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## Agreement of the Verb with the Subject

A verb must agree with its subject in number and person. The problem arises when the verb is wrongly matched with the noun near it instead of with its proper subject. **Rule 1 **

When two subjects are joined by as well as or with, the verb should be matched with the first subject.

For Example :

Incorrect : The teacher with his students are busy

Correct : The teacher with his students is busy

Incorrect : He as well as I am in the wrong

Correct : He as well as I is in the wrong**Rule 2**

When two subjects are joined by Neither - nor or Either - or, the verb should be matched with the second subject.

For Example :

Incorrect : Neither he nor I is in the wrong.

Correct : Neither he nor I am in the wrong

Incorrect : Either you or he are guilty

Correct : Either you or he is guilty

Note : When one of the subjects joined by Either - or or Neither - nor is in the plural, the verb should be in the plural, and the plural subject should be placed near the verb.

Incorrect : Neither the teacher nor the students is busy

Correct : Neither the teacher not the students are busy

Incorrect : Either Ram or his friends is guilty

Correct : Either Ram or his friends are guilty. **Rule 3**

When two subject are joined by and, the verb should be in the plural.

For Example :

Incorrect : He and I is good friends

Correct : He and I are good friends

Note : However, if two nouns constitute one idea or are used as one phrase, the verb should be in the singular.

Incorrect : Bread and butter are his favourite food

Correct : Bread and butter is his favourite food

Incorrect : The teacher and guide are dead

Correct : The teacher and guide is dead

Note : Sometimes, the use of the article in such sentences also determines the verb

For Example :

Incorrect : The teacher and the guide is dead

correct : The teacher and the guide are dead**Rule 4**

Either, Neither, Each, Everyone, Many a take singular verb.

For Example :

Incorrect : Either of the students are guilty

Correct : Either of the students is guilty

Incorrect : Each of the students have been well provided

Correct : Each of the students has been well provided

Incorrect : May a politician have been found to be corrupt

Correct : Many a politician has been found to be corrupt

Incorrect : Everyone of the politicians are guilty

Correct : Everyone of the politicians is guilty **Rule 5**

Nouns qualified by each or every even when joined by and take a singular verb

For Example :

Incorrect : Each boy and girl were given separate seat

Correct : Each boy and girl was given a separate seat

Incorrect : Every boy and girl were offered hospitality

Correct : Every boy and girl was offered hospitality **Rule 6**

Collective nouns like Jury, Army, Committee, Assembly etc. take a singular verb when they subscribe to one view. If they express divergent views, the verb should be used in the plural.

For Example :

Incorrect : The Jury are unanimous in their judgment

Correct : The Jury is unanimous in its judgment

Incorrect : The jury is divided in its judgment

Correct : The jury are divided in their judgment

Incorrect : The assembly has disagreed on the question of discipline

Correct : The assembly have disagreed on the question of discipline **Rule 7**

Some nouns like mathematics, politics, news, innings, wages, summons, advice, scenery, furniture, machinery, poetry, information, vacation etc take a singular verb.

For Example :

Incorrect : Mathematics are a difficult subject.

Correct : Mathematics is a difficult subject.

Incorrect : Politics are his bread and butter.

Correct : Politics is his bread and butter.

Incorrect : New machinery for the factory have arrived.

Correct : New machinery for the factory has arrived.

Incorrect : We have just returned after long vacations.

Correct : We have just returned after a long vacation. **Rule 8**

Some nouns like scissors, trousers, measles, spectacles, tongs, riches etc take a plural verb.

For Example :

Incorrect : The scissors is blunt.

Correct : The scissors are blunt.

Incorrect : My spectacles has been stolen.

Correct : My spectacles have been stolen.

Incorrect : Your trousers is torn.

Correct : Your trousers are torn.**Rule 9**

Some collective nouns like cattle, poultry, gentry, vermin take a plural verb.

For Example :

Incorrect : The cattle is grazing.

Correct : The cattle are grazing.

Incorrect : The gentry is absent.

Correct : The gentry are absent.

Note People can be used both as singular and plural For example

(I) The Indians are supposed to be emotional people

(II) Peoples of different races and religions live in India. **Rule 10**

Some nouns like sheep, deer, salmon, swine etc retain the same form in the singular and the plural.

For Example :

The sheep is grazing (singular)

The sheep are grazing (plural)**Rule 11**

When a plural noun refers to a specific quantity or amount considered as a whole, it generally takes a singular verb, dozen, score, hundred (when preceded by numerals) : a five rupee note.

For Example :

Incorrect : Five hundred miles are a long distance.

Correct : Five hundred miles is a long distance.

Incorrect : I purchased two dozens mangoes.

Correct : I purchased two dozen mangoes.**Rule 12**

When a plural noun is a proper noun referring to a single entity, it takes a singular verb.

For Example :

Incorrect : The United States of America act like a bully.

Correct : The United States of America acts like a bully.

Incorrect : 'The three Musketeers' were written by A. Dumas.

Correct : 'The Three Musketeers' was written by A. Dumas.

## Proper Usage of Verbs

**Rule 1**

In a compound sentence, a single verb can fit in with two subjects, provided the form of the verb agrees with the subject.

For Example :

His mouth was open, his nose twitching.

But the following sentence is incorrect:

His mouth was open, his eyes staring.

It should be written as :

His mouth was open, his eyes were staring.

Note the difference in the verb form. **Rule 2**

Two auxiliary verbs can be used with one principal verb, provided the form of the principal verb suits both the auxiliaries.

For Example :

I never can or will hurt a fly.

But the following sentence is incorrect:

He never has and never will refuse a bribe.

