Simple and Compound interest

Simple Interest

It is the sum which is paid by the borrower to the lender for using the money for a specific time period. The money borrowed is called the Principal. The rate at which the interest is calculated on the principal is called Rate of Interest. The time for which the money is borrowed is Time and the total sum of principal and interest is called the Amount.
Simple Interest
If P = Principal, R = Rate per cent per annum T = Number of years, SI = Simple Interest and A = Amount. Then,

Here, the interest is calculated on the original principal ie, the principal to calculate the interest remains constant throughout the time period. The interest earned on the principal is not taken into account for the purpose of calculating interest for later years.

Compound Interest

In compound interest, the interest is added to the principal at the end of each period and the amount thus obtained becomes the principal for the next period. The process is repeated till the end of the specified time.
If P = Principal,
R = Rate per cent per annum
Time = Number of years,
CI = Compound Interest
A = Amount. Then,
When the interest is compounded annually

Important Formulae

1. If the rate of interest differs from year to year ie, R1 in the first year, R2 in the second year, R3 in the third year.
Then
2. When the principal changes every year, we say that the interest is compounded annually. Then,

3. When the principal changes as per every six months, we say that the interest is compounded half yearly or semi-annually. Then,

4. When the principal changes every three months, we say that the interest is compounded quarterly. Then,

5. When the principal changes after every month, we say that the interest is compounded monthly. Then,

6. When the interest is compounded annually but time is in fraction say year.

7. The difference between the simple interest and compound interest for 2 year (or terms) is given by the formula

Where D is the difference, P is the principal and R is the rate of interest.
8. Present worth of x ` due n years, hence is given by

Probability

                                           Probability

Probability is used to indicate a possibility of an event to occur. It is often used synonymously with chance.

• In any experiment if the result of an experiment is unique or certain, then the experiment is said to be deterministic in nature.
• If the result of the experiment is not unique and can be one of the several possible outcomes then the experiment is said to be probabilistic in nature.

Various Terms Used in Defining Probability

(i) Random Experiment: Whenever an experiment is conducted any number of times under identical conditions and if the result is not certain and is any one of the several possible outcomes, the experiment is called a trial or a random experiment, the outcomes are known as events.
eg, When a die is thrown is a trial, getting a number 1 or 2 or 3 or 4 or 5 or 6 is an event.
(ii) Equally Likely Events: Events are said to be equally likely when there is no reason to expect any one of them rather than any one of the others.
eg, When a die is thrown any number 1 or 2 or 3 or 4 or 5 or 6 may occur. In this trial, the six events are equally likely.
(iii) Exhaustive Events: All the possible events in any trial are known as exhaustive events. eg, When a die is thrown, there are six exhaustive events.
(iv) Mutually Exclusive Events: If the occurrence of any one of the events in a trial prevents the occurrence of any one of the others, then the events are said to be mutually exclusive events. eg, When a die is thrown the event of getting faces numbered 1 to 6 are mutually exclusive.

Classical Definition of Probability

If in a random experiment, there are n mutually exclusive and equally likely elementary events in which n elementary events are favourable to a particular event E, then the probability of the event E is defined as P (E)

• If the probability of occurrence of an event E is P(E) and the probability of non-occurrence is P, then,
  the sum of the probabilities of success and failure is 1. Also, 0 ≤ P(E) ≤ 1 and 0 ≤ P ≤ 1.
• If P(E) = 1, the event E is called a certain event and if P(E) = 0, the event E is called an impossible event.
• If E is an event, then the odds in favour of E are defined as P(E) : P(E) and the odds against E are defined
as P(E): P. Hence, the odds in favour of E are the odds against E are

Addition Theorem on Probability

Independent and Dependent Events

• Simple Event : An event which cannot be further split is called a simple event. The set of all simple events in a trial is called a sample space.
• Compound Event : When two or more events occur in relation with each other, they are called compound events.
• Conditional Event: If El and E2 are events of a sample space S and if E2 occurs after the occurrence of El, then the event of occurrence of E2 after the event El is called conditional event of E2 given El. It is denoted by E2/El.

