In our earlier articles, we have introduced you to the basics of Vedic Maths. We hope, by now, you are well-versed with them. We will now talk about the general method of multiplication. This method is useful when numbers are far apart in the number series and the choice of base (base method) can’t be made.

**General Method of Multiplication**

**Example 1:**

Multiply 34 by 86 (two digit number by two digit number)

**Step 1:**

Put the numbers one below the other, like this:

3 4

8 6

Here, we get only two columns since both of them are two digit numbers.

**Step 2: **

Now, multiply the digits in the first column (the first column is the one formed from the unit’s place of both the numbers).

4 X 6 = 24

Put 4 at the unit’s place of the answer (as shown below) while 2 is to be carried-over

3 4

X 8 6

—————————–

4 Carry-over = 2

**Step 3:**

Now cross-multiply both the columns, that is, multiply the digit at unit’s place of row 1 with the digit at ten’s place in row 2. Similarly, multiply the digit at ten’s place of row 1 with the digit at unit’s place of row 2, that is,

4 X 8 = 32

3 X 6 = 18

Add 32 and 18 (32 + 18 = 50)

Add the previous carry-over to 50, that is, 50 + 2 = 52

Put 2 at the ten’s place in the answer, 5 is the new carry-over

3 4

X 8 6

——————————

2 4 Carry-over = 5

**Step 4:**

Now, multiply the digits of the ten’s place column with each other, that is, 3 X 8 = 24

Add previous carry-over 5 to it

24 + 5 = 29

This 29 will be written as:

3 4

X 8 6

——————————————–

2 9 2 4

2924 is the answer

In our next segment, we will show you how to use the same method for 3-digit (or more) numbers. Till then, practice it well.

**SEE ALSO:**

**1. Vedic Maths – 3: Base Method**

**2. Vedic Maths: For Fast Calculations – Part 2**

**3. Vedic Maths: For Fast Calculations – Part 1**

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com**** or connect with us on ****Facebook **** or ****Twitter **** ******

**We continue with our series on common mistakes in English. In this part, we will mainly focus on double negatives. Often, you must have seen writers using two negatives together. Is it right to do so? Does it make sense? And why can’t we just be direct? We try to answer these questions**.

You must have come across sentences where the writer has used two negatives. It is often done for effect. But remember, in the majority of cases, two negatives in the same sentence do not make for emphasis. Instead, it simply cancels each other.

For example, it is incorrect to write, *He didn’t give me no call*.

The correct statement should be – *He didn’t give me a call/any call.*

This rule applies not only to all the negative words beginning with n (like no, not, never, none, nowhere, nobody) but also to words which have a negative force, such as scarcely/hardly. Example –

Incorrect: *I didn’t hardly understand him*

Correct: *I hardly (or scarcely) understood him*

‘Neither’ is always accompanied by ‘nor’. Similarly, ‘either’ is balanced by ‘or’. Example –

*He gave me neither any money nor food.*

*He didn’t bring either my luggage or papers.*

Since the two negatives cancel each other, they are often used to write a positive statement. For example –

*She owed me no small sum* (here, the writer wants to say ‘a large amount,’ hence ‘no small sum’).

Similarly, we can write – *Not for nothing have I travelled all this way* (the writer is emphasising on the gains of the travel).

Sometimes, it is a good idea to switch a negative to the beginning of the sentence. Example – *Not for a moment did I think before accepting the job offer*.

*Can you think of more such mistakes? Well, send them across and we will keep on updating the list.*

**YOU MAY LIKE TO READ:**

1. **Common mistakes in English – 1**

2. **Common mistakes in English – 2**

3. **Common mistakes in English – 3**

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com**** or connect with us on ****Facebook ****or ****Twitter **** **

In this article, we are going to talk about Base Method. In this method, we identify a ‘base’. Base is a number that is a simple number (usually multiples of 10) and must be close to both the numbers given. For example, 13 x 14 – In this, the base will be 10.

Now let’s try to solve some problems using the base method.

Ex 1: Multiply 14 by 16

Step 1:

Identify the base. Since 10 is close, this will be the base.

Step 2:

The difference between number 14 and base is (+4). Since the difference is positive, we call it surplus (the negative difference is called deficit).

Similarly, the difference between number 16 and base is (+6).

Step 3:

Add either (+4) to 16 or (+6) to 14. This process is called cross addition. Either way, we get 20 as answer.

Now, multiply 20 with base, i.e, 20 x 10 = 200

Step 4:

Now multiply the surpluses, i.e, 4 and 6 in this example.

(+4) x (+6) = (+24)

Step 5:

Add the answers obtained in step 3 and 4, i.e.,

200 + 24 = 224

This is our answer.

Simple! Isn’t it?

Now let’s try one more.

**Example 2:**

Multiply 32 x 36

Step 1: Base will be 30

Step 2: 32 – 30 = (+2)

36 – 30 = (+6)

Hence, surpluses are (+2) and (+6)

Step 3: Cross addition gives us 38. By multiplying 38 with 30, we get 1140

Step 4: (+2) x (+6) = (+12)

Step 5: 1140 + (+12) = 1152. This is our answer.

**Example 3:**

Now let’s try this when we have deficits.

Multiply 28 x 27

Step 1: Since 20 will not be close to the numbers, we will choose 30 as our base.

