Car Number Puzzle

Find the Car Number

Three friends were returning from college when they saw a speeding car hitting a man while he was crossing the road and fleeing away. They shouted in desperateness  to seek help from the passerbyes but failed.

In the meantime they noticed the registration number of the car and tried to remember in their own style. They rushed the person to the nearby hospital where he was admitted. The doctor called the police as this was a case of road accident and a case was registered. 

In order to do proper investigation the police started inquiring the three friends and asked them if they could provide the registration number of car so that the driver who had hit the person can be caught up. All the three friends tried a lot to remember the car number but failed so they altogether decided to zero upon the number of the car as they have remembered the car number with some mathematical properties behind it.

1^st Student: -- Sir, the first two digits of car number were same.

2^nd Student: - Sir, the last two digits of car number were same.

3^rd Student:- The four digit number is a  square number.

Can you guess the car number to help the police to nab the guilty?

Send me your valuable comments on rkthakur1974@gmail.com and try to reason this out with algebraic proof ? See you next day till then open up your mind.

Ans :- 7744

Multiplication without Papers

Multiplication without paper

Multiplication can be made an interesting exercise by learning some tricks.Don’t you think that you should have knowledge of some super-duper trick which could have been saved your time and multiplication could have been done mentally without using pen and paper?

There are hundreds of such magical trick which help to do calculation mentally and without pen and paper. Do you know what is more interesting here that there are absolutely no chance of mistake?  Let me ask you a simple question first: -

 What is 100 times 456789?

I am really in a serious mood and not joking guys despite the fact that I know you can do this multiplication in fraction of second by just putting two zeros after 456789 and you will return back with your answer 4567890. The fact of the matter is that when you are asked to multiply a number by the multiple of 10, like 10, 100, 1000, 10000… you just put the number of zeros in front of the number to be multiplied by noticing how many numbers of zeros are in multiplier which is the multiple of 10’s. Likewise, the method discussed here will let you get the answer in seconds.

Hey guys, are you ready for the journey of multiplication fantasy? If your answer is a big YES, then here I begin.

1.      When the sum of unit digit is 10 and ten’s digit are same

Rule: - a) The answer will have two parts. Multiply the unit parts. The product should have exactly two digits. If you get the lesser digit put a zero before.

            b) Second Part = Ten’s digit x (Ten’s digit + 1)

Example: - 42 × 48 =?

Solution: - Here the sum of unit digit is 10 (2 + 8 = 10) and ten’s digit in both multiplicand and multiplier are same.

Unit Part = 2 × 8 = 16

Second Part = 4 × 5 = 20

Hence, 42 × 48 = 2016

Example: - 74 × 76 =?

Solution: - Sum of unit digit = 10 (4+ 6 = 10) and ten’s digit in both multiplicand and multiplier are same (7).

Unit Part = 4 × 6 = 24

Second Part = 7 × 8 = 56

Hence, 74 × 76 = 5624

Example: - 99 × 91 =?

Solution: - Sum of unit digit = 10 (9+ 1= 10) and ten’s digit in both multiplicand and multiplier are same (9).

Unit Part = 9 × 1 = 09 (In order to fulfill the condition of two digits in the product 0 is put before 9)

Second Part = 9 ×10 = 90

Hence, 99 × 91 = 9009

2.      Sum of ten’s digit is 10 and unit digit are same

Rule:-The answer will have two parts

a)      LHS = Product of ten’s digit + Unit digit

RHS = Product of Unit digits

Always remember there should be exactly two digits in the RHS. In case there is a single digit in RHS, put a zero before the product you get

Example: - 84 × 24 =?

Solution: - Sum of ten’s digit = 10 (8+2 = 10)           and Unit digit in multiplicand and multiplier are same (4)

LHS = Product of ten’s digit + Unit digit = 8×2 + 4 = 20                                                 
     RHS = Product of Unit digit = 4 × 4 = 16                                                                                         Hence, 84 × 24 = 2016   

Example: - 93 × 13 =?

Solution: - Sum of ten’s digit = 10 (9+1 = 10)           and 

Unit digit in multiplicand and multiplier are same (3)

LHS = Product of ten’s digit + Unit digit = 9 × 1 + 3 = 12                                               

RHS = Product of Unit digit = 3 × 3 = 09                                                                                                                

(one Zero is put due to less number of digit in the product)                                              

Hence, 93 × 13 = 1209                       

Example: - 75 × 35 =?