‘Smart’ Facts

• When a die is rolled six events occur. They are {1, 2, 3, 4, 5 and 6}
• When two dice are rolled 36 events occur. They are [(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)]
• When a coin is tossed 2 events occur. They are {H, T}
• When two coins are tossed 4 events occur. They are {HH, HT, TH, T T}
• When three coins are tossed 8 events occur. They are {HHH HHT, HTH, HT T, T HH, THT, T TH, T T T}
• In a pack of 52 cards there are 26 red cards and 26 black cards. The 26 red cards are divided into 13 heart cards and 13 diamond cards. The 26 black cards are divided into 13 club cards and 13 spade card. Each of the colours, hearts, diamonds, clubs and spades is called a suit. In a suit, we have 13 cards (ie, A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3 and 2)

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Alligation or Mixture

                                                      Aligation  or Mixture
                                                          

 1. In what ratio must a grocer mix two varieties of tea worth Rs. 60 a kg and Rs. 65 a kg so that by selling the mixture at Rs. 68.20 a kg he may gain 10%?
1. 3 : 2
2. 6 : 7
3. 3 : 5
4. 4 : 5
5. 4 : 3

2. How many kilograms of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing Rs. 7 per kg so that there may be a gain of 10% by selling the mixture at Rs. 9.24 per kg?
1. 38 kg
2. 43 kg
3. 54 kg
4. 63 kg
5. 48 kg

3. Two vessels A and B contain spirit and water mixed in the ratio 5 : 2 and 7 : 6 respectively. Find the ratio in which these mixture be mixed to obtain a new mixture in vessel C containing spirit and water in the ratio 8 : 5?
1. 5 : 3
2. 9 : 4
3. 5 : 6
4. 7 : 9
5. 3 : 7

4. A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3 : 5?
1. 9 litres, 8 litres
2. 6 litres, 6 litres
3. 6 litres, 7 litres
4. 7 litres, 5 litres
5. 4 litres, 5 litres

5. One quality of wheat at Rs. 9.30 per kg is mixed with another quality at a certain rate in the ratio 8 : 7. If the mixture so formed be worth Rs. 10 per kg, what is the rate per kg of the second quality of wheat?
1. Rs. 12
2. Rs. 10.60
3. Rs. 10.80
4. Rs. 15
5. Rs. 13

6. Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:
1. Rs. 187
2. Rs. 172
3. Rs. 175.50
4. Rs. 180
5. Rs. 185

7. A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is:
1. 460 kg
2. 660 kg
3. 600 kg
4. 640 kg
5. 450 kg

Time and Distance(MCQ)

       
1. A boy goes to his school from his house at a speed of 3 km/hr and returns at a speed of 2 km/hr. If he takes 5 hours in going and coming, the distance between his house and school is:
1. 8.5 km
2. 5.5 km
3. 6 km
4. 9 km
5. 7 km

2. The average speed of a train in the onward journey is 25% more than that in the return journey. The train halts for one hour on reaching the destination. The total time taken for the complete to and for journey is 17 hours, covering a distance of 800 km. The speed of the train in the onward journey is:
1. 50 km/hr
2. 53 km/hr
3. 52 km/hr
4. 56.25 km/hr
5. 46 km/hr

3. A man on tour travels first 160 km at 64 km/hr and the next 160 kin at 80 km/hr. The average speed for the first 320 km of the tour is:
1. 35.55 km/hr
2. 38 km/hr
3. 71.11 km/hr
4. 75 km/hr
5. 72 km/hr

4. A boy rides his bicycle 10 km at an average speed of 12 km/hr and again travels 12 km at an average speed of 10 km/hr. His average speed for the entire trip is approximately :
1. 10.4 km/hr
2. 10.8 km/hr
3. 12 km/hr
4. 14 km/hr
5. 13 km/hr