It should be written as:

He never has refused and never will refuse a bribe.

Note the difference in the verb form.**Rule 3**

One auxiliary verb can be used with two principal verbs, provided its form suits both principal verbs.

For Example :

Several victims have been killed and several buried.

But the following sentence is incorrect:

A new secretary has been elected and the old resigned.

It should be written as :

A new secretary has been elected and the old has resigned.

Again note the difference in the verb form.**Rule 4**

The verb lay and lie are often incorrectly used. The verb lay always takes an object and the three forms of the verb are : lay, laid, laid.

The verb lie (recline) does not take an object and the three forms of the verb are lie, lay and lain.

For Example :

Lay the table.

The table was laid by the servant.

Let the child lie.

The child lay on the bed.

The body has lain for two hours.

Note: Another meaning of the verb lie is "to tell a falsehood". The three forms of this verb are lie, lied, lied. This should not be confused with the lie mentioned above.

For Example :

She is fond of telling lies.

Does she lie often?**Rule 5**

The verb rise and raise are also often incorrectly used.

The verb rise means 'to ascend', 'to go up.'

The three forms of the verb are: rise, rose, risen. It does not take an object.

For Example :

The plane rose very quickly.

The mountains rise above the land.

My uncle has risen in life.

The verb raise means 'to lift up', 'to increase.

The three forms of the verb are : raise, raised and raised.

For Example :

She raised her leg.

I requested the manager to give me a raise.

Do not raise your voice. **Rule 6**

Note carefully the use of the following verbs:

Ring, sing, sink, begin, show, flow, hang, awake and fly

The past tense and the past participle of these verbs are often mixed up.

The three forms of these verbs are :

Ring, Rang, Rung.

Sing, Sang, Sung.

Sink, Sank, Sunk.

Begin, Began, Begun.

Show, Showed, Showed/Shown.

Flow, Flowed, Flowed.

Hang, Hung, Hung (a picture)

Hang, Hanged, Hanged (a criminal)

Fly, Flew, Flown.

For Example :

The visitor rang (not rung) the bell.

The ship sank (not sunk) without a trace.

The show has begun (not began).

The river has overflowed (not overflown) its banks

The criminal was hanged (not hung)

The picture was hung (not hanged). **Rule 7**

The use of shall and will

Many of the precise distinctions concerning the use of shall and will have passed out of informal speaking and writing. But formal writing still prohibits the arbitrary use of these two forms of the verb.

Shall is used in the first person and will in the second and third person to express simple futurity.

For Example :

I shall We shall

You will They will

However, in order to express determination, compulsion, threat, willingness, command or promise, reverse the order of shall and will. Use will in the first person and shall in the second and third person.

For Example :

I will go tomorrow, come what may.

You shall do this work.

I will try and improve my performance.

They shall go by this afternoon.

Note

(I) Will or shall should not be used twice in the same sentence if both actions refer to the future.

For Example :

Incorrect : I shall reach the office if the bus will come in time.

Correct : I shall reach the office if the bus comes in time.

(II) In asking questions, will is not used in the first person.

For Example :

Shall I go ?

Shall we go ?**Rule 8**

The use of should and would. Should is the past tense of shall and generally follows the same rules that apply to shall.

Would is the past tense of will and generally follows the same rules that apply to will.

Both should and would have special uses too. Should is used in all three persons to express obligation. Both ought and should are used interchangeably to express obligation.

For Example :

I should go if I were you.

You should do a good deed everyday.

You ought to be courteous.

Would is used in all three persons to express habitual action, determination and willingness.

For Example :

We would go for a walk every evening.

I would not run away from responsibility.

He would try to do his best. **Rule 9**

Use of ought

ought is usually followed by to.

For Example :

You ought to go home now.

ought is used to express :

(a) duty or moral obligation :

The rich ought to help the poor.

(b) probability :

He ought to have come back home by now.

(c) desirability :

You ought to pray before every meal.**Rule 10**

May and Might

As a principal verb, may expresses possibility or permission.

For Example :

Let's go, he may be home now.

May I leave now.

As an auxiliary verb, may expresses a wish or purpose.

For Example :

May you succeed in life.

We take medicine so that we may be cured.

Might is the past tense of may

For Example :

He asked if he might leave.

It is also used to express a weak possibility or a polite suggestion .

For Example :

You might find the purse with the peon, but I doubt it.

You might make a suggestion if you want.**Rule 11**

Can and Could

(i) Can expresses ability or capacity

For Example :

I can walk ten miles.

Can you solve this problem

(ii) Con also expresses permission.

For Example :

You can go

In this sentence can has the same meaning as may. The difference is that may is used to express possibility in affirmative sentences. Can is used in interrogative or negative sentences. .

For Example :

It may be true.

Can this be true ?

It cannot be true.

Could is the past tense of can. It also acts as a principal verb when it expresses its own meaning.

For Example :

Inspite of his illness, he could do well in the examination.

Could here relates to ability.

## Proper Usage Of Nouns and Pronouns

**Rule 1.**

A Pronoun should agree with its antecedent in person, number and gender.

For Example :

Every father must bring up his children properly.

All students should show their home work.

One must do one's duty.

Each girl should contribute her share. **Rule 2**

Each, every, either, neither, many a, any, anybody, everybody, everyone, take a singular pronoun.

For Example :

Incorrect : Everybody in the bus were injured.

Correct : Everybody in the bus was injured.

Incorrect : Each boy paid their own share.

Correct : Each boy paid his own share.

Incorrect : Everyone are happy with their effort.

Correct : Everyone is happy with his effort. **Rule 3**

Anyone, everyone, each, everybody etc. take a pronoun which agrees with the antecedent gender. However, when gender is not mentioned, then masculine pronoun is used.

