400 puzzle -- Part 2

Problem 1:

Math gurus may use series formula to solve it.(series: 1,3,5,7,9,11.....upto 18250 terms)

A worker earns a 5% raise. A year later, the worker receives a 2.5% cut in pay, & now his salary is Rs. 22702.68

What was his salary to begin with?

Solution :

Assume his salary was Rs. X

He earns 5% raise. So his salary is (105*X)/100

A year later he receives 2.5% cut. So his salary is ((105*X)/100)*(97.5/100) which is Rs. 22702.68

Hence, solving equation ((105*X)/100)*(97.5/100) = 22702.68  
X = 22176

Answer :

                   Rs.22176

-------------------------------------------------------------------------------------------------

Problem 2:

At 6'o a clock ticks 6 times. The time between first and last ticks is 30 seconds. How long does it tick at 12'o. Solution :

It is given that the time between first and last ticks at 6'o is 30 seconds.
Total time gaps between first and last ticks at 6'o = 5
(i.e. between 1 & 2, 2 & 3, 3 & 4, 4 & 5 and 5 & 6)

So time gap between two ticks = 30/5 = 6 seconds.

Now, total time gaps between first and last ticks at 12'o = 11
Therefore time taken for 12 ticks = 11 * 6 = 66 seconds (and not 60 seconds)

Answer : 

                     66 seconds

-------------------------------------------------------------------------------------------------

Problem 3:

500 men are arranged in an array of 10 rows and 50 columns according to their heights.

Tallest among each row of all are asked to come out. And the shortest among them is A.

Similarly after resuming them to their original positions, the shortest among each column are asked to come out. And the tallest among them is B.

Now who is taller A or B ?

Solution :

As it is mentioned that 500 men are arranged in an array of 10 rows and 50 columns according to their heights. Let's assume that position numbers represent their heights. Hence, the shortest among the 50, 100, 150, ... 450, 500 is person with height 50 i.e. A. Similarly the tallest among 1, 2, 3, 4, 5, ..... 48, 48, 50 is person with height 50 i.e. B

Now, both A and B are the person with height 50. Hence both are same.

Answer : 

               No one is taller, both are same as A and B are the same person. 
-------------------------------------------------------------------------------------------------

Problem 4:

In Mr. Mehta's family, there are one grandfather, one grandmother, two fathers, two mothers, one father-in-law, one mother-in-law, four children, three grandchildren, one brother, two sisters, two sons, two daughters and one daughter-in-law.

How many members are there in Mr. Mehta's family? Give minimal possible answer.

Solution :

There are 7 members in Mr. Mehta's family. Mother & Father of Mr. Mehta, Mr. & Mrs. Mehta, his son and two daughters.

Mother & Father of Mr. Mehta
|
|
Mr. & Mrs. Mehta
|
|
One Son & Two Daughters

Answer : 

               7

-------------------------------------------------------------------------------------------------

Problem 5:

When Alexander the Great attacked the forces of Porus, an Indian soldier was captured by the Greeks. He had displayed such bravery in battle, however, that the enemy offered to let him choose how he wanted to be killed. They told him, "If you tell a lie, you will put to the sword, and if you tell the truth you will be hanged."

The soldier could make only one statement. He made that statement and went free. What did he say?

Solution :

The soldier said, "You will put me to the sword."

The soldier has to say a Paradox to save himself. If his statement is true, he will be hanged, which is not the sword and hence false. If his statement is false, he will be put to the sword, which will make it true. A Paradox !!!
-------------------------------------------------------------------------------------------------

Problem 6:

A person wanted to withdraw X rupees and Y paise from the bank. But cashier made a mistake and gave him Y rupees and X paise. Neither the person nor the cashier noticed that.

After spending 20 paise, the person counts the money. And to his surprise, he has double the amount he wanted to withdraw.