5. A car travels the first one-third of a certain distance with a speed of 10 km/hr, the next one-third distance with a speed of 20 km/hr, and the last one-third distance with a speed. of 60 km/hr. The average speed of the car for the whole journey is:
1. 18 km/hr
2. 34 km/hr
3. 35 km/hr
4. 39 km/hr
5. 40 km/hr

6. A motorist covers a distance of 39 km in 45 minutes by moving at a speed of x kmph for the first 15 minutes, then moving at double the speed for the next 20 minutes and then again moving at his original speed for the rest of the journey Then, x is equal to:
1. 31.2
2. 36
3. 42
4. 54
5. 50

7. Mary jogs 9 km at a speed of 6 km per hour. At what speed would she need to jog during the next 1.5 hours to have an average of 9 km per hour for the entire jogging session?
1. 9 kmph
2. 13 kmph
3. 12 kmph
4. 15 kmph
5. 11 kmph

8. A car travelling with 5/7 of its actual speed covers 42 km in 1 hr 40 min 48 sec. Find the actual speed of the car.
1. 17 km/hr
2. 32 km/hr
3. 31 km/hr
4. 35 km/hr
5. 45 km/hr

9. A man can reach a certain place in 30 hours. If he reduces his speed by 1/15th, he goes 10 km less in that time. Find his speed.
1. 7 km/hr
2. 5 km/hr
3. 52 km/hr
4. 8 km/hr
5. 35 km/hr

Time and Distance

Time and Distance

Relation between Time, Speed and Distance
Distance covered, time and speed are related by
Time = ...(i)
Speed = ...(ii)
Distance = Speed × Time ...(iii)

• Distance is measured in metres, kilometres and miles.
• Time in hours, minutes and seconds.
• Speed in km/h, miles/h and m/s.

1. To convert speed of an object from km/h to m/s multiply the speed by . 5/18
2. To convert speed of an object from m/s to km/h, multiply the speed by . 18/5
Average Speed
It is the ratio of total distance covered to total time of journey.
Average Speed =
General Rules for Solving Time & Distance Problems


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Time and W ork (MCQ)

1. A can do a piece of work in 10 days; B in 15 days. They work for 5 days. The rest of the work was finished by C in 2 days. If they get Rs. 1500 for the whole work, the daily wages of B and C are:
1. Rs. 160
2. Rs. 225
3. Rs. 255
4. Rs. 350
5. Rs. 275

2. A and B together can complete a work in 12 days. A alone can complete it in 20 days. If B does the work only for half a day daily, then in how many days A and B together will complete the work?
1. 13 days
2. 16 days
3. 15 days
4. 22 days
5. 18 days

3. 12 men can complete a piece of work in 4 days, while 15 women can complete the same work in 4 days. 6 men start working on the job and after working for 2 days, all of them stopped working. How many women should be put on the job to complete the remaining work, if it is to be completed in 3 days?
1. 15
2. 19
3. 23
4. data inadequate
5. None of these

4. Twelve children take sixteen days to complete a work which can be completed by eight adults in twelve days. Sixteen adults started working and after three days ten adults left and four children joined them. How many days will they take to complete the remaining work?
1. 3 days
2. 5 days
3. 6 days
4. 9 days
5. None of these

5. 10 women can complete a work in  7 days and 10 children take 14 days to complete the work. How many days will 5 women and 10 children take to complete the work?
1. 5 days
2. 8 days
3. 7 days
4. cannot be determined
5. None of these

6. Sixteen men can complete a work in twelve days. Twenty-four children can complete the same work in eighteen days. Twelve men and eight children started working and after eight days three more children joined them. How many days will they now take to complete the remaining work?
1. 5 days
2. 4 days
3. 10 days
4. 9 days
5. None of these