For Example :

Incorrect : Every student must show their homework.

Correct : Every student must show his home work.

Incorrect : Anyone can ask for their turn.

Correct : Anyone can ask for his turn.

Note One and everyone take different pronouns. One is used throughout.

For Example :

Incorrect : One should do his duty.

Correct : One should do one's duty.

Incorrect : Everyone should do one's duty.

Correct : Everyone should do his duty.**Rule 4**

When the verb form to be (is, am, are, was, were) is to be complemented by a pronoun, it should be in the nominative form, i.e. I, you, he, she.

For Example :

Incorrect : It was him who came in the morning.

Correct : It was he who came in the morning.

Incorrect : I am taller than her.

Correct : I am taller than she.

Incorrect : Is it me you are looking for?

Correct : Is it I you are looking for?

Incorrect : You are smarter than him.

Correct : You are smarter than he. **Rule 5**

A pronoun in its objective from (him, her, me) should be used as an object of a verb or a preposition.

For Example :

Incorrect : He has helped I and my father.

Correct : He has helped me and my father.

Incorrect : She, who appeared so considerate, turned out to be cruel.

Correct : Her, who appeared so considerate, turned out to be cruel.

Incorrect : Between you and I, you are a crook.

Correct : Between you and me, you are a crook.

Incorrect : Except he all were present.

Correct : Except him all were present.**Rule 6**

If three persons are used in a sentence, the order should be Second person, Third person and First person.

For Example :

Incorrect : I, you and he can leave at 5 p.m.

Correct : You, he and I can leave at 5 p.m.

Incorrect : If Ram and you are going, I and Sham shall accompany you.

Correct : If you and Ram are going, Sham and I shall accompany you. **Rule 7 **

Each other should be used for referring to two persons or things, one another for referring to more than two.

For Example :

Incorrect : Husband and wife exchanged vows with one another.

Correct : Husband and wife exchanged vows with each other.

Incorrect : People should love each other.

Correct : People should love one another. **Rule 8**

Yours is usually used before words ending in -ly, otherwise your is used

For Example :

Incorrect : I am, your obediently.

Correct : I am, Yours obediently.

Incorrect : I remain, Yours affectionate husband.

Correct : I remain, your affectionate husband.**Rule 9**

Who and Whom are often used incorrectly.

Who is to be used in Nominature (subjective) case.

Whom is to be used in objective case.

For Example :

Incorrect : There are some whom I think are very smart.

Correct : There are some who I think are very smart.

Incorrect : Whom do you think was there?

Correct : Who do you think was there?

Incorrect : Who do you wish to address?

Correct : Whom do you wish to address?

Incorrect : Who the gods love die young.

Correct : Whom the gods love die young.**Rule 10**

A Reflexive pronoun (addition of self) is used as an object of a verb, and refers back to the same subject.

For Example :

The culprit hanged himself.

If we write "the culprit hanged him", 'him' would refer to somebody else and not to the culprit himself. The meaning thus would change.

When you wrong me, you actually wrong yourself.

Some people always talk about themselves.

I plan to go myself.

We ourselves are to be blamed.

## Profiting from profit and loss

**Introduction:**

Percentage may be one of the most popular mathematical concepts around but as a pure motivating force, it is difficult to beat profit and loss. Such has been their power that for centuries, human beings have done everything from the noble to the utterly dishonest just to make a few dollars more. And it is not just business. In fact, almost all economic activity revolves around the concept of profit. After all, it is impossible to spend more than you earn, is it not?

It is also impossible to take a competitive examination without coming across at least one question on profit and loss. So if you are not too good on profit and loss, you will be incurring a loss in these examinations. Which would be a pity as there is nothing really difficult about profit and loss.

Like most commercial mathematics, profit and loss is fundamentally simple in nature. All one needs is the knowledge of basic mathematical operations, fractions, and percentage. Add lots of common sense to that mixture, and you will be ready to make a neat little profit whenever an exercise on profit and loss comes up.

**Nature of exercises**- What you need to solve them
- Some strategies
- Points to keep in mind
- Sample questions with solutions
- Send us a question
- Feedback

## Nature of exercises:

- The rate of profit or loss

- The amount of profit or loss

- The cost price

- The selling price

- The rate or amount of discount offered

- The quantity to be sold or manufactured to make a certain amount of profit or loss

## What you need to solve them:

At its very core, profit and loss is dreadfully simple. If the selling price of an article exceeds the cost of making it, a profit is made. If on the other hand, the cost is greater than the selling price, a loss has been incurred. And as we all now, discounts are basically reductions in selling price to boost sales. It is as basic as that.

The formulae that are going to come into play are as follows:

- Profit = Selling price - Cost price

- Loss = Cost price - Selling price

- Discount = Selling price - amount of discount

- Profit per cent = [Profit/ Cost price] x 100

- Loss per cent = [Loss / Cost price] x 100

- Discount per cent = [Discount/ Selling price] x 100

- Selling price = Cost price x [100 + profit per cent] / 100, if there has been a profit.

- Selling price = Cost price x [100 - Loss per cent] /100, if there has been a loss.

- Cost price = Selling price x [100/ 100 + Profit per cent], if there has been a profit

- Cost price = Selling price x [100/ 100 - Loss per cent], if there has been a loss.

- Discounted price = Selling price x [100 - Discount per cent] / 100

## Some strategies:

- As there are basically five terms involved in these questions - profit, loss, discount, cost price and selling price - I always found that it helped me immensely to keep five slots mentally prepared for them and slip the amounts or rates concerned into these slots. Believe me, if you can remember your basic information, half the work is done.

