started running towards chittagong from dhaka at a constant speed. After they crossed each other, it took Samia 40 minutes to reach Dhaka, and it took Ashfaq 90 minutes to reach chittagong. What is the ratio of samia's speed to ashfaq's speed?

A) 4:9
B) 9:4
C) 3:2
D) 13:8
E) cannot be determined

We can plug in the answers, which represent the ratio of S's speed to A's speed.
After the two meet, (S's time to finish traveling) : (A's time to finish traveling) = 40:90 = 4:9.

Note the following:
When elements travel TOWARD each other, they WORK TOGETHER to cover the distance between them, so we ADD their rates.

Answer choice C: S:A = 3:2.
Let S's rate = 3 miles per hour and A's rate = 2 miles per hour.
Let the distance = 30, which is a multiple of S's rate, A's rate, and their combined rate (since 3+2 = 5 miles per hour).

Combined rate for S and A = 3+2 = 5 miles per hour.
Time for S and A to meet = d/r = 30/5 = 6 hours.

Time for S to travel the entire distance = d/r = 30/3 = 10 hours.
Since S travels for a total of 10 hours, and the time for S and A to meet is 6 hours, the time for S to finish after S and A meet = 10-6 = 4 hours.

Time for A to travel the entire distance = 30/2 = 15 hours.
Since S travels for a total of 15 hours, and the time for S and A to meet is 6 hours, the time for A to finish after S and A meet = 15-6 = 9 hours.

(S's time to finish) : (A's time to finish) = 4:9.
Success!

The correct answer is C.

Hi das.ashmita...For question 3) - I am an impatient little fellow. Once I saw that A) was satisfying the equation I didnt check for C).

For question 4) - Thank You.

arifaisal wrote:1) An amount of money was divided between some people in such a way that if there had been 4 more people, everyone would have got tk 16 less. However, if there had been 4 less people, everyone would have got tk 16 less. However, if there had been 4 less people, everyone would have got tk. 24 more. How many people were there in the group?

A) 32
B) 24
C) 20
D) 16
E) none

Number * average = total.
Regardless of the number of people, the total amount of money here does not change.
Thus, the change in the number and the change in the average must be RECIPROCALS:
If the new number is 6/5 of the old number, then the new average must be 5/6 of the old average, so that the new total is the same as the original total.

Let x = the average per person.
To determine the required number of people, we can plug in the answers.

Answer choice C: 20
When there are 4 more people, (new number)/(old number) = 24/20 = 6/5.
Thus, new average = (5/6)x.
Since the average decreases by 1/6, and the decrease is equal to $16, we get:
(1/6)x = 16.
x = 96.

When there are 4 fewer people, (new number)/(old number) = 16/20 = 4/5.
Thus, new average = (5/4)x.
Since the average increases by 1/4, and the increase is equal to $24, we get:
(1/4)x= 24.
x = 96.
Success!

The correct answer is C.

Algebraically:

Let N = the number of people and A = the average per person.
Sum = NA.
Since the sum never changes, in each of the cases below, (new number)(new average) = NA.

If there had been 4 more people, everyone would have got 16 less:
(N+4)(A-16) = NA
NA -16N + 4A - 64 = NA
-16N + 4A = 64.

If there had been 4 fewer people, everyone would have got 24 more:
(N-4)(A+24) = NA
NA + 24N - 4A - 96 = NA
24N - 4A = 96.

Adding the two equations, we get:
(-16N + 4A) + (24N - 4A) = 64+96
8N = 160
N = 20.