Step 2: 28 – 30 = (-2)

27 – 30 = (-3)

Step 3: Cross addition gives us 25 (28 – 3) and (27 – 2)

25 x 30 = 750

Step 4: (-2) x (-3) = (+6)

Step 5: 750 + (+6) = 756

Hence, 28 x 27 = 756

**NB: **

**1. If the numbers are in 80s or 90s, it is better to have 100 as base for easier calculations. **

**2. Base method is useful where the two numbers to be multiplied are close in the number series. What if the numbers are 38 x 74 or 43 x 168. We will deal with that in our next part. Till then, practice and master the base method.**

**SEE ALSO: **

**Vedic Maths: For Fast Calculations – Part 2**

**Vedic Maths: For Fast Calculations – Part 1**

**Do you feel strongly about something? Have a story to share? Write to us at ***info@thepeepertimes.com** or connect with us on Facebook or ***Twitter *** *

We all love to speak in English language. It makes us feel good. However, when we speak it wrongly, the impression that we intend to create on our audience is nullified. In fact, it leaves them horrified.

We have compiled a list of 10 common mistakes that we often notice in usage of English language among Indians.

**1) Past tense after ‘did’**

This is the most common mistake people make. Remember, ‘did’ should never be followed by the past tense.

Incorrect: *He did gave me the gift.*

Correct: *He did give me the gift.*

**2) Difference between take exam and give exam**

A student never gives an exam. He takes an exam. A teacher gives an exam.

Remember it this way – A teachers gives papers to students; a student takes a paper from a teacher.

**3) Return back/ Repeat again or revert back**

Return means ‘go back,’ so no need to add ‘back’ to the word.

Similarly, repeat means ‘doing again,’ hence no need to use again with ‘repeat.’

And, revert means ‘go back’ or ‘return to.’ Do we need to repeat the logic?

**4) One of the common mistakes we make is to use singular noun after ‘one of the’**

Incorrect: *Agatha Christie is one of the well-known author in the world.*

Correct: *Agatha Christie is one of the well-known authors in the world.*

**5) Incorrect: ****Write between 500 to 700 words****.**

It is wrong to say ‘between 500 to 700.’ Between requires two values or objects, hence correct usage is ‘between 500 and 700.’

Correct: *Write between 500 and 700 words*.

**READ ALSO: ****Common Mistakes in English – 2**

**6) Comprise means ‘consist of,’ so no need to repeat ‘of.’**

Incorrect: *Our country comprises of 29 states.*

Correct: *Our country comprises 29 states.*

**7) Can and May**

‘Can’ indicates an ability to do something while ‘may’ signifies asking for permission for doing something.

Incorrect: *Can I take this book?*

Here, you are questioning your ability to take a book. Instead, use *“May I take this book?” *as you want to ask for someone’s permission.

**8) Cousin and Cousin Brother/Sister**

How often we hear ‘my cousin sister or brother.’ Cousin is used for both the sexes – male or female. Hence, it is incorrect to say *“He is my cousin brother”*. *“He is my cousin”* is enough.

**9) Order for/Discuss about**

Incorrect: *Let’s order for dinner*

Correct: *Let’s order dinner*

You order something, not ‘order for’ – hence the preposition ‘for’ is unnecessary.

Similarly, you discuss something and not ‘discuss about.’

Incorrect: *We will discuss about the issues.*

Correct: *We will discuss the issues.*

However, one can say – *We will hold discussion about the issues.*

**10) Mr and Mrs**

They should be used either with the full names or last names. Using them with a first name is wrong. For example, Mr Barrack Obama or Mr Obama is correct. Mr Barrack is wrong.

Also, nowadays it is outdated to use Mrs or Miss for females. Use ‘Ms’ instead. Whether a woman is married or single is no one’s business.

**Can you think of more such mistakes? Well, send them across and we will keep on updating the list.**

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com**** or connect with us on Facebook or ****Twitter **** **

Many students get goose bumps when asked to study Chemistry – often dreading Organic Chemistry – hundreds of formulae, periodic table, etc. In short, the carbon and its combinations appear tough to even remember, let aside clubbing them with other compounds. But just like Maths requires clarity of fundamentals and everything gets simplified, so does Organic Chemistry. Once you get your fundamentals right, believe us, you will never forget it all your life. It is that simple.

So let’s get our fundamentals clear, from the beginning:

**What’s Chemistry?**

Chemistry is a branch of Science which deals with behaviour of substances in nature. Substances show physical and chemical behaviour; and they all become a part of Chemistry.

All substances on earth exist in the elementary form or compound or complex form. Elementary forms are the simplest forms of substances with no other constituent. These are called elements and there are more than a hundred elements known so far. The whole matter around is formed of only these elements, in various combined forms.

Every element is made up of small identical particles called atoms which are the combining agents of elements. These elements are grouped and arranged systematically in Periodic table and the students in secondary class and onwards are required to know in detail about the arrangement of elements, which covers a great part of details of behaviour of elements.

The atoms of various elements join chemically under different energy conditions to give different compounds; and all are studied under laws of chemical combination. All chemical description is well remembered when written in symbolic forms. For example, in symbolic form, Sodium + Chlorine = Sodium Chloride can be written as Na + Cl = NaCl. Similarly, Aluminium + Bromine = Aluminium Bromide can be written, in symbolic form, as Al + 3Br = AlBr_{3}. The number of atoms joining together to give a molecule of compound formed depends upon the combining capacity of the atoms called as valency which is clearly given by a number with positive (+) or negative (–) sign shown in symbolic representation of combining substances. In the above two equations, the combining capacity of sodium and chlorine atoms is one each and therefore the equation becomes simple as Na1^{+}+ Cl1^{–} = NaCl. But in case of second reaction, one Aluminium atom has 3 times combining capacity with Bromine and therefore one atom of Aluminium reacts with 3 Bromine atoms to give AlBr_{3}, so the equation can be shown as 1Al^{+++} + 3Br^{–} = AlBr_{3}