- A terrific way to prepare for these exercises is to mentally work out the amounts and rates of discounts whenever you come across an advertisement for a discount sale in a newspaper or periodical. Also work out how much you gain from a discount - that is your profit. Similarly, the next time you hear of a price increase, just try to figure out how much you are losing. As I mentioned earlier, profit and loss are a part of our lives. If you get accustomed to calculating them, questions on profit and loss will not pose any problems to you.

- There is no substitute for memorising the formulae. Some smart alecks claim that remembering a single formula is enough as the others are derived from it. The fact is that you do not have time to sit about deriving formulae in the examination hall. Formulae are mugging up territory. Fortunately, there are not too many of them to remember.

## Points to keep in mind:

- Profit and loss per cent is ALWAYS computed by dividing the profit or loss made by the cost price. I know that sounds elementary and yet candidates quite often stumble into the error of basing the profit or loss on the selling price. It is only discount which is calculated on selling price.

- Sometimes questions mention more than one discount on the same article. In these cases, calculate the initial rate of discount on the original selling price and the subsequent rate of discount on the selling price minus the original discount. See the sample questions in this regard.

- Occasionally, questions make a reference to 'marked price'. Do not let this confuse you - it is in fact the selling price. Marked price quite literally means the price marked on the article and that is the selling price. Similarly, 'gain' is the same as profit.

- Do remember to distinguish between the quantity of profit or loss and profit or loss expressed in terms of percentage. It has been observed that people sometimes mix up the two concepts. For instance, if a question mentions that a person made a 10% loss on the sale of an article costing Rs 45, many times candidates simply subtract 10 from 45, forgetting that 10 is actually a percentage figure! This is an error that creeps in when one works in a hurry, so be careful.

- As percentage is an intrinsic part of questions on profit and loss, the points that need to be kept in mind while handling percentage exercises need to be kept in mind here as well. To view them, click here (link to 'Points to keep in mind' section of percentage)

## Sample questions with solutions:

1. A sold his watch for Rs 190, thus bearing a loss of 5%. The cost price of the article is

a. Rs 237.50 b. Rs. 220 c. Rs. 210 d. Rs. 200 e. None of these

Solution:

1. What you need to find here is the cost price. You are given the rate of loss and the selling price.

2. Now, cast your mind back to the formulae mentioned in the 'What you need to know' section. The formula for determining the cost price when a loss has been incurred is = Selling price x [100/ 100 - Loss per cent],

3. Inserting the values of loss and selling price in this formula, we get,

Cost price = Rs.190 x [100/100-5]

Cost price = Rs.190 x [100/95]

Cost price = Rs. 200

The cost of the article is Rs. 200 and the correct answer is d.

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2. A man sells two shirts for Rs. 990 each. On one he gains 10% and on the other he loses 10%. What is his total gain or loss?

a. 1% gain b. 1% loss c. No gain or loss d. 5% gain

Solution:

1. Now, do not go rushing in and assume that as the selling price is the same and the profit and loss rates are equal as well, the profit and loss would have cancelled each other out and the man makes neither a profit or a loss. Things are not as simple as that here.

2. Here we have been given two selling prices and the rates of profit [gain] and loss. What we need to find is whether the transaction yielded the person a gain or loss.

3. First we need to find the cost price of the two shirts. Using the formula for determining cost price, we get Cost of the shirt on which a profit was made = Selling price x [100/ 100 + Profit per cent], or 990x [100/110] = Rs.900 Cost of the shirt on which a loss was made = Selling price x (100/ 100 - Loss per cent), or 990 x [100/90] = Rs.1100

4. Therefore the total cost of producing the two shirts is Rs [1100 + 900] = Rs.2000

5. But the total selling price of the two shirts is = Rs. [990 + 990] = Rs 1980

6. As the cost price is greater than the selling price, a loss has been incurred. This loss is equal to Rs. [2000 -1980] = Rs. 20

7. To calculate the percentage of the loss, we use the formula [Loss / Cost price] x 100, or [20/2000] x 100 = 1%.

So the man made a loss of 1% and the correct answer is b.

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3. Peter bought 2000 bananas at Rs 9 per dozen. He sold some of them for Rs. 1 each and the remainder at Rs. 0.80 each. He made a profit of Rs. 150. How many bananas did he sell at Rs.1 each?

a. 550 b. 600 c. 50 d.250

Solution:

1. Here we have been given the cost price, the selling prices and the amount of profit. What we need to find is the quantity sold at a particular price.

2. The cost of bananas is Rs 9 per dozen. So the cost of a single banana would be Rs. [9/12] or Rs. 0.75.

3. So, the cost of 200 bananas would be = Rs. [2000 x 0.75] = Rs. 1500

4. Now, Peter being an astute businessman, made a profit of Rs 150. So the total selling price of the bananas must have been = Rs. [1500 + 150] = Rs. 1650

5. So, if the number of bananas sold for Rs 1 each is b, then the bananas sold at Rs. 0.80 would be 2000 - b, as there were a total of 2000 bananas,

6. Now, as per the conditions given in the question

[b x 1] + [2000 - b] x 0.80 = Rs. 1650

b+1600-0.80b = 1650

0.20 b= [1650-1600]

0.20b = 50

b= 50 /0.20 or 5000/20

b=250

Peter sold 250 bananas as Rs 1 each and the remaining 1750 at Rs.0.80 each. The correct answer is d.

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4. Two successive discounts, the first of 20% and the second of 25% are equivalent to a single discount of

a. 22 ½% b. 36% c.40% d.45%

[RBI exam, 1980]

Solution:

First of all, do not make the mistake of simply adding the two rates of discounts and arriving at 45%. This is because the second rate of discount was offered on the selling price, which had been reduced by the earlier discount.