7. Twenty-four men can complete a work in sixteen days. Thirty-two women can complete the same work in twenty-four days. Sixteen men and sixteen women started working and worked for twelve days. How many more men are to be added to complete the remaining work in 2 days?
1. 18 men
2. 24 men
3. 36 men
4. 50 men
5. None of these

8. 5 men and 2 boys working together can do four times as much work as a man and a boy. Working capacities of a woman and a boy are in the ratio:
1. 1 : 2
2. 2 : 1
3. 1 : 3
4. 6 : 7
5. 1 : 4

9. If 12 men and 16 boys can do a piece of work in 5 days; 13 men and 24 boys can do it in 4 days, then the ratio of the daily work done by a man to that of a boy is:
1. 2 : 1
2. 3 : 1
3. 3 : 2
4. 6 : 7
5. 1 : 4

Time and Work

Work
Work to be done is generally considered as one unit. It may be digging a trench constructing or painting a wall, filling up or emptying a tank, reservoir or a cistern.
General rules to be followed in the problems on Time and Work
1. If Acan do a piece of work in n days, then work done by Ain 1 day is 1/n. ie,if a person can do some work in 12 days, he does 1/ 12th of the work in one day.
2. If A s 1 day’s work = 1/n, then A can finish the whole work in n days. ie, if a person’s one day work is 1/10, then he can finish the whole work in 10 days.
3. If A is thrice as good a workman as B, then ratio of work done by A and B = 3 : 1. ie, if a man works three times as fast as a woman does, then when the work is complete, 3 parts of the work has been done by the man and 1 part by the woman.
4. If A is thrice as good a workman as B, then ratio of time taken by A and B = 1 : 3. ie, if the woman takes 15 days to complete the work, then the man takes 5 days to complete the same work.
5. If two persons A and B can individually do some work in a and b days respectively, then A and B together can complete the same work in ab (a + b) days.
6. The fundamental rules on variation also apply in Time and Work.
(i) Work and men are directly proportional to each other ie, if the work increases, the no. of men required to do it, also increases, if the work is to be completed in the same number of days.
(ii) Men and days are inversely proportional, ie, if the number of men increases, the number of days required to complete the same work decreases and vice versa.
(iii) Work and days are directly proportional, ie, if the work increases, the number of days required also increases, if the work is to be completed with the same number of men and vice versa.

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Unitary method(MCQ)

1. Cost of 24 pens is Rs. 96. Find the cost of 16 such pens.
(1) Rs. 66
(2) Rs. 64
(3) Rs. 62
(4) Rs. 68
(5) None of these

2. Cost of 8 dozen bananas is Rs. 180. How many bananas can be purchased for Rs. 30?
(1) 16 bananas
(2) 24 bananas
(3) 14 bananas
(4) 22 bananas
(5) None of these

3. A man pays Rs. 13000 as rent for 4 months. How much does he pay for the whole year, if rent per month remains the same?
(1) Rs. 48000
(2) Rs. 39000
(3) Rs. 46000
(4) Rs. 47000
(5) None of these

4. 23 envelopes cost Rs. 57.50. How many envelopes can be bought for Rs. 127.50.
(1) 55
(2) 56
(3) 51
(4) 53
(5) None of these

5. 20 men can reap a field in 20 days. When should 5 men leave the work, if the whole field is to be reaped in 24 days after they leave the work?
(1) 2 days
(2) 4 days
(3) 3 days
(4) 5 days
(5) None of these

Unitary method

Unitary Method

Direct Proportion

Two quantities are said to be directly proportional, if on the increase in one the other increases proportionally or on the decrease in one the other decreases proportionally.
eg, More the numbers of articles, more is the cost.
More the number of workers, more is the work done.
Less the number of articles, less is the cost.
Less the number of workers, less is the work done.

Indirect Proportion

Two quantities are said to be indirectly proportional, if on the increase in one the other decreases proportionally or on the decrease in one the other increases proportionally.
eg, More the number of workers, less is the number of days required to finish a work. More the speed, less is the time taken to cover a certain distance.
Less the number of workers, more is the number of days required to finish a work. Less the speed, more is the time taken to cover a certain distance.