**Nomenclature of compounds**

Learning Chemistry is easier once student gets familiar with naming of compounds, that is, the nomenclature of compounds. This topic though a big subject in itself needs brief description here, which when understood well, creates sense of satisfaction and also a sense of completeness of understanding of subject topics. There are no hard and fast rules for giving names to compounds, but in true scientific sense, the substances are named according to their constituents and their linkages. Inorganic compounds which are formed by *electrovalent* bonds between constituents, exist in limited numbers; and organic compounds which are formed by *covalent* bonds between constituents get differentiated even by change of position of same constituent in molecular space resulting in the formation of a number of compounds with same molecular formula. This property called Isomerism causes abundance of compounds in Organic Chemistry. Constituents may exist in the form of an atom or a group of atoms, dominating properties of compounds, which are called functional groups. For example,

- -F, –Cl,-Br, -I as Halides
- -CN as nitriles or cyanide
- -CHO as aldehydes
- -COOH as carboxylic acids
- -NH
_{2}as amines, etc

Compounds are thus classified according to functional groups which each compound of the class carries as suffix to its name, e.g, inorganic compound like Metallic halides (Inorganic) – Na, K, Cl, etc., with halogen as NaCl, NaBr, KCl – Sodium Chloride, Sodium bromide; or Metallic cyanide, CN, as NaCN, KCN – Sodium Cyanide, Potassium Cyanide, etc.

+ Organic compounds like Alkyl halides or Alkyl Cyanides

**Saturated & Unsaturated Hydrocarbons**

Organic compounds are essentially hydrocarbons and their derivatives. Hydrocarbons are so named as being made of Hydrogen and Carbon, combined in such a way to satisfy the valencies of each combining atom. If, all carbon atoms of compound have free valencies satisfied by H-atoms, the compounds are called saturated hydrocarbons and if lesser valencies of C-atoms satisfied, the compounds are called unsaturated hydrocarbons.

The saturated hydrocarbons are named, in general, as alkanes. ‘Alk’ showing number of C-atoms and ‘ane’ showing the saturated nature of compound, for example,

1) For hydrocarbon with one C atom, 4 H-atoms will be required to satisfy the tetra-covalency of C-atom. So the formula showing one C-atom fully satisfied with Hydrogen can be CH_{4}

This CH_{4 }is named as Methane, prefix ‘meth’ standing for one carbon and suffix ‘ane’ standing for saturated nature.

Similarly, for a compound with 2-C atoms, the formula will be C_{2}H_{6}

The two free valencies on carbon atoms joining to form bond between them.

Similarly, for each addition of C atom in chain, we can have compounds.

We can have a formula for 3-C atoms, saturated hydrocarbon as C_{3}H_{8}

This is named Propane where prefix is ‘Prop’ showing 3-Carbon atoms and suffix is ‘ane’ denoting the saturated nature. Thus, we can name all saturated hydrocarbons (alk-ane) changing the prefix part according to the nature of C-atoms in one molecules as Butane, Pentane, Hexane and so on.

For unsaturated hydrocarbons, with 2H atoms less, another bond is introduced between 2-atoms and the double bond thus formed shows 2H less in compound. This is represented by suffix ‘ene’ in hydrocarbon

Thus for hydrocarbon with 2H less, the formulae will be as;

This is called Eth-ene. ‘Eth’ showing number of C atoms and suffix ‘ene’ showing double bond.

If however, there is further 2H less in a compound, it will again be reflected by additional bond between the C-atoms. The triple bond is represented by suffix ‘ine’ or ‘yne’. Thus for C-atoms, we can have

Thus from saturated hydrocarbon with 2-C atoms, we can have unsaturated form like:

Compound with more than one bond – double or triple bonds can be written as… with di, tri etc, to suffixes ad di-ene or tri-ene etc.

**Cyclic compound**

Similarly, the closed chain compounds are named as cyclic compound –

Cyclo-alkane

Or, cyclo-alkene

Cyclo-alkine

For example,

Double or triple bonds can be introduced into cyclo-compounds and the compounds named accordingly as cyclo-alkenes or cyclo-alkines. For example, cyclo-propane with one double bond can give a compound C_{3}H_{4 }and will be called as cyclo-propene. If more than one double bonds are present in cyclo-alkanes,the compounds can be named as cyclo-alk-(di,tri,tetra)-ene(s)

Similarly, cyclo-alkanes with triple bonds can be written as cyclo-alka-(di/tri/tetra)-ines

With Isomerism, the possibilities of having more number of compounds, makes nomenclature of the compounds a little more complicated and the students are advised to practice as much as possible to understand and remember well.

**Still confused? Do you have any questions about Organic Chemistry? Write to us your questions/confusion at info@thepeepertimes.com**

**Do you feel strongly about something? Have a story to share? Write to us at ***info@thepeepertimes.com** or connect with us on Facebook or ***Twitter *** *

*In Part 1, we taught you how to multiply two-digit numbers where the units digit is same and the sum of the numbers at tens digit is 10?*

**Now, let’s interchange them. **

**Method 3:**

How do we find the product of two-digit numbers where tens digit is the same and the sum of the units digits is 10?

For example,

34 x 36

Note the sum of the units digits (4+6) is 10 and the tens digits of the two numbers is the same, i.e, 3

**Step 1:**

Multiply the tens digit with its next number.

3 x 4 = 12

**Step 2:**

Multiply the units digits

4 x 6 = 24

**Step 3:**

Combine the answers obtained in step 1 and 2 to get the correct answer:

1224 (34 x 36 = 1224)

Simple! Isn’t it?

**Method 4:** Multiplication by factors

**Example 1: 44 x 14**

Since we know 14 = 2 x 7, multiplication by 14 can be easily achieved. Hence, in order to multiply any number by 14, we will first multiply it by 2 and then multiply the answer by 7.