Now, let us assume that the selling price of the article mentioned is Rs. 100. Why Rs. 100? Well, because it is easier to calculate percentages on an assumed price of Rs. 100 and also because using algebra would be really messy here.

So, if the selling price is Rs. 100, the price after the first discount would be = Selling price x [100 - Discount per cent] / 100 or, 100 x [100-20] /100 = Rs. 80

Rs. 80 would be price at which the second discount would have been given. So the selling price after the second discount would be = Selling price x [100 - Discount per cent] / 100 or 80 x (100-25)/100 = Rs. 60.

Now, the selling price after the two discounts is Rs. 60. So a single rate of discount equal to the two discounts would be = [Discount/ Selling price] x 100 or [100-60] /100 x 100 = 40%

A single discount of 40% would thus be equal to successive discounts of 20% and 25%. The correct answer is c.

## Send us a question

Is there a profit and loss -based question that you cannot solve? Or a question that you would like to share with other candidates? If so, then do mail them to us, along with details of the examination paper / sample paper in which you found them. We will try to solve them and send the solution to you. Any questions that you send will also be uploaded to the Sample Questions section of the web site and you will be given credit for contributing them.

## Feedback

Did this section help you in your preparation for competitive examinations? Do let us know if we are doing a decent job or an inept one. Mail us your opinions and suggestions at editor@enableall.org.

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Questions on profit and loss are always descriptive in nature. These inevitably revolve around the determination of:

## The cent per cent solution

**Introduction:**

Few mathematical concepts are as popular as per cent. Be it your marks in the examination, the discount at the local market or the growth of the Indian economy, it is a fair chance that the figures mentioned will be followed by the term 'per cent'. In fact, so ingrained is the concept of percentage in human society that even corrupt middlemen demand payments [or cuts] in terms of per cent!

Now, if per cent is so popular, you can scarcely expect it to steer clear of competitive examinations, can you? Sure enough, questions pertaining to percentage are an integral part of competitive examinations all over the world. If there is a question paper on quantitative aptitude or mathematics, you can be sure that an exercise on percentage will be lurking somewhere within it. What's more, per cent is at the core of many other concepts such as profit and loss, rate of growth, simple and compound interest - all of which play their role in competitive exams.

Fortunately, it does not take much to master percentage. If you know your fractions, elementary algebra, and basic mathematical operations [good old add, subtract, divide and multiply], you will be able to handle percentage problems in no time at all.

**Nature of exercises****What you need to solve them****Some strategies****Points to keep in mind****Sample questions with solutions****Send us a question****Feedback**

## Nature of exercises:

As I mentioned in the introduction, the concept of percentage is used in a variety of questions. However, for the sake of simplicity, we will here deal with questions that pertain directly to percentage. Fear not, we will be tackling profit and loss, interest and other related concepts too in the days that come. But now is the time for basics.

Questions on percentage could take the form of a simple equation or might be elaborately described.

Generally, questions on percentage are found in the following flavours:

- Calculating the percentage [yes, sometimes it is actually as simple as that]
- Measuring the change in percentage
- Determining the change in quantities involved
- Finding the original value of the quantities

## What you need to solve them:

The charm of percentage related questions is that most of them boil down to simply using the basic percentage formula.

But before we get cracking, let me just refresh your minds about per cent. The term per cent is derived from the Latin term 'per centum', which literally means 'per hundred' or 'out of hundred'. So any fraction with a denominator of a hundred is a percentage. Any fraction can be expressed in terms of percentage by multiplying it with 100. Thus ¼ can be expressed in per cent as ¼ x100 = 25%.

Similarly, you can convert a percentage figure into a fraction by simply putting 100 in the denominator and the per cent figure in the numerator and reducing it to its lowest terms. Thus 25% is 25/100 or ¼.

When you are dealing with an increase or decrease, the formula to use is: Per cent increase or decrease = [amount of increase or decrease/ earlier amount] x 100

A tricky use of percentage is when it is used to compare two quantities. For instance, if we say that Bob is earning 20% more than David, what we mean is that Bob's income exceeds David's by 20/100 x Bob's income. You can understand the concept better by looking at sample question 3.

Simple enough, isn't it? Believe it or not, that is all you need to solve questions on per cent.

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## Some strategies:

There are no secret methods to unravel the solutions of percentage exercises. One has to know one's mathematics and the formulae to use. It is as simple as that. Of course, there are ways in the process can be made a bit easier. In the case of descriptive questions, I used to make it a point to collect all the numerical information at one place before getting down to the operations. Collecting the information also help me make sure that I was not making any mistake with the figures and helped me understand the question more clearly.

Holmes. So I used to imagine that I was Sherlock Holmes, the information in the question were the clues and the answer was the solution to some terrific mystery. Call me childish To be honest, I used to be [and still am], a huge fan of the detective Sherlock but it seemed to work. It made me focus on the problem and if you can do that, you are on your way.

## Points to keep in mind:

When dealing with purely numerical questions involving different mathematical operations, remember to follow the BODMAS rule for the sequence of different mathematical operations. Quite simply, this is:

BO: Bracket Operations; do the mathematical operations within the brackets first of all.

D: Division; after the bracket operations, do the division.

M: Multiplication; once the division is done, you can proceed to multiplication

A: Addition; after multiplication, you can move on to addition.

S: Subtraction; when all else has been done, you can subtract.

If that sounds a bit complex, do try out this sample question given below:

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[Sample question for BODMAS rule:

What is 25x[10+2]/6+17-4?