Chain Rule

When a series of variables are connected with one another, that we know how much of the first kind is equivalent to a given quantity of second, how much of the second is equivalent to a given quantity of the third and so on. The rule by which we can find how much of the last kind is equivalent to a given quantity of the first kind is called the Chain Rule

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order of magnitude(MCQ)

1. Ramesh gave milk to have to his three sons Harish, Shayam and Ajay in three pots of the shape hemisphere, cube and cuboid. If radius of hemisphere pot is 5 cm, side of cubic pot is 5 cm and sides of cuboid pot are 5 cm × 5 cm × 6 cm, then who will get more milk?

(1) Harish
(2) Shayam
(3) Ajay
(4) Equal to all
(5) None of these

2. The velocity of sound in first medium is 320 m/s and in second medium is 1152 km/h. In which medium velocity of sound is maximum?
(1) First
(2) Second
(3) Equal in both
(4) Can’t be determined
(5) None of these

3. A minister of a village gave land to his four sons A, B, C and D. He gave them 5 hectare, 12 acre, 1600 sq m and 20 sq hectometre land respectively. Who got the maximum land?
(1) A
(2) B
(3) C
(4) D
(5) None of these

4. Aakash, Amar and Sawan has the lengths 20 decimetre, 1.7 m and 180 cm respectively. Who is the shortest?
(1) Aakash
(2) Amar
(3) Sawan
(4) All are of equal length
(5) None of these

5. Rotation period of four planets Mercury, Venus, Saturn and Earth are respectively 3784320 thousands, 4572720 thousand min, 166 yr and 365 days, 5 h, 56 min, 4 s. Which planet has the least rotation period?

(1) Mercury
(2) Venus
(3) Saturn
(4) Earth
(5) None of these

Oder of magnitude

Orders of Magnitude

Example 1: Ajay, Akshay and Saroj cover a distance of , 33500 m and 290 hactometre respectively in an hour. Who has the maximum speed?
Solution. Distance covered by Ajay = = 67 × 1000 m = 33500 m
Distance covered by Akshay = 33500 m
Distance covered by Saroj = 290 hactometre
= 290 × 100 m = 29000 m
Since, distance covered by Ajay and Akshay are maximum and equal. Hence, Ajay and Akshay have maximum speed.

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Partnership

When two or more than two persons run a business jointly, they are called partners in the business and the deal between them is known as partnership.

Partnership is of two types

1. Simple Partnership
2. Compound Partnership

1. Simple Partnership: When investments of all the partners are for the same period of time, the profit or loss is distributed among the partners in the ratio of their original investments.
Suppose A and B invest ` p and ` q respectively for a year in a business, then at the end of the year. Share of A’s profit (loss) : Share of B’s profit (loss) = p : q.
2. Compound Partnership: When investments of all the partners are for different period of time, then equivalent capitals are calculated for a unit of time and the profit or loss is divided in the ratio of the product of time and investment.
Suppose A and B invest ` p and ` q for x months and y months respectively, then Share of A’s profit (loss): Share of B’s profit (loss) = px : qy.

Partners are of two types

• (i) Working Partner, and
• (ii) Sleeping Partner

(i) Working Partner: A partner who manages the business is called a working partner.
(ii) Sleeping Partner: A partner who only invests the money is called a sleeping partner.

Suppose A and B invest ` p and ` q respectively for a year in a business, then at the end of the year. Share of A’s profit (loss) : Share of B’s profit (loss) = p : q.
2. Compound Partnership: When investments of all the partners are for different period of time, then equivalent capitals are calculated for a unit of time and the profit or loss is divided in the ratio of the product of time and investment.
Suppose A and B invest ` p and ` q for x months and y months respectively, then Share of A’s profit (loss): Share of B’s profit (loss) = px : qy.