**Step 1:**

44 x 14 = 44 x (2 x 7)

**Step 2:**

First multiply 44 by 2, i.e,

44 x 2 = 88

**Step 3:**

Multiply the number obtained in step 2 with 7 to get the answer, i.e.,

88 x 7 = 616

Answer: 44 x 14 = 616

**Example 2:**

**Step 1:**

79 x 81 = 79 x (9 x 9)

**Step 2:**

Multiply 79 by 9, i.e,

79 x 9 = 711

**Step 3:**

Multiply the number obtained in step 2 with 9 to get the answer, i.e.,

711 x 9 = 6399

Answer: 79 x 81 = 6399

**Method 5:** When the multiplication involves 25 or 50

**Example: 44 x 25**

We know that 25 is one-quarter of 100, that is, 100 /4 = 25. So instead of multiplying by 25, we will multiply the number by 100 and then divide the answer by 4.

**Step 1:**

44 x 25 = 44 x 100/4

**Step 2:**

Multiply the number by 100

44 x 100 = 4400

**Step 3:**

Divide the answer obtained in Step 2 with 4

4400/4 = 1100

The number obtained is the required answer, i.e, 44 x 25 = 1100

Similarly, when multiplying with 50, the fraction will become 100/2. The rest of the method will remain the same.

**SEE ALSO:**

1. Vedic Maths: For Fast Calculations – I

2. Vedic Maths: For Fast Calculations – Part 3

**Do you feel strongly about something? Have a story to share? Write to us at ***info@thepeepertimes.com** or connect with us on Facebook or ***Twitter *** *

This is the fifth and the final article in our series on Vedic Maths. We hope, by now, you have mastered all the techniques that we discussed in our earlier articles. As mentioned in our previous article, this write-up will teach you how to use the General Method of Multiplication for 3-digit (or more) numbers. So, here we go:

**General Method of Multiplication **

Multiply 341 by 752 (three digit number by three digit number)

**Step 1:**

As done in two-digit numbers, put the numbers one below the other, like this:

3 4 1

7 5 2

**Step 2: **

Now, multiply the digits in the first column (the first column is the one formed from the unit’s place of both the numbers).

3 4 1

X 7 5 2

——————————————–

2 No carry-over

**Step 3:**

Now cross-multiply unit’s and ten’s column, that is, 4X2 = 8; and 1X5 = 5. Add the answers (8 + 5) = 13. Put 3 in the ten’s column and 1 is carried-over.

3 4 1

X 7 5 2

——————————————–

3 2 Carry-over = 1

**Step 4:**

Cross-multiply third column with the first one, that is, 3X2 = 6 and 1X7 = 7. Add the two (6 + 7) = 13

Now, multiply the numbers of second column with each other, that is, 4X5 = 20

Add the above obtained answers with the carry-over, like this:

13 + 20 + 1 = 34

Put 4 in the answer and 3 is the new carry-over

3 4 1

X 7 5 2

——————————————-

4 3 2 Carry-over = 3

**Step 5:**

Now that all the three columns have been used, we will start eliminating the columns. Eliminate unit’s place column and cross-multiply second and third column with each other, that is, 3X5 = 15; and 7X4 = 28

Add the answers and the carry-over obtained in step 4

15 + 28 + 3 = 46 (Now, 4 is the new carry-over)

3 4 1

X 7 5 2

———————————————————

6 4 3 2 Carry-over = 4

**Step 6:**

Now, eliminate the ten’s place column as well. We are left with only one column, that is, the third one. Multiply the digits with each other (3X7) = 21. Add the carry-over of step 5 to 21, like this:

21 + 4 = 25

Since we are left with no more columns, the answer obtained (25) will be written as it is:

3 4 1

X 7 5 2

————————————————————————————

2 5 6 4 3 2 This is your answer.

You can repeat the above method with four digit number or above as well.

For being able to do your calculations fast, master all the techniques discussed under this topic here.

**SEE ALSO:**

**1. Vedic Maths – 4: General Method of Multiplication**

**2. Vedic Maths – 3: Base Method**

**3. Vedic Maths: For Fast Calculations – Part 2**

**4. Vedic Maths: For Fast Calculations – Part 1**

**Do you feel strongly about something? Have a story to share? Write to us at ***info@thepeepertimes.com** or connect with us on ***Facebook *** or ***Twitter *** *****

We continue with our series on common mistakes in English. In this part, we will mainly focus on mixed doubles. Sometimes, you may have found yourself confused whether to use singular verb with words like everybody, crowd, etc., or not? Or whether to use neither without nor, either without or, and so on? So here, we try to answer such questions:

**Beware of mixed doubles**

1. Words such as everybody, nobody, either, neither are followed by a singular verb, for example, Everybody has his good qualities.

*Note: The possessive adjective (his) in the above sentence is also singular.*

2. When using expressions like ‘sort of’ and ‘kinds of’, make sure you have them either all in the singular or all in the plural. Examples,

- This sort of movie interests me.
- I do not like those kinds of movies (it is incorrect to say – those kind of movies).

3. Depending on the sense in which they are being used, words like crowd, number, company, enemy can take either a singular or a plural verb. Example:

- The number of casualties is high.
- A number of people were involved in the attack.
- The crowd was moving towards us (sense of one vast mob).
- The crowd were shouting anti-government slogans (sense of many people).

4. A singular verb follows two or more subjects joined by ‘and’ representing a single idea, example – Eggs and bacon is the favourite English breakfast.

5. Most words ending in -ics such as mathematics, physics, politics are treated as singular. Example, Politics is a tricky career.