Find X and Y. (1 Rupee = 100 Paise)

Solution : As given, the person wanted to withdraw 100X + Y paise. But he got 100Y + X paise. After spending 20 paise, he has double the amount he wanted to withdraw. Hence, the equation is         2 * (100X + Y) = 100Y + X - 20         200X + 2Y = 100Y +X - 20         199X - 98Y = -20         98Y - 199X = 20 Now, we got one equation; but there are 2 variables. We have to apply little bit of logic over here. We know that if we interchange X & Y, amount gets double. So Y should be twice of X or one more than twice of X i.e. Y = 2X or Y = 2X+1 Case I : Y=2X Solving two equations simultaneously 98Y - 199X = 20 Y - 2X = 0 We get X = - 20/3 & Y = - 40/2 Case II : Y=2X+1 Solving two equations simultaneously 98Y - 199X = 20 Y - 2X = 1 We get X = 26 & Y = 53 Now, its obvious that he wanted to withdraw Rs. 26.53 Answer :                 Rs. 26.53 ------------------------------------------------------------------------------------------------- Problem 7:

The game of Tic-Tac-Toe is being played between two players. Only the last mark to be placed in the game as shown.

Who will win the game, O or X? Can you tell which was the sixth mark and at which position? Do explain your answer. 
Assume that both the players are intelligent enough.

Solution :

The 7th mark must be placed in square 5 which is the win situation for both X and O. Hence, the 6th mark must be placed in a line already containing two of the opponents marks. There are two such possibilities - the 6th mark would have been either O in square 7 or X in square 9.

As we know both the players are intelligent enough, the 6th mark could not be O in square 7. Instead, he would have placed O in square 5 and would have won.

Hence, the sixth mark must be X placed in square 9. And the seventh mark will be O. Thus O will win the game.

Answer :  

                O will win the game. The sixth mark was X in square 9. -------------------------------------------------------------------------------------------------

Problem 8:

 At the Party:

1.  There were 9 men and children.

2.  There were 2 more women than children.

3.  The number of different man-woman couples possible was 24. Note that if there were 7 men and 5 women, then there would have been 35 man-woman couples possible.

Also, of the three groups - men, women and children - at the party:

4.  There were 4 of one group.

5.  There were 6 of one group.

6.  There were 8 of one group.

Exactly one of the above 6 statements is false.

Can you tell which one is false? Also, how many men, women and children are there at the party

Solution :

Assume that Statements (4), (5) and (6) are all true. Then, Statement (1) is false. But then Statement (2) and (3) both can not be true. Thus, contradictory to the fact that exactly one statement is false.

So Statement (4) or Statement (5) or Statement (6) is false. Also, Statements (1), (2) and (3) all are true.

From (1) and (2), there are 11 men and women. Then from (3), there are 2 possible cases - either there are 8 men and 3 women or there are 3 men and 8 women.

If there are 8 men and 3 women, then there is 1 child. Then Statements (4) and (5) both are false, which is not possible.

Hence, there are 3 men, 8 women and 6 children. Statement (4) is false.

Answer : 

               Statement (4) is false. There are 3 men, 8 women and 6 children.
-------------------------------------------------------------------------------------------------

Problem 9:

There is a shortage of tubelights, bulbs and fans in a village - Kharghar. It is found that

·   All houses do not have either tubelight or bulb or fan.

·   exactly 19% of houses do not have just one of these.

·   atleast 67% of houses do not have tubelights.

·   atleast 83% of houses do not have bulbs.

·   atleast 73% of houses do not have fans.

What percentage of houses do not have tubelight, bulb and fan?

Solution :

Let's assume that there are 100 houses. Hence, there should be total 300 items i.e. 100 tubelights, 100 bulbs and 100 fans.

From the given data, we know that there is shortage of atleast (67+83+73) 223 items in every 100 houses.

Also, exactly 19 houses do not have just one item. It means that remaining 81 houses should account for the shortage of remaining (223-19) 204 items. If those remaining 81 houses do not have 2 items each, there would be a shortage of 162 items. But total of 204 items are short. Hence, atleast (204-162) 42 houses do not have all 3 items - tubelight, bulb and fan.