Partners are of two types

• (i) Working Partner, and
• (ii) Sleeping Partner

(i) Working Partner: A partner who manages the business is called a working partner.
(ii) Sleeping Partner: A partner who only invests the money is called a sleeping partner.

Partnership MCQ

1. A and B entered into a partnership with investments of Rs. 15000 and Rs. 40000 respectively. After 3 months A left from the business. At the same time C joins with Rs. 30000. At the end of 9 months they got Rs. 7800 as profit. Find the share of B.
(1) Rs. 4800
(2) Rs. 600
(3) Rs. 2400
(4) Rs. 1200
(5) None of these

2. A started a business with Rs. 18000. After 4 months B joins with Rs. 24000. After 2 more months C joins with Rs. 30000. At the end of 10 months C received Rs. 1850 as his share. Find the total profit.
(1) Rs. 7955
(2) Rs. 7030
(3) Rs. 8510
(4) Rs. 6845
(5) None of these

3. Three partners started a business with Rs. 80000. At the end of the year they receive Rs. 1800, Rs. 3000 and Rs. 4800 as profit. Find the investment of the second person.
(1) Rs. 25000
(2) Rs. 40000
(3) Rs. 15000
(4) Rs. 32000
(5) None of these

4. A and B together invested Rs. 12000 in a business. At the end of the year, out of a total profit Rs. 1800 A’s share was Rs. 750. What was the investment of A?
(1) Rs. 5000
(2) Rs. 10000
(3) Rs. 12000
(4) Rs. 15000
(5) None of these

5. A started a business with a capital of Rs. 10000 and 4 months later, B joined him with a capital of Rs. 5000. What is the share of A in the total profit of Rs. 2000 at the end of the year?
(1) Rs. 800
(2) Rs. 1000
(3) Rs. 1500
(4) Rs. 1800
(5) None of these

6. A, B and C enter into a partnership. A contributes 320 for 4 months, B contributes Rs. 510 for 3 months, and C contributes Rs. 270 for 5 months. If the total profit is Rs. 208, find the profit share of the partner A.
(1) Rs. 76.50
(2) Rs. 64
(3) Rs. 67.50
(4) Rs. 46
(5) None of these

Profit/loss(MCQ)

1. A trader mixes three varieties of groundnuts costing Rs. 50, Rs. 20 and Rs. 30 per kg in the ratio 2 : 4 : 3 in terms of weight, and sells the mixture at Rs. 33 per kg. What percentage of profit does he make?

1. 8%
2. 3%
3. 11%.
4. 12%
5. 10%

2. A dairyman pays Rs. 6.40 per litre of milk. He adds water and sells the mixture at Rs. 8 per litre, thereby making 37.5% profit. The proportion of water to milk received by the customers is:

1. 1 : 10
2. 1 : 12
3. 2 : 7
4. 3 : 20
5. 2 : 5

3. A fruitseller has 24 kg of apples. He sells a part of these at a gain of 20% and the balance at a loss of 5%. If on the whole he earns a profit of 10%, the amount of apples sold at a loss is:

1. 9.8 kg
2. 8 kg
3. 9.6 kg
4. 12.4 kg
5. 9.7 kg

4. Two-third of a consignment was sold at a profit of 5% and the remainder at a loss of 2%. If the total profit was Rs. 400, the value of the consignment (in Rs.) was
1. 13,000
2. 17,000
3. 15,000
4. 40,000
5. 16,000

5. A trader purchases a watch and a wall clock for Rs. 390. He sells them making a profit of 10% on the watch and 15% on the wall clock. He earns a profit of Rs. 51.50. The difference between the original prices of the wall clock and the watch is equal to:

1. Rs. 111
2. Rs. 150
3. Rs. 110
4. Rs. 130
5. Rs. 115

6. Albert buys 4 horses and 9 cows for Rs.13,400. If he sells the horses at 10% profit and the cows at 20% profit, then he earns a total profit of Rs.1880. The cost of a horse is:
1. Rs. 2200
2. Rs. 2000
3. Rs. 2700
4. Rs. 3200
5. Rs. 2100

7. A man purchases two clocks A and B at a total cost of Rs. 650. He sells A with 20% profit and B at a loss of 25% and gets the same selling price for both the clocks. What are the purchasing prices of A and B respectively?