Solution:

As per the BODMAS rule,

1. BO: Bracket Operations, so first we tackle the terms within the brackets which is [10+2]=12.

2. D: Division, we move on then to division which is 12/6=2

3. M: Multiplication, multiply 2 by 25=50

4. A: Addition, add 17, 50+17=67

5. S: top it all off by subtraction, 67-4=63>

All this may seem rather elementary but believe me, it is important. For instance, if you had carried out your addition before multiplication, you would have added 2 to 17 and got 19. Multiplying this with 25 would give you 475 and finally subtracting 4 would have made for a final answer of 471 - a solution that is nowhere near the correct answer of 63.

In mathematical operations, order is sacrosanct. ]

I have always found that breaking down a descriptive question into the form of a mathematical equation is the toughest part of a per cent exercise. The sooner you can do this, the quicker you will on your way to a solution. For instance, in the first sample question, we broke down the question to a simple equation of 33m /100 = 231. Solving the equations is a matter of basic mathematics but if you get the equation wrong, you can bid your marks farewell.

Always try to reduce fractions to their lowest level before doing any operations. Let us face it, it is easier to deal with 1/3 than with 25/75.

When percentages are given in decimal points, remember that each decimal point adds a further zero to the denominator. So, while 6% is 6/100, 6.5% is NOT 65/100 but 65/1000. Similarly, 6.52 will 652/10000.

Because in most questions pertaining to percentage can be broken down into an equation, some candidates try to fit in the alternatives given as answers into the equation and choose the correct fit as the answer. I would not recommend trying this as it takes up way too much time.

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## Sample questions with solutions

1. 33% marks are required to pass an examination. A candidate who gets 210 marks fails by 21 marks. The total marks in the examination are:

a. 500 b. 600 c.700 d.800

[RBI examination, 1985]

**Solution:**

1.The candidate got 210 marks and failed by 21 marks. This means that the marks needed to pass the exam are [210+21]=231.

2.As the 33% marks are needed to pass the exam, it follows that 33% of the total marks is 231.

3.Assuming the total marks to be m, we get

33/100 x m = 231

33m/100=231

m=231x 100/33

m=700

Therefore the total number of marks in the examination are 700 and the correct answer is c.

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## Send us a question

Is there a percentage-based question that you cannot solve? Or a question that you would like to share with other candidates? If so, then do mail them to us, along with details of the examination paper / sample paper in which you found them. We will try to solve them and send the solution to you. Any questions that you send will also be uploaded to the Sample Questions section of the web site and you will be given credit for contributing them.

## Feedback

Did this section help you in your preparation for competitive examinations? Do let us know if we are doing a decent job or an inept one. Mail us your opinions and suggestions at editor@enableall.org.

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Alternative approach:

We can also solve this question by resorting to algebra at an earlier stage. Here is how it is done:

1. Let the total number of marks be m.

2. Then in order to pass, a person has to obtain 33% or 33/100 x m marks, which amounts to 33m/100

3. As per the question, the candidate got 210 marks and was 21 marks short of passing. So it follows that

33m/100 - 21 = 210

33m/100 = 210+21

33m/100 = 231

m= 231x 100/33

m =700

As you can see, the solution is the same and even the equations are the same. It is only the start that is different.

Both approaches work. Pick the one you are most comfortable with.

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2. A's salary is 20% below B's salary. By how much per cent is B's salary above A's salary?

a. 16 2/3 b. 25 c. 20 d. 33 1/3

Solution:

1. First of all, resist the temptation to say that as A is earning 20% less than B, B must be earning 20% more than him. He does not.

2. Now, assume that B earns Rs. 100 [it is simpler than assuming than using algebra, trust me. Also notice that here we are concerned with the percentage and not the actual amount]

3. So, A's salary is 20% less than B, or in other words is 80% [100-20] of what B gets.

4. So if B's salary is 100, A's salary will be 80% of Rs.100, which will be Rs.80.

5. Then it follows that B's salary will [100-80]/80 more than A's salary.

6. [100-80] / 80 gives us 20/80 or ¼ or 25%.

7. The answer is 25% or b.

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3. A reduction of 20% in the price of oranges enables a man to buy 5 oranges more for Rs. 10. The price of an orange before reduction was

a. 20 paise b.40 paise c. 50 paise d. 60 paise

[Auditors', 1982]

Solution:

1. Do not head for the algebra yet. Just call up your reserves of common sense. Consider the facts:

There has been a 20% cut in the price of oranges.

The man spends Rs 10 on oranges.

2. Therefore after the price cut, the man needs to spend 20% less to get the same number of oranges that he used to get before the price cut. 20% of Rs. 10 is Rs. 2. The man saves Rs 2 due to the price cut,

3. These Rs.2 enable him to buy 5 additional oranges.

4. So the price of each orange right now will be 2/5 = Rs 0.40, or 40 paise.

5. Right, now let us go for the algebra. Let the price before the reduction be p.

6. Then p - 20%= 40 paise

7. p - 20/100 = 40 paise

p - 1/5=40 paise

4p/5 = 40 paise

p = [40x5/4 ] paise

p = 50 paise.

The price of oranges before the reduction in price was 50 paise per orange. The correct answer is c.

## Getting a sense of directions

**Introduction**

Somewhere in almost every competitive examination will come a question that will ask you to track movements and direction. You will be confronted by a character who has been given a licence to wander (the licence to kill had already been handed out to Mister. Bond!). He will go south, north, east and west, and turn left and right for no reason whatsoever. And at the end of it all, you will be asked to find his exact location and how far he has travelled.