*Can you think of more such mistakes? Well, send them across and we will keep on updating the list.*

**YOU MAY LIKE TO READ:**

1. **Common mistakes in English – 1**

2. **Common mistakes in English – 2**

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com****or connect with us on Facebook or ****Twitter **

Vedic Maths involves mental calculation techniques that, if practised well, can reduce your calculation time drastically. Many institutes teach this as a part of their curriculum while preparing for entrance examinations. We recommend you to try this once and judge for yourself the effectiveness of these techniques or tricks.

In Vedic Maths, there are various methods of multiplication. Regular practice will enable you to choose the one that best suits a particular problem. Below are some simple multiplication methods that can really improve your calculation speed.

**Method 1:**

How to multiply when one number ends with 9?

**Example 1: 33 x 29**

Step 1: Since 29 ends with 9, we will focus on this number.

29 is succeeded by 30. So we will multiply the given number (in this case 33) by 30.

33 x 30 = 990

Step 2: The difference between 30 and 29 is 1. This 1 needs to be multiplied with 33 which gives us 33 as answer. We will now subtract 33 from 990 to get the correct answer.

990 – 33 = 957

957 is our answer (33 x 29 = 957)

Isn’t is simple?

Try this with other examples. Below is an example when one number is a two-digit and the other a three-digit.

**Example 2: 59 x 437**

Step 1: 60 x 437 = 26220

Step 2: 26220 -437 = 25783

Answer: 59 x 437 = 25783

**Method 2:**

How do we multiply two-digit numbers where the sum of the numbers at tens digit is 10 and the units digit is same?

**Example 1: 37 x 77**

In the above example, the units digits is the same (7) and the sum of the numbers at tens digits (3 + 7) is 10.

Step 1: Multiply the tens digits

3 x 7 = 21

Step 2: Add units digit to the answer obtained in step 1

21 + 7 = 28

Step 3: Multiply the units digits

7 x 7 = 49

(We need a two-digit answer in this step. In case, the answer is not a two-digit number, then we will add zero in the tens place)

Step 4: Combine the answers obtained in step 2 and 3. The combined figure is the answer.

2849 (37 x 77 = 2849)

**Example 2: 83 x 23**

Step 1: Multiply the tens digits

8 x 2 = 16

Step 2: Add units digit to the answer obtained in step 1

16 + 3 = 19

Step 3: Multiply the units digits

3 x 3 = 9 Since we need a two-digit answer in this step, we will add zero before 9. So, 3 x 3 = 09

Step 4: Combine the answers obtained in step 2 and 3. The combined figure is the answer.

1909 is the answer (83 x 23 = 1909)

Does these not sound too simple? Vedic Maths offers several such techniques. We will discuss some more techniques in our subsequent articles. Till then, practice them well.

**SEE ALSO:**

Vedic Maths: For Fast Calculations – Part 2

Vedic Maths: For Fast Calculations – Part 3

**Do you feel strongly about something? Have a story to share? Write to us at ***info@thepeepertimes.com** or connect with us on Facebook or ***Twitter *** *

Ever wondered why a non-drinker is called teetotaller in English language? Or why do we use the term ‘white elephant’ for defining something useless yet costly? And what do you mean by ‘egg someone on’? Does it mean throwing eggs at a person? Well, these and many such words have interesting stories behind their origin. Read on to find out…

**1) Why do we call someone who doesn’t drink a teetotaller?**

In earlier days, many virtuous people claimed ‘total abstinence’ from alcohol but behaved contrary. Thus, a need was felt for a word that did justice to those who were truly steadfast in their rejection of all forms of alcohol. To meet this requirement, England native Dick Turner coined the word ‘teetotal,’ emphasising on the initial constant of total. Many say ‘tee’ in the word actually stands for ‘tea’ for Turner recommended people to have tea as a substitute for alcohol in times of temptation. So pleased was Turner with this discovery that when he died, he had instructed his loved ones to inscribe this invention on his teetombstone.

**2) White elephant**

White elephants were considered good luck omen in Siam. It is believed that the King of Siam would punish his courtiers, who annoyed him, by gifting them a white elephant. The courtier was then burdened with something which was useless, unwanted, would cost him a fortune but had to be taken care of if he loved his life.

**3) Why does sub rosa mean secret?**

This is a Latin expression which means ‘under the rose’. It is said that in order to hide the goings-on of Venus, Cupid bribed the God of Silence (Harpocrates) with the first ever created rose. Also, medieval dining halls usually had a rose carved in the ceiling. This was to remind the guests that whatever conversation takes place at the table should not be repeated anywhere else.

**4) What do you mean by ‘egg someone on’? **

It certainly does not mean to throw eggs at someone. ‘Egg someone on’ means to incite a person until he goes into some course of action, which is usually a rash and regrettable one.

**5) Why do we refer to elite as the upper crust?**

Canadian humourist Thomas Haliburton first used these words in this sense in Sam Slick of Slickville (1835). While some say was trying to describe the hard exterior of aristocrats or their insolence, others think the term was coined because the upper crust of a loaf of bread was considered the most desirable.

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com**** or connect with us on ****Facebook ****or ****Twitter **** **

**Maths educator Rupesh Gesota does not believe in teaching his students by simply jotting down the textbook method on blackboard. Instead, he believes in the following teaching mantra– How can students be guided towards discovery by asking them right questions, so that they are able to detect and correct mistakes on their own? **

**In this article, he brings alive his class by detailing about the interactions he had with his students while teaching them one of the dreaded mathematics topics, Arithmetic Progression (AP). We are reproducing it here with the hope that it will help students learn the method in a fun-filled way while at the same time, inspire many educators to invoke curiosity and innovativeness in their teaching methods.**

I have been playing Maths with a bunch of marathi-medium municipal school students after their school-hours from some time now.