1. Rs. 550, Rs. 660
2. Rs. 250, Rs. 400
3. Rs. 378, Rs. 375
4. Rs. 300, Rs. 350
5. Rs. 350, Rs. 400

8. The C.P. of two watches taken together is Rs. 840. If by selling one at a profit of 16% and the other at a loss of 12%, there is no loss or gain in the whole transaction, then the C.P of the two watches are respectively:

1. Rs. 360, Rs. 480
2. Rs. 500, Rs. 360
3. Rs. 360, Rs. 460
4. Rs. 400, Rs. 440
5. Rs. 350, Rs. 450

9. On selling a chair at 7% loss and a table at 17% gain, a man gains Rs.296. If he sells the chair at 7% gain and the table at 12% gain, then he gains Rs.400. The actual price of the table is:
1. Rs. 2100
2. Rs. 1900
3. Rs. 2200
4. Rs. 2400
5. Rs. 2300

Profit/loss

Cost Price
The price at which an article is purchased is called the cost price or CP.
Selling Price
The price at which an article is sold is called the selling price or SP.
Formulae
Gain or Profit = SP – CP
Loss = CP – SP
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percentage (MCQ)

1. In a competitive examination in State A, 6% candidates got selected from the total appeared candidates. State B had an equal number of candidates appeared and 7% candidates got selected with 80 more candidates got selected than A. What was the number of candidates appeared from each State?
1. 7600
2. 8000
3. 8300
4. 4000
5. 5000

2. The price of a car is Rs. 3,25,000. It was insured to 85% of its price. The car was damaged completely in an accident and the insurance company paid 90% of the insurance. What was the difference between the price of the car and the amount received?
1. Rs. 60,000
2. Rs. 49,000
3. Rs. 76,375
4. Rs. 82,250
5. Rs. 75,000

3. The sum of the number of boys and girls in a school is 150. If the number of boys is x, then the number of girls becomes x% of the total number of students. The number of boys is:
1. 51
2. 65
3. 60
4. 95
5. 70

4. In an examination of n questions, a student replied 15 out of the first 20 questions correctly. Of the remaining questions, he answered one-third correctly. All the questions have the same credit. If the student gets 50% marks, the value of n is:
1. 30
2. 67
3. 50
4. 82
5. 60

5. The salaries of A and B together amount to Rs. 2000. A spends 95% of his salary and B, 85% of his. If now, their savings are the same, what is A’s salary?
1. Rs. 850
2. Rs. 1350
3. Rs. 1500
4. Rs. 1800
5. Rs. 1550

6. A’s marks in Biology are 20 less than 25% of the total marks obtained by him in Biology, Maths and Drawing. If his marks in Drawing be 50, what are his marks in Maths?
1. 60
2. 47
3. 63
4. 55
5. cannot be determined

7. In an examination, there are three papers and a candidate has to get 35% of the total to pass. In one paper, he gets 62 out of 150 and in the second 35 out of 150. How much must he get, out of 180, in the third paper to just qualify for a pass?
1. 50.5
2. 88
3. 70
4. 71
5. 75

8. In a History examination, the average for the entire class was 80 marks. If 10% of the students scored 95 marks and 20% scored 90 marks, what was the average marks of the remaining students of the class?
1. 60
2. 72
3. 75
4. 85
5. 77

9. A scored 30% marks and failed by 15 marks. B scored 40% marks and obtained 35 marks more than those required to pass. The pass percentage is:
1. 33%
2. 40%
3. 34%
4. 48%
5. 35%

Percentage

‘Per cent’ means ‘per hundred’. It is given by % symbol. Here x% means x per hundred . Thus, any percentage can be converted into an equivalent fraction by dividing it by 100.
                    
