Mathematics Problems

We first had the problem of   "17 Horses and 3 brothers".  Then there was the problem of "Necklace broke in Love-making".  Some friends have asked me to enroll them in the Mathematics class.  Some of them have put a condition that the problems to be solved in the class should be only like the ones similar to breaking of Necklace in Love making.

In view of the persistent demand for enrolling in the class, classes have to be started.  But in order to screen the applicants, there should be an entrance test.  When there are tests nowadays for parents to admit kids to LKG or UKG, we should necessarily have an entrance test.

Here are 12 Mathematics Problems.  Some are very simple.  Some are a little complex.  None of them are difficult. The entrance test begins now and you have 15 minutes to answer these questions.


It is an interesting exercise.  Pick up a paper and pencil if you want to. Quickly solve the problems.

01.     50  ÷ 2 = 100 ÷ x    What is the value of  x?

02.      - 8 = 2 - x           What is the value of  x?

03.      52- x²   + x = 10  What is the value of x?

04.      3  (π -0.14)  = ?

05.      1221  ÷  111 = ?

06.      198 divided by 66 = ?

07.      √4 = ?

08.    1/8 multiplied by 96/2 = ?

09.      6 x 2 = ?

10.     102413 - 102412  = ?

11.     √64 = ?

12.   630 ÷  126 = ?


This is not just a set of 12 problems.  There is something more to it! 

 After you have solved all the problems, click here for answers.

Mathematical Clock

Recently I visited the house of a friend  and found a wall clock, and it was a wall clock with a difference.  Instead of  numbers on the hour places, there are mathematical problems.  An image of the clock is given below:

The twelve questions given in the entrance test are the same as the questions on the clock.  Only the order of Questions is changed.  Naturally answers should be 1 to 12.

If you have answered the questions correctly, you will receive a communication for admission along with details of fee to be remitted etc.

Our next meeting will be in the Mathematics class!

17 Horses and 3 Brothers

This is a very old story of a super rich man and his three sons.  Of his vast properties and possessions that included seventeen horses.  A story some may have heard or read, but can be re-heard or  re-read one more time.  It is not just a story, but more of a mathematical problem.

Once upon a time, there lived a very rich man.   Who became rich by sheer hard work, and more importantly by honest means.   He had three sons and he loved them very much.  The sons also loved and respected their father equally.   The sons were as honest and hard working as the father.  The family lived happily.

The rich man became very old and realised that his end was near.  He wrote a Will dividing his properties and possessions among the three sons. He called all the three sons together and handed it over to them.  He knew that the sons could be trusted and there was no need for an attorney or witnesses for the Will.  He told the sons to open the Will after his death and distribute the properties and possessions as detailed in the Will.   He told them that there may be some differentiation in dividing the properties among the three brothers.  One may get more land than the others.  Another may get more buildings than the other two.  The third may get more movables assets than immovable properties.  The division was based on economies of scale and preservation of property value as against dividing each item equally among the three brothers and thereby resulting in loss in net value.  Sound Economic principles.   He assured them that the division made in the Will, though may not look equitable on the face of it, was actually fair and all the three brothers would get a third of the overall value of the estate.  The sons believed in the judgement of the father and the father believed in the understanding the sons would show.  And more importantly, in the understanding the three daughters-in-law would show.   Difficult to believe that such an ideal family ever existed, but it is true.  At least in this story.

The super rich man died in the normal course and peacefully.   The customary last rites were performed by the heirs and all the relatives, friends, well wishers and the not-so-well-wishers of the family participated in them.  On the last day, just before dispersing, the elders in the gathering offered their services for supervising and dividing the property among the three brothers.  The brothers politely declined the offer and  told them that they would themselves do it as suggested by their father before his death.  The well wishers were indeed happy and left after remarking that the rich man was really lucky for having such children.  The not-so-well-wishers were disappointed at denial of an opportunity to fish in troubled waters.  Nevertheless, they also repeated that the rich man was really lucky to have such sons and they also left.

All the assets were distributed as directed in the will.  The only item left to be distributed were the horses.  The will stated that they should be divided thus:  Half to the elder son, One Third to the middle son and One Ninth to the Youngest son.  The three sons tried their best to solve the division issue, but failed in their efforts.  For the first time in their lives, fissures appeared and they started discussing the issue agitatedly.

An old friend of the rich old man, himself very old but still alive, was going on his horse when he heard the heated discussion among the brothers.  He came to them and asked what the problem was.  In deference to the old man and a good friend of their father, they explained the problem of division of the horses.  He thought for a moment and told them that he would divide them as per the wishes of their father, if they all agreed to his method.  When they agreed, he added his horse to the pool and told them he would divide the pool of 18 horses.  He gave 9 horses to the elder brother, being half of the 18 horses.  The middle brother was given 6 horses being one third of the 18 horses.  The youngest got 2 horses being one ninth of 18 horses.  That still left one more horse, the one belonging to the wise old man.   He thanked them for their co-operation, mounted on his horse and rode away!

The brothers stood there, amazed at the simple way in which the wise man had solved the tough problem.   They realised that this was not just a division of horses, but the last lesson their father wanted to teach them.  They took the advice of the old wise man whenever the need arose, as long as he was around.