As I can now see them approach and solve quite challenging (out of the textbook) problems comfortably, I decided, for a change, to pick up their textbook for a while and see what unfolds… Of course, I cannot make them solve the problems from the school textbook of their age (they have surpassed it long back). So, I directly picked up their SSC (grade-10) Algebra textbook (they have just cleared their grade-7 and grade-8 exams). The first chapter was about Sequence (AP, GP). I was aware that this would be a cakewalk for them. So, I just rushed through the terminologies and asked them to derive the formula for nth term of Arithmetic Progression (AP).*(For those who are unaware or have forgotten what AP is, let me give you a couple of examples, instead of its definition) *

*i) 2, 5, 8, 11, 14,…… *

*ii) 3, -7, -17, -27,….*

*iii) 5, 10, 12, 17, 19,….*

*iv) 2, 20, 200, 2000,…*

*Only (i) and (ii) are APs while (iii) and (iv) are not APs… (common difference (d) = 3 in the 1st case and d = -10 in the 2nd case) *

So, I asked them to come up with a formula for nth term of AP, that is, in a given AP, if you wish to ‘quickly’ find the 200th term, 5369th term, etc., without listing down all the middle terms, then how can you do that using the formula?

They didn’t take more than half-a-minute to derive at the required result. It was correct.

Tn = T1 + (n-1) d

So now, I immediately asked them to derive the formula for Sum of first nth terms of AP (Sn).

The book was in my hand and hence I saw its method. I was wondering if this method will click to them or will they do something else, but I was very sure they will crack it. And in almost a minute, all hands went up.

No wonder, they had used a different method… But the main thing to wonder was they all had used the same method.

I forgot to take the snap of their work. So I am typing down what they did:

Sn = T1 + T2 + T3 +T4 +…

= T1 + (T1 + d) + (T1 + 2d) + (T1 + 3d) +….

= nT1 + d + 2d + 3d +….

= nT1 + d (1 + 2 + 3 +….)

= nT1 + d (n-1) n / 2

= n [T1 + d (n-1)/2]

If you observe, they have used the formula for sum of first few (n-1) natural numbers to derive expression for Sn.

However, the textbook derivation does not use this method and formula (in fact, the formula for sum of first n natural numbers is derived later in the book using the formula for Sn)… And that’s also fine.

So, I first appreciated them for their accuracy and then challenged them to get the expression for Sn without using the formula for sum of first few natural numbers. I asked them – what if you didn’t know the formula that you have used in this derivation?

To this, one of them replied, “Sir, we would have derived it the way we had done it long back.”

“Hmm… And how did you do that?”

“By pairing up, first and last numbers, second and second-last, and so on.”

“Yes, then why don’t you use the same idea here too?”

They looked at me, puzzled…

So after a minute, I just showed them the method given in the textbook.

They studied it and then said, “Sir, isn’t this the same pairing method?”

I then gave them some textbook problems just for practice (were they problems?). Luckily, the last one among them was a little better.

**Problem: 4 terms are in AP. Product of 1st and 4th terms is 45. Sum of 2nd and 3rd terms is 18. Find the terms. **

I would suggest you to (try to) solve this problem before reading further…

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Since the textbook was in my hand, I saw the method. However, let me also confess that I would have used the same method because this is what was directly taught to me through ‘drill and kill’.

Thankfully, I recovered from being an instructor and didn’t do this damage anymore.

And you know what? This not only saves my effort but also helps me learn interesting methods from my students.

**Method-1: By Jitu**

He argued that since 3 x d x t1 and 45 are multiples of 3, hence (t1)^{2} should also be multiple of 3. Hence, it can take only two values: 9 and 36 to get the required sum 45. Substituting (t1)^{2} as 36 does not satisfy the equation and substituting as 9 does satisfy. He solved further to arrive at the required AP.

He was the first to complete the problem. To engage him, I gave him another similar problem, but of 3 consecutive numbers – Sum of 3 numbers is 27 and their product is 504; and he was quick to solve this one too.

**Method-2 – By Sahil (original 4 numbers problem)**

He too argued similarly about the addition equation. Since ‘2a’ and ‘3d’ sum up to an even number, and ‘2a’ is already even, so ‘3d’ too should be even. Hence, ’d’ should be even. Did you understand this?

He then found the possible values for d. He also said that ‘d’ cannot be 0 since that would make all terms equal, which is in turn not possible as 45 (given as product of two terms) is not a perfect square. He also said that ’d’ cannot be ‘6’ because that would make a=0 leading to the product of 1st and 4th terms as 0, thus contradicting the given information.

He then found the corresponding 2 values for a, substituted each pair in the equation.

(a, d) = (6, 2) didn’t satisfy but (3, 4) did satisfy. He then got the required sequence.

**Method-3 – by Kanchan**

She first told me that Sum of 2nd and 3th terms would be same as the Sum of 1st and 4th terms.

When I asked her why, she said – it’s obvious. 1st term is less than 2nd term by the same amount as the 4th term is more than 3rd term.

Did you get this?

I was happy she could visualise this relation; however, I challenged her to prove it while she was writing on the board, and if you notice, she has done it partially.

What drove me crazy was her next move. She went into Quadratic Equation.

“We know the sum of two numbers; we know the product of same two numbers… So we can make an equation, and find its roots…”

Hope you will take some time to study her solution.

*(let me mention that quadratic equation is formally taught in 10th, that too in a dry way… she has not only used quadratics while in 8th but has also applied that knowledge in another topic altogether!)*

And her method, gave us both the possible sequences – 3, 7, 11, 15 and 15, 11, 7, 3 which was not the case with other methods.

Finally, I asked them if they wanted to know how it’s solved in the textbook. And they all were curious to know.