Solution: - Sum of ten’s digit = 10 (7+3 = 10)           and 

Unit digit in multiplicand and multiplier are same (5)

LHS = Product of ten’s digit + Unit digit = 7× 3 + 5 = 26                                                    

RHS = Product of Unit digit = 5 × 5 = 25                                                                                         

Hence, 75 × 35 = 2625                                                   

3.      When the difference between the digits to be multiplied is 1

When two consecutive numbers are to be multiplied, follow this technique for quicker multiplication.

a)      Square the smaller/ greater number.                                                                              

    b) Add / subtract the smaller/ larger number to the previous result

Example: - Multiply 23 by 24

Solution: - 

a) Square the smaller number = 23 x 23 = 529                                                                                  
 b) Add the smaller number to the previous result =529 + 23 = 552                                              Hence, 23 × 24 = 552

Example: - Multiply 75 by 76

Solution: - 

a) Square the smaller number = 75 x 75 = 5625                                                                                
  b) Add the smaller number to the previous result =5625 + 75 = 5700                                         Hence, 75 × 76 = 5700                                                    

                                                                   

4.      When the difference between the digits to be multiplied is 2

a)      Find the mean and square it. The mean of two numbers a and b = a + b / 2

b)      Subtract 1

Example: - Multiply 25 by 27

Solution: - a) Mean = 25 + 27 / 2 = 26. (Mean can be found by adding 1 to smaller or subtracting 1 from the larger number in this case.)

              b) Square the mean = 26 ² = 676

            c) Subtract 1 from the last result = 676 – 1 = 675

Hence, 25 x 27 = 675

Example: - Multiply 74 by76                                                                                                
Solution: - a) Mean = (74 + 76)/ 2 = 75. (Mean can be found by adding 1 to smaller or subtracting 1 from the larger number in this case.)

              b) Square the mean = 75 ² = 5625

            c) Subtract 1 from the last result = 5625 – 1 = 5624

Hence, 74 x 76 = 5624

5.  When the difference between the digits to be multiplied is 3

Rule: - 

a) Add 1 to the smaller number and square it                                                                                  

  b) Subtract 1 from the smaller number and add to the previous result                           

Example: - Multiply 28 by 31

Solution: - 

Here the smaller number is 28.                                                                                                    

  a) Add 1 to it = 28 + 1 = 29                                                                                                                   b) Square it = 29² = 841                                                                                                                   

  c) Subtract 1 from the smaller number and add to the previous result = 841 + 28 – 1 = 868 

Hence, 28 × 31 = 868

Example: - Multiply 24 by 27

Solution: -

 Here the smaller number is 24.                                                                                                     

 a) Add 1 to it = 24 + 1 = 25                                                                                                                   b) Square it = 25² = 625                                                                                                                    

 c) Subtract 1 from the smaller number and add to the previous result = 625 + 24 – 1 = 648 

Hence, 24 × 27 = 648

6.  When the difference between the two numbers to be multiplied is 4

Rule: 

a) Take the mean of numbers. If the numbers are a and b; where a >b then mean is either a – 2 or b + 2                                                                                                                                                              b) Take the square of mean number.                                                                                                         

 c) Subtract 4 from the result.       

Example: - Multiply 65 by 69?

Solution: Here the difference between the numbers to be multiplied is 4. Take its mean.                             

a) Mean = 65 + 2 = 67                                                                                                                            b) Square it = 67² = 4489                                                                                                                

   c) Subtract 4 from the previous result = 4489 – 4 = 4485                                                     

 Hence, 65 × 69 = 4485

Example: - Multiply 84 by 88?

Solution: Here the difference between the numbers to be multiplied is 4. Take its mean.                             

a) Mean = 84 + 2 = 86                                                                                                                            b) Square it = 86² = 7396                                                                                                               

 c) Subtract 4 from the previous result = 7396 – 4 = 7392                                                     

 Hence, 84 × 88 = 7392

7.  When the difference between the two numbers to be multiplied is 6

Rule: a) Take the mean of numbers.                                                                                                   

b) Take the square of mean number.                                                                                                          c) Subtract 9 from the result.       