Sounds familiar? Welcome to the world of direction-based exercises. Love them or hate them, you cannot afford to neglect them, as they are an integral part of most competitive examinations today. And believe us,they are not as difficult as they seem to be.

All one needs is some common sense allied at times with some very basic Mathematics

## Contents

- Nature of exercises
- What you need to solve them
- Some strategies
- Points to keep in mind
- Sample questions with solutions
- Send us a question
- Feedback

## Nature of exercises

As their very name indicates, direction-based questions revolve around the direction taken by a person or an object. We will term this person or object the 'subject' as he / she or it is at the centre of the problem.

Most direction-based exercises consist of a description of a route taken by the subject. As this is after all an exercise to test your ability, this route often involves three to four turns in different directions after travelling a certain distance. In most cases, you will be asked to find the following

- How far is the subject from the place where it started its journey
- In which direction is the subject from the place where it started its journey
- In which direction did the subject start its journey

## What you need to solve them

As we said in the introduction, all that is needed to solve these questions is common sense along with some basic mathematics. And the mathematics is not always necessary!

More specifically, you need the following to tackle direction-based questions:

**1. A sense of direction:**You need to have a clear idea of north, south, east and west. Let us face it the subject of the question does not seem to have any sense of direction. If you do not have it either, you stand little chance of solving the question.

**2. An ability to visualise:**One can always trace the movements of the subject by drawing a diagram showing its approximate movement. However, this can be time consuming and even inconvenient for some candidates. A solution is to develop an ability to visualise the movement of the subject. One is still drawing a diagram, only this time it is in one's own mind. It does take a lot of practice to get used to this but believe us it is well worth the effort.

**3. Some basic mathematics:**Here comes the dreaded mathematical angle! Well, it is actually rather simple geometry. The subject more often than not ends up moving in a certain geometrical shape - it could be a rectangle, a square, a parallelogram or a triangle. In these cases, it does help to keep the following in mind:

a. If the subject moves in a square shaped path: All the sides of a square are equal.

b. If the subject moves in a rectangular path: The opposite side of a rectangle are equal

c. If the subject moves in a path similar to a parallelogram: The opposite sides of a parallelogram are equal, just like a rectangle.

d. If the subject moves in a triangular path: As in most cases, the subject moves in a straight line, there is a fair chance that its path will take the form of a right triangle. A right triangle, incidentally, is a triangle in which one of the angles is 90 degrees in measure.

e. If the subject's path [or some part of it] is indeed in the form of a right triangle, you can use Pythagoras' Theorem to find out the measurements of one of its sides, provided the measurements of the other two sides are given [which they usually are]. If you have forgotten the Pythagoras Theorem, relax, because we are explaining it once again below.

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Quite simply, this theorem states that in a right triangle, the square of the hypotenuse [the side opposite the right angle and also the longest side of the triangle] will be equal to the sum of squares of the other two sides.

So, if in a right triangle ABC,

BC= the hypotenuse

And AB and AC are the other two sides,

BC2= AB2 + AC2

Or, BC = v[AB2 + AC2]

The theorem also enables us to find the measurements of the other sides of a triangle, if one is given the measurement of the hypotenuse and any other side. To take the example of our triangle ABC once again

If BC2= AB2 + AC2

Then, AB2 = BC2 - AC2

Or AB = v [BC2 - AC2]

Similarly, AC2= BC2-AB2

Or AC= v [BC2-AB2]

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## Some strategies

Your ability to visualise the movement of the subject will determine how successfully you solve direction-based questions. Mind you, visualising is not the easiest task at the best of times, especially if the subject moves in a wide range of directions. Now, unless you are born with a magnetic compass inside your head, tracking movements can be a bit of a pain. Hence the need for some strategies for visualisation. While every person has his or her own special way of visualising movements, here are a few strategies that can help in this regard:

1. The front-to-back strategy: Quite simply, the area in front of you is north, the area behind you is south, the area to the right is east and the area to the left is west. This strategy is the simplest and we will be using it to solve our sample questions.

2. The crosshair strategy: Consider a cross made by a vertical line and a horizontal line. Beginning with the top of the vertical line, move clockwise and label each end point of the lines as north, east, south and west. The place where the lines intersect is the place from which the subject starts travelling and the four segments show different directions.

3. The clock strategy: Quite simply, 12 o'clock is north, 3 o'clock is east, 6 o'clock is south and 9 'clock is west.

4. The face strategy: One can use one's own face to keep a rough track of directions. If your nose is the point of origin, your forehead is north, your right cheek is east, your chin is south and your left cheek is west.

5. Tackling left and right: Some exercises state that the subject moved to its right or to its left. One can tackle these by assuming that one is standing at the point where the turn is needed and taking a turn towards the side stated.

Remember, these strategies only give you a sense of direction. There is no real strategy for visualising how far the subject has moved. That one is up to you - the strategies can only point out the direction of the subject's movement.

Feel free to adopt any of them. Whichever it is, do practice it thoroughly as nothing aids effective visualisation more than practice. And if you do have a strategy of your own which you would like to share, mail it to us at Feedback@enablingdimensions.comWe will post it on the site, and give you due credit for it.

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## Points to keep in mind:

1. Sometimes questions contain references to directions such as north-east, south-east, north-west and south west. These directions refer to the regions that cannot be strictly labelled as east west, north or south but fall in between them. For instance, north-west would refer to a direction between north and west! Generally, these directions are meant to indicate the fact that the subject is not travelling in a straight line but is moving diagonally.

2. Unless otherwise specified, it must be remembered that the subject always begins its journey from the very centre.

3. Geographical directions like north, south, east and west are always taken with reference to the centre. However, a turn to the right or the left is made with reference to the subject's current position.