So, I simply copy pasted this method on the board, leaving for them to analyze and understand…. And they could easily do so….

I waited for them to study this approach. And then –

“So if I give you another problem, but this time of 3 consecutive numbers in AP, then how will you solve it?”

“We will then take those terms as a-d, a, a+d”

“Oh! You will use this textbook method?”

“Yes sir, this is an interesting and even an efficient one!”

I could see them smiling.

“So I should have directly showed this method to you, isn’t it?”

“No…. then how would we think and discover our own methods?” Vaishnavi reacted.

*(All the above conversations happened in Marathi. These students are from Marathi medium municipal school, based*

Today, the world is going to witness the longest lunar eclipse of this century. Scientists are excited. Not only because it is going to be extra-ordinary in nature but also because eclipses are the time when they may discover something new. Just like hundreds of years ago, when they discovered new elements?

**Why does lunar eclipse occur?**

Lunar eclipse occurs when the Earth comes between the Sun and the Moon and all three are perfectly aligned. This formation prevents sunlight from reaching the Moon.

**Why does moon appear red this time as against black in usual eclipses of moon?**

Today, a cosmic phenomenon “the lunar eclipse” is occurring in nature, a complete eclipse, extra-ordinary in nature from usual eclipses, with moon looking red, as against black in usual eclipses of moon. This, of course is due to extra-ordinary position of moon with respect to earth and sun happening once in a century or more; and the redness on moon being caused by the scattered Sun’s light passing through moon’s atmosphere where all six colours but the seventh colour (Red) of white light (VIBGYOR) are absorbed, and red being reflected. This makes moon look red.

**Did you know?**

During the phenomena duration, scientists go high up in space with instruments like spectrometers, etc., to examine the scattered light and try to find the changes if any occurring or even discovering new things thereby. The discovery of Zero group elements of Periodic table like Helium (He), Neon (Ne), Argon (A), etc., has been made by examining the spectrum of the light.

**Why you (common people/students) should be excited too?**

- Today is going to be the longest lunar eclipse of this century. It will last for 1 hour and 43 minutes (total 103 minutes).
- The phenomenon will be visible in most parts of India. You can view it through naked eyes.
- The eclipse will start in India at 11.44 pm on Friday night.
- The entire celestial event, from start to finish, will last for about four hours.
- The moon will turn red (also called blood moon) or ruddy-brown colour.
- Planet Mars will appear bigger and brighter today.
- Mars will be seen making closest approach to Earth in 15 years. It will be brightest on July 31.

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com**** or connect with us on ****Facebook**** or ****Twitter**

We continue with our series on common mistakes in English. In this part, we will mainly focus on homophones. Often described as the most confusing words in English, homephones are words that sound the same but have different meaning. But, let’s start with apostrophes.

**1) Misplaced apostrophes**

This is one of the most familiar mistakes in the English language. Many people use an apostrophe to form the plural of a word. Remember, apostrophes are never used to make a word plural. They indicate either possession (when indicating something belonging to one person, the apostrophe is placed before the ‘s’ – example, John’s pen; however, when indicating something belonging to more than one person, the apostrophe is put after the ‘s’ – example, the boys’ horse) or the omission of letters or numbers (e.g. don’t; 26 Jan ’98).

**2) There, their, they’re**

Well, this is a common error people make. ‘There’ refers to that place or position which is not here, e.g. “We went to Sydney and stayed there.” It can also be used to state something, such as, “There is no doubt who is the best student.”

‘Their’ indicates possession – something belonging to or associated with the people or things previously mentioned, e.g. “Their belongings are missing.”

They’re – it is short for “they are.” For example, “They are going to be here anytime,” can also be written as, “They’re going to be here anytime.”

**3) Your/you’re**

This confuses many, and yet it is so simple. ‘Your’ indicates something belonging to you. For example, “Your books are on the table.”

‘You’re’ is short for ‘you are’. For example, “You are (you’re) beautiful”

**4) Its/it’s**

There we go again. Many people think the apostrophe in ‘It’s’ indicates possession. No, it is a short for ‘it is’. For example, “It’s raining outside.”

‘Its’ means ‘belonging to or associated with a thing previously mentioned’ but used when you are not talking about a person. Example – “Does a baby in its mother’s womb cry?”

**5) Then/than**

Confusing for many. Remember, ‘Than’ is used in comparisons. Example – He is better than his brother.

‘Then’ means ‘at that time’ or ‘afterwards/after that’. Example – We’ll go to the bank first, then the restaurant.

**YOU MAY LIKE TO READ: ****Common mistakes in English – 1**

**6) Me/myself/I**

The best way to remember it is:

When you are referring to yourself and somebody else, put the latter’s name first in the sentence. Then check what will sound right with it – ‘me’ or ‘I’. For example, with the sentence “Meera and I are going to office”, you wouldn’t say “Meera and me are going to office” Hence you wouldn’t use ‘me’ here.

However, you will use ‘myself’ if you have already used ‘I’, making you the subject of the sentence. For example, I’ll handle it myself.

**7) i.e. and e.g.**

i.e. means ‘that is’.

E.g. means ‘for example’.

They are usually used in informal writing.

**8) Compliment/complement**

Complement – It means ‘a person or thing that completes something’ or ‘a thing that contributes extra features to something else in such a way as to improve or emphasize its quality.’ For example, “The two of them complemented each other well; he was a good guitarist, and she a good singer.”

Compliment – It is a polite expression of praise or admiration. Example – “The dress looks nice on you.”

**9) Fewer/less**

While ‘fewer’ refers to items one can count individually, ‘less’ refers to a commodity which can’t be counted individually. For example, There are fewer cakes left in the shop.

Towards its north, you will find less water in this river.