Example: - Multiply 42 by 48?

Solution: Here the difference between the numbers to be multiplied is 6. Take its mean.                             

a) Mean = 42 +3 = 45                                                                                                                             b) Square it = 45² = 2025                                                                                                                 

  c) Subtract 9 from the previous result = 2025 – 9 = 2016                                                      

Hence, 42 × 48 = 2016

8.  When the difference between the two numbers to be multiplied is 8

Rule: a) Take the mean of numbers. If the numbers are a and b; where a >b then mean is either a – 4 orb+4                                                                                                                                                         

    

b) Take the square of mean number.                                                                                                          c) Subtract 16 from the result.       

Example: - Multiply 64 by 72?

Solution: Here the difference between the numbers to be multiplied is 8. Take its mean.                             

a) Mean = 64 + 4 = 68                                                                                                                            b) Square it = 68² = 4624                                                                                                                

   c) Subtract 16 from the previous result = 4624 – 16 = 4608                                                

    Hence, 64 × 72 = 4608

9.  When the difference between the two numbers to be multiplied is 10

Rule: a) Take the mean of numbers. If the numbers are a and b; where a >b then

 mean is either a – 5 or b + 5                                                                                                                                                    

b) Take the square of mean number.                                                                                                          c) Subtract 25 from the result.       

Example: - Multiply 84 by 94?

Solution: Here the difference between the numbers to be multiplied is 10. Take its mean.                           

a) Mean = 84 + 5 = 89                                                                                                                            b) Square it = 89² = 7921                                                                                                                

   c) Subtract 25 from the previous result = 7921 – 25 = 7896                                                 

   Hence, 84 × 94 = 7896

Courtesy :- From Maths made Easy - Rupa Publication by Rajesh Kumar Thakur

(Book available on different website  / e- books also available)

Rajesh Kumar Thakur

rkthakur1974@gmail.com

What is Mathematics

What is Mathematics?

According to Webster’s New Collegiate Dictionary – Mathematics is the science of numbers and their operations, inter relations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations and generalizations.

The word mathematics comes from the Greek word mathema and its meaning is – to learn. If I ask you what is mathematics then you will simply say mathematics is all about calculations and nothing else. But this is one of the complicated questions to answer. In simple words, Mathematics is a language that expresses relationship. According to Bertrand Russel, the famous British mathematicians and philosopher “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” This includes Logic, Measurement, Algebra, Geometry, Calculus etc. this language allows us to understand our universe and to solve problems in it. 

Mathematics as a discipline contains two things—

 1. Theory

 2.  Logic

Both are interrelated. Nobody is certain where did mathematics come from? But mathematics may have been created much before the first words were spoken by humans. Our ancestor might have surely encountered problem like –

·         Hunger

·         An attack by wild animals

·         Aggression by another group

So they may have used some logic to determine the best place to find food and satisfy their hunger. They could have applied mathematical theory to determine if they had enough rocks to repel the attacking animals. They might have developed strategies to encounter the aggression by another group. All this is enough to prove that mathematics is the backbone of life.

Mathematics is significant

Mathematics as an expression of the human mind reflects the active will, the contemplative reason and desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Simon Blackburn in his book The Big Question writes—“It is a vast and multifaceted subject. It covers such a brand spectrum of activity that it appears scarcely possible to classify all its manifestation within a single subject. At the one end of the spectrum it defines the nuts and bolts of counting, time and money that enable daily life to chug away. At the other end, it can sum a sealed world in which great ivory- towered minds manufacture puzzles of mammoth complexity”

How important mathematics is in our life can be understood with the help of these examples.

1      

       Imagine going to market to buy food with a Rs 1000 note and bought food for Rs. 543.87 with tax. You wait for your balance but have no idea how much it should be. This sound like a child in nursery school with little understanding of counting. This is also about the time children start to see money and are excited about handling it but have no idea about mathematics.