4. When asked how far the subject has to travel to return to the point from which its journey started, assume that the subject will be taking the shortest possible route [a straight line] from his current location.

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## Sample questions with solutions

[All these questions have been solved using the front-to-back strategy]

A person walks 6 kilometers to the north, then turns towards the east and walks 4 kilometers. He then turns to the south and walks a further 6 kilometers. How far will he have to travel to reach the place where he started his journey and in which direction?

a. 4 kilometers south b. 5 kilometers west c. 4 kilometers west d. 16 kilometers north e. None of the above

**Solution:**

1. The objective is to find the direction as well as distance.

2. The person walks 6 kilometers north, that is, in front.

3. He then turns towards east and walks 4 kilometers. The east is on the right hand side.

4. He then turns to the south and walks a further 6 kilometers. South is the area behind you, so the subject actually is walking back in the direction where he started out from [remember, he first went towards the north - in front of you].

5. Now that you have mapped the movement, take a good look at it. The subject goes 6 kilometers, turns right to travel a further 4 kilometers and then walks south for 6 kilometers.

6. The subject's path is taking the shape of an incomplete rectangle. Two opposite sides [the north path and the south path] are equal, both being 6 kilometers.

7. The distance between the subject's current location and his point of origin can be considered to be the fourth side of the rectangle. As the opposite sides of a rectangle are equal, this will be equal to the distance travelled east. Thus, the subject will have to travel the same distance he had travelled to the east to reach his point of origin. Only this time he will have to go in the opposite direction.

8. The subject will therefore have to travel 4 kilometers to the west to reach the place from where he started his journey. The answer is c.

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A person walks 20 meters to the north. He then turns to the right and walks another 30 meters. Once again, he turns to the right and walks 35 meters. He then turns left and walks 15 meters. Finally he turns left yet again and walks another 15 meters. In which direction is he from the starting point?

a. East b. West c. North d. South e. None of these

**Solution:**

1. The objective is to judge the direction and not the distance.

2. The subject walks 20 meters to the north, that is, in front of him.

3. He then turns to the right, that is, the east, and walks 30 meters.

4. He turns to the right again and walks 35 meters. This is the tricky part because he is moving to his right and not your right hand side. Turn yourself to the right and then turn again to the right. You will find that you are facing the area that was behind you when you started the turns. This means the subject has moved 35 meters to the south.

5. The subject then turns to the left and walks 15 meters. Remember the he is now facing south, so his left will be towards east [when he faced north, his right was towards the east!]

6. Finally, the subject who is now facing the east, turns left yet again and walks a further 15 meters. Turn yourself to the east [or your right] and you will find that your left hand is towards the north. So when the subject moves to his left, he heads 15 meters to the north.

7. Now take a final look at the subject's movements in geographical terms - remember we have to discover where he is in geographical terms from the place where he started. He has gone 20 meters to the north, then 30 meters to the east, followed by 35 meters to the south, 15 meters to the east and finally 15 meters to the north.

8. Visualise the directions and you will see that when the subject in his third move went south [35 meters], he actually went south of the place from where he started his journey. Remember that he had advanced only 20 meters to the north and then headed 30 meters to the east. So he actually went [35 minus 20] 15 meters south of the point from where he started out.

9. After heading south, the subject moved 15 meters to the east and then 15 meters to the north. By moving 15 meters to the north, he actually moved back into line with the place from where he had started the journey [remember, he had gone 15 meters south of it]. But in which direction? Well, he had gone 15 meters east before that. So he is actually to the east of the point from where he started out. The answer is A.

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Kevin drives his car 10 kilometers to the east. He then takes a right turn and travels 2 kilometers. Once again he turns to the right and goes a further 1 kilometer. In which direction is he from the starting point?

a. North b. East c. South-east d North-east e. None of the above

**Solution:**

1. Once again, the objective here is to find the direction rather than the distance.

2. To begin with, Kevin drives the car 10 kilometers to the east [that is, his right].

3. He then turns to the right and travels 2 kilometers. The right when he is facing east will be towards the south, so Kevin travels 2 kilometers to the south.

4. Finally he turns right again and drives for 1 kilometer. If he is facing the south, his right hand will be towards the west, so he drives 1 kilometer to the west.

5. Now take a final look at Kevin's movements in geographical terms - remember we have to discover where he is in geographical terms from the place where he started. He has gone 10 kilometers to the east, then gone 2 kilometers to the south and finally one kilometer to the west.

6. Visualise the directions and you will see that Kevin goes to the east, then to the south and then to the west. But he has travelled further east [10 kilometers] than he has in the west [1 kilometer]. So his direction is a mixture of south and east, that is, the south-east. The answer is c.

## Send us a question

Is there a direction-based reasoning question that you cannot solve? Or a question that you would like to share with other candidates? If so, then do mail them to us, along with details of the examination paper / sample paper in which you found them. We will try to solve them and send the solution to you. Any questions that you send will also be uploaded to the Sample Questions section of the web site and you will be given credit for contributing them.

## Feedback

Did this section help you in your preparation for competitive examinations? Do let us know if we are doing a decent job or an inept one. Mail us your opinions and suggestions at editor@enableall.org

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## Numerical Aptitude and Logical Reasoning

Reasoning is a process of thinking that is coherent and logical in nature.

Logical reasoning is the process of pure thinking, where we demonstrate that if certain statements are accepted as true, then other statements can be shown to follow from them. It involves observing data, recognizing patterns, and making generalizations from the observations.

Any competitive exam today has a section on logical reasoning having questions on directions, percentage and profit and loss, so we have compiled a guide to assist you in negotiating with each of the three.

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