**10) Know/no**

Often mixed up as the two sound the same. Know means ‘be familiar with’ or ‘be aware of something’. For example, “I know he is being honest.”

‘No’ indicates rejection. It is the opposite of ‘yes’. Also, ‘no’ with a full stop after it (no.) means ‘number’. For example, “No. of times people visited the site: 150.”

*Can you think of more such mistakes? Well, send them across and we will keep on updating the list.*

**Do you feel strongly about something? Have a story to share? Write to us at ****info@thepeepertimes.com**** or connect with us on Facebook or ****Twitter **** **

Many of us hate Science, labelling it a subject of studious people. Who wants to study boring formulae or spend hours in labs. But believe it or not, science is so intertwined in our existence, in our everyday life, that we often don’t realise that our actions are governed by simple scientific logics.

Below are 10 such instances where we use science in our everyday living:

**1)** **When we lean forward while climbing a hill**

When we lean forward, the centre of gravity of our body also shifts forward, stabilizing our standing body position, and thus helping us in climbing.

**2)** **Usage of blotting paper for absorbing ink or liquid**

Blotting paper is porous by nature. The pores act as its capillaries. When liquid substance comes in contact with blotting paper, it is sucked by the paper due to capillary action.

**3) When we pour hot liquid in a thick glass tumbler, it breaks. Why?**

Often, you must have heard mothers advising you against pouring hot liquid in a thick glass tumbler, else it will break. That’s because when we pour hot liquid in a thick glass tumbler, its inner surface expands due to heat while the outer surface remains comparatively cool. This unequal expansion causes breakage.

**4) Conversion of milk into curd**

Milk contains a protein called casein. When curd is added to milk, the lactic acid producing bacteria present in curd cause coagulation of casein, thus converting milk into curd.

**5) Ice floats on water. Why?**

Density of ice (nine volumes of water is equal to 10 volumes of ice) is lesser than that of water, hence it floats.

**6) Ever wondered why petrol fire is not extinguished by throwing water on it?**

There are two reasons for that. One, petrol floats on water and keeps burning. Two, the heat generated by the fire is so intense that it results in decomposition of water upon contact.

**7) Whenever we see a straight stick half immersed in water, it appears bent. Why?**

That’s because of refraction. As the light rays coming from the immersed portion of the stick emerge from water medium to air medium, they suffer refraction. The refracted rays thus enter eyes and the object (the immersed part) appears bent.

**8) How does a drinking straw work?**

Straw has a hole inside. As we draw liquid from one end, the air in the straw is sucked creating vacuum which is filled by the rise of liquid facilitated by the pressure of atmosphere outside.

**9) During summers, water kept in an earthen pot is colder than outside. Ever wondered why?**

Earthen pot has small pores through which water oozes out slowly and keeps evaporating from the outer surface, and thereby producing cooling effect.

But do you know that during rains, water in an earthen pot does not cool. This is because evaporation of water does not take place quickly as the air outside is moist and cool.

**10) Why do we not get hurt while cutting our nails?**

Our nails are made up of dead protein called keratin, and have no connection with blood vessels or nerves. Thus, cutting them do not affect our nerve system in any way, and we do feel hurt.

**SEE ALSO:**

**1. Agni-5 – A weapon of peace**

**2. Ten famous scientists of India**

**Do you feel strongly about something? Have a story to share? Write to us at ***info@thepeepertimes.com** or connect with us on Facebook or ***Twitter *** *

We all know that English is one of the most flexible and adaptable languages in the world. But what makes the language even more intriguing is that many of its commonly used words have a rather interesting (even funny) background? Read on to get surprised…

**1) Boycott**

Did you know that the first person ever to be boycotted was an Englishman named Boycott. Charles Cunningham Boycott (1832-97), who was an English land agent at Woolwich, came into conflict with the Land League agitation. He had to seek police protection as his men, whose sympathies were with the agitators, refused to work for him. The kind of treatment he received at that time is today known by his name, Boycott.

**2) Bootlegging**

Smuggling of illicit liquor is popularly known as bootlegging. The origin of the word goes back to the days when liquor was concealed in large sea-boots by smugglers. And the practice was revived by smugglers in America when prohibition was in force there.

**3) Fifth column**

The Spanish Civil War gave English language the expression ‘fifth column.’ The Nationalist General, Mola, who laid seige to Madrid, is said to have coined this phrase when he declared: “I have four columns operating against Madrid and a fifth inside, composed of my sympathizers.”

**4) Quisling**

A Norwegian leader called Mr Quisling collaborated with the Nazis during the World War II, thus giving English language a new word. Quisling nowadays means a person who betrays his or her own country by aiding an invading enemy.

**5) Cinchona**

In the seventeenth century, the Countess of Chinchona, wife of the ruler of Peru, was diagnosed with malaria. She was cured when administered a medicine prepared from the bark of a South American tree. This led to an extensive planting of this tree, which today bears the name Cinchona.

**6) Chauvinism**

Nicholas Chauvin was excessively devoted to his Emperor Napolean. Hence, the term ‘chauvinism’ – which means unreasonable and exaggerated pride in one’s country with corresponding contempt towards other nations.

**7) Daltonism**

John Dalton (1760-1844), famous for his atomic theory of matter, is also remembered for giving the English language the word, Daltonism (which means colour-blindness). Dalton and his brother were both colour-blind and published papers on the causes of this ailment.

**8) Sabotage**

A word of French origin. During the French Revolution, cultivators are said to have destroyed their crops by treading upon them with heavy wooden shoes called ‘sabot’ so that the King’s men do not profit by seizing them.

**9) Silhoutte**

This word is derived from the name of Etienne de Silhoutte (1709-67). Silhoutte belonged to France and had a particular fancy for such pictures.

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