2

          In the early morning the office goers wakes up with the sound of the alarm and starts his day with the number on clock face. He may rush to his duty by the bus with number 149, 26,861,990—or by bicycle, scooter or car. Suppose you are going to your destination by car and you are driving at a speed of 50 Km/h but suddenly on the way you find a heavy traffic jam and it kills your 20 minutes. Now in order to reach your destination you have to accelerate your speed. What conclusion can you draw from this example? Distance, Speed and Time are the mathematical terms and in order to reach your office in time you are actually using the formula Distance = Speed x Time

Bus showing the number on it

3

     Have you ever thought how does your motorcycle meter show you the kilometer you cover? The wheel of a car, motorcycle has certain diameter and in 1revolution it covers its own distance and thus it tells you the distance. If the diameter of a wheel is 56 cm then in one revolution it will cover the distance = 2 x ²²/₇ x 56 = 352 cm, so in order to cover a distance of 1 Km = 1000m = 100000cm it will have to make 100000/352 = 284.09 = 284 revolution approximately.

Speedometer

4      Mathematics is the one skill everyone needs to master. Suppose you go to a grocery shop and ask a shopkeeper the rate of capsicum and he tells you it is Rs 38 per kg.  If you buy 250g of it how much will you pay? Certainly you need to pay Rs 9.50 to the shopkeeper but if you don’t know the mathematical calculation may be the shopkeeper asks you to pay Rs 12 and you may get cheated.  Just doing the basic essentials is dependent on your ability to do mathematics.

5

       During the month of February every tax payers starts calculating income tax to be paid that includes the money to deposit in different insurance companies and PPF, medical policies etc so that you pay minimal tax to the government. All this is nothing but the manipulation of mathematics.

The above examples are sufficient to show that mathematics plays an important role in life and everyone should read mathematics. In school, mathematics helps students understand other subjects better. When a student excels in mathematics, other subjects automatically become easier. Some shop attendants still use rudimentary techniques like counting sticks to calculate their sales. Although one can avoid simple mathematics in school, numbers will always knock at one's door because of its role of counting in our everyday life. Many people still face difficulty in dealing with numbers. These include difficulty in counting votes, failure to calculate menstrual cycles, failure to do proper accounting and failure to keep time. Fagil Mandy, an education consultant and Uganda National Examination Board, says mathematics is a subject of life because life lives on calculation. "You calculate how much to eat, sleep, and do other things in life. All this is mathematics," he says.

The origin of Mathematics goes back to more than 5000 years old but it was the Sumerian who developed mathematics as a formal area of teaching and learning. The development of reading, writing and formal mathematics 5000 years ago allowed the codification of mathematics knowledge, formal instruction in mathematics and mathematics was born. The primitive men used sign language to indicate the number he wanted to use. The journey first began with quipu which they were initially using for counting the number of sheep.

        Quipu

Later they began to count with their fingers, used sticks, pebbles but they had not developed any number system to write numbers. The earliest written number so far discovered was used in ancient Egypt around 3000BC. They were using special pictograph to write down the numbers. The Egyptian number system was based on decimal system and they called their numerals as hieroglyphs.

                                    Egyptian Numeral System

The Babylonian also had by that time invented their own numerical system based on 60 and called the writing system as Cuneiform. Cuneiform is made of two words cuneus and forma meaning a wedge and shape respectively.

  Babylonian Numeral System

The Chinese, the Romans all developed their own numeric system and had different symbols for each of the numbers but none of the numeric system had invented the symbol for zero. The Hindus were the first to introduce a symbol for zero and also the place value system which was later known to the world with the Latin translation of Arabic text written by Al-Khwarizmi. The indigenous discovery of zero by the Indians in between 2nd century BC and 6th century was echoed by the Arab writers and when Leonardo Fibonacci translated the Arabic text into Latin in the 12th century it brought a great revolution in the mathematical arena. (Read my blog on Origin of Zero for more details)

Mathematics is the bone of most of the science subject Engineering, Architect, Planning, Physics, Chemistry and now even in Biology. The way mathematics has spread its wings that it has become an integral part of subjects like Economics, Music, Arts etc. It has also the wide association with Nature and it is common to note mathematical shapes in nature. Mathematics plays important role in developing business. During the Second World War mathematics subject operation research has played important role in making strategy for the decision making authorities. In one sentence – Mathematics is the oxygen for all subjects.

Source :- From my book --- Enrich Your Maths Skill published by Prabhat Prakashan available on Internet )

Rajesh Kumar Thakur

rkthakur1974@gmail.com