Is not this story a better way of teaching "Fractions" in "Mathematics" to the children?   One could use Apples or Chocolates instead of horses to give a practical touch.  There are many such riddles in ancient Indian books like "Leelavati".  Some are simple, some more complex and some, they say, have not yet been solved.

The Question is wrong

This happened when I was appearing at the SSLC (10th standard) examination. Our High School was in a good location and had large number of rooms and other infrastructure facilities were also available. Classes from VIII standard to X standard were conducted in the school and each class had at least four sections of about 50 students each. That made for nearly six hundred students. About 150 to 200 students would sit for this examination from our school. Many of the High Schools in the villages had 10 or 12 students taking the examination and the biggest school had about 50 students. Being a taluk headquarters and better connected by roads, it was an ideal place to act as a center for conducting the examination.

We often hear about the demand for seats to merit candidates. The voice for such a demand emanates from the Urban and educated sections. Because the rural and uneducated sections do not have much voice in making the demands. No doubt rural areas are politically important and strong and throw up a large number of representatives. But the representatives once elected mostly remain in the cities and the basic requirements of the villages is still inadequate. Students and women suffer more than others. Even today students in some rural areas are made to sit on the floor to write the examinations because the schools do not have required number of desks! Students from villages have to travel by bus or walk two or three miles before reaching the examination center. Many of them may not even have a breakfast when they arrive at the center. Some may say it is good because they do not fall asleep due to full stomachs. Once the examination starts all are equal. Questions papers and method of valuation are the same. Where is the level playing field?

Our school being the examination center for students of many nearby village schools cast an extra responsibility on our school authorities. Question paper bundles would arrive from the district headquarters a week before commencement of the examination and kept in the Locker room of the local bank. Head Master of the school was the Examination Superintendent and had to bring the bundles of question papers each day for the next day's subjects and keep in a room with door locked and sealed in the presence of two other officials and opened the next day in their presence. Two other senior teachers of the taluk were assigned as Assistant Superintendents to support him and run the system. Each day was a tense day for these officials as bundles have to be opened and arranged room wise and subject wise, answer papers bundled, sealed and dispatched  and everything had to go like clock work.

I was a student of Mathematics. The system at that time was that every student had to study Mathematics for 100 marks paper and this paper was called General Mathematics. Students had an option to take either Science or Arts as Optional subjects. Among the optional science subjects was another paper of Mathematics of 100 marks. In other words, arts students were to write one paper of Mathematics for 100 marks and Science students had to write two papers of Mathematics for 100 marks each. One common paper of General Mathematics and one paper of Optional Mathematics. Art students used to have subjects like History, Geography and Political Science. Naturally Optional Mathematics had a higher syllabus content and tougher than the General Mathematics paper.

There was and still is a general consensus that students taking up Optional Mathematics were or are more intelligent than arts students. Nothing can be farther from truth. During the course of our lives we have all experienced that much of what is learnt in school is useless for many of us. I was a student of Chemistry in college and ended up as a Bank Manager, fighting with Accountancy and Banking Law and had to study these subjects outside formal education. Many arts students did very well in their life and brilliant mathematics students often had to work as their employees! 

On the day of Optional Mathematics paper I went to school as usual and the examination started at 10 AM. It was a paper for 3 hours. After receiving the question paper and reading it, I found that one question for four marks was wrong. I told the hall Supervisor that the question was wrong and that I want to talk to my Mathematics teacher - the one who taught "Anything multiplied by Zero is Zero". He was one of the Assistant Superintendent  for the examination and was a busy man. Student from any other school could not have the privilege of calling his teacher. The Hall Supervisor sent words for calling the teacher. I kept reminding him every half an hour. Around 12 noon the teacher, now Assistant Superintendent , came almost running and asked me what the problem was. I showed him the question and told him it is wrong. He saw the question for a few seconds and told me it is correct and advised me to think calmly and answer. I started arguing with him but left hurriedly to attend to other work after again advising me to think calmly and answer. I did not have the maturity at that time to understand that he was an Assistant Superintendent and discharging a different role and also that we were in an examination hall. I could not answer the question and left after completing the paper except that question.

When I came out of the hall he was waiting for me and took me aside. He asked me to read the question. The question was : "A bird flies at 60 miles an hour while flying with the wind and 20 miles an hour while flying against the wind. What is the speed at which it is flying?". What is the principle on which it is based, he asked. I told him it was based on Simultaneous equations. How, he asked.   x+y = 60 and x-y = 20, I replied. Then what is the speed at which the bird was flying, he asked.  40 miles per hour, I answered. Where is the question wrong, he asked.  They have not taken the weight of the bird into account, I said. Why is it required?, he asked. I did not have an answer. I had used super intelligence and brought an extraneous issue into the question. In the bargain lost 4 marks!

Over the years I have realised that we commit the same silly mistake many times in our real life as well.  Many questions that confront us in daily life are actually correct. We know the answers too. Often the correct answers. Instead of answering the questions in a simple way that we already know, we complicate the question by bringing in irrelevant issues. And allow the question to become tougher and more demanding. Thus we fail to answer the questions and in the bargain lose peace of mind also!