Thus, 42% houses do not have tubelight, bulb and fan.

Answer :  

              42% houses do not have tubelight, bulb and fan.
-------------------------------------------------------------------------------------------------

Problem 10:

Mr. Subramaniam rents a private car for Andheri-Colaba-Andheri trip. It costs him Rs. 300 everyday.

One day the car driver informed Mr. Subramaniam that there were two students from Bandra who wished to go from Bandra to Colaba and back to Bandra. Bandra is halfway between Andheri and Colaba. Mr. Subramaniam asked the driver to let the students travel with him.

On the first day when they came, Mr. Subramaniam said, "If you tell me the mathematically correct price you should pay individually for your portion of the trip, I will let you travel for free."

How much should the individual student pay for their journey?

Solution:

Note that 3 persons are travelling between Bandra and Colaba.

The entire trip costs Rs. 300 to Mr. Subramanian. Hence, half of the trip costs Rs. 150.

For Andheri-Bandra-Andheri, only one person i.e. Mr. Subramaniam is travelling. Hence, he would pay Rs. 150.

For Bandra-Colaba-Bandra, three persons i.e Mr. Subramaniam and two students, are travelling. Hence, each student would pay Rs. 50.

Answer : 

                The individual student should pay Rs. 50 for their journey.
-------------------------------------------------------------------------------------------------

Problem 11:

Substitute digits for the letters to make the following Division true

                   O U T

           -------------

   S T E M | D E M I S E

           | D M O C

           -------------

               T U I S

               S T E M

              ----------

                 Z Z Z E

                 Z U M M

                --------

                   I S T

Note that the leftmost letter can't be zero in any word. Also, there must be a one-to-one mapping between digits and letters. e.g. if you substitute 3 for the letter M, no other letter can be 3 and all other M in the puzzle must be

Solution :

C=0, U=1, S=2, T=3, O=4, M=5, I=6, Z=7, E=8, D=9

It is obvious that U=1 (as U*STEM=STEM) and C=0 (as I-C=I).

S*O is a single digit and also S*T is a single digit. Hence, their values (O, S, T) must be 2, 3 or 4 (as they can not be 0 or 1 or greater than 4).

Consider, STEM*O=DMOC, where C=0. It means that M must be 5. Now, its simple. O=4, S=2, T=3, E=8, Z=7, I=6 and D=9.

                   O U T                        4 1 3

           -------------                -------------

   S T E M | D E M I S E        2 3 8 5 | 9 8 5 6 2 8

           | D M O C                    | 9 5 4 0

           -------------                -------------

               T U I S                      3 1 6 2

               S T E M                      2 3 8 5

              ----------                   ----------

                 Z Z Z E                      7 7 7 8

                 Z U M M                      7 1 5 5

                --------                     --------

                   I S T                        6 2 3

Also, when arranged from 0 to 9, it spells CUSTOMIZED.
-------------------------------------------------------------------------------------------------

Problem 12:

At what time after 4.00 p.m. is the minutes hand of a clock exactly aligned with the hour hand?

Solution :

Assume that X minutes after 4.00 PM minute hand exactly aligns with and hour hand.

For every minute, minute hand travels 6 degrees.
Hence, for X minutes it will travel 6 * X degrees.

For every minute, hour hand travels 1/2 degrees.
Hence, for X minutes it will travel X/2 degrees.

At 4.00 PM, the angle between minute hand and hour hand is 120 degrees. Also, after X minutes, minute hand and hour hand are exactly aligned. So the angle with respect to 12 i.e. Vertical Plane will be same. Therefore,

6 * X = 120 + X/2
12 * X = 240 + X
11 * X = 240
X = 21.8182
X = 21 minutes 49.5 seconds

Hence, at 4:21:49.5 minute hand is exactly aligned with the hour hand.

Answer :  

              4:21:49.5