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Problem on Ages (MCQ )

1. A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of the son is:

1. 24 years
2. 26 years
3. 28 years
4. 22 years
5. 27 years

2. Eighteen years ago, a father was three times as old as his son. Now the father is only twice as old as his son. Then the sum of the present ages of the son and the father is:

1. 102
2. 76
3. 105
4. 108
5. 103

3. A person’s present age is two-fifth of the age of his mother. After 8 years, he will be one-half of the age of his mother. How old is the mother at present?

1. 44 years
2. 42 years
3. 40 years
4. 55 years
5. 44 years

4. Tanya’s grandfather was 8 times older to her 16 years ago. He would be 3 times of her age 8 years from now. Eight years ago, what was the ratio of Tanya’s age to that of her grandfather?

1. 6 : 1
2. 1 : 5
3. 7 : 3
4. None of these
5. 11 : 53

5. The age of father 10 years ago was thrice the age of his son. Ten years hence, father’s age will be twice that of his son. The ratio of their present ages is:

1. 8 : 3
2. 7 : 3
3. 11 : 2
4. 13 : 4
5. 7 : 9

6. Four years ago, the father’s age was three times the age of his son. The total of the ages of the father and the son after four years, will be 64 years. What is the father’s age at present?

1. 35 years
2. 36 years
3. 46 years
4. Data inadequate
5. None of these

7. One year ago, Promila was four times as old as her daughter Sakshi. Six years hence, Promila’s age will exceed her daughter’s age by 9 years. The ratio of the present ages of Promila and her daughter is:

1. 8 : 3
2. 11 : 5
3. 12 : 5
4. 11 : 2
5. 13 : 4

8. The sum of the present ages of a father and his son is 60 years. Six years ago, father’s age was five times the age of the son. After 6 years, son’s age will be:

1. 15 years
2. 16 years
3. 19 years
4. 20 years
5. 22 years

9. The total age of A and B is 12 years more than the total age of B and C. C is how many years younger than A?

1. 12
2. 26
3. C is elder than A
4. 13
5. None of these

Ratio & Proportion

                                                                  Ratio

The ratio of two quantities a and b is the fraction a/ b and is expressed as a : b. Here a is the first term or antecedent and b is the second term or consequent. Since the ratio expresses the number of times one quantity contains the other, it is an abstract (without units) quantity.

A ratio remains unaltered if its numerator and denominator are multiplied or divided by the same number. eg, 4 : 3 is the same as (4 × 10) : (3 × 10) ie, 40 : 30.

“A ratio is said to be a ratio of greater or less inequality or of equality according as antecedent is
greater than, less than or equal to consequent”.
• If a > b, then a : b is called a ratio of greater inequality (eg, 4 : 3, 5 : 2, 11 : 3, ...)
• If a < b, then a : b is called a ratio of less inequality (eg, 3 : 4, 2 : 5, 3 : 11, ...)
• If a = b, then a : b is called a ratio of equality (eg, 1 : 1, 3 : 3, 5 : 5, ...)

From this we find that

    If a > b and some positive number is added to each term of a : b, then the ratio is diminished. If a > b, then (a + x) : (b + x) < a: b.
    If a < b and some positive number is added to each term of a : b, then the ratio is increased. If a < b, then (a + x) : (b + x) < a : b.
    If a = b and some positive number is added to each term of a : b, then the ratio is unaltered. If a = b, then (a + x) : (b + x) = a : b The price at which an article is sold is called the selling price or SP.


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Ratio & Proportion (MCQ )

1. A sum of money is divided among 160 males and some females in the ratio 16 : 21. Individually each male gets  Rs. 4 and each female Rs. 3. The number of females is:

1. 280
2. 298
3. 292
4. 293
5. 300

2. In a factory men, women and children were employed in the ratio 8 : 5 : 1 to finish a job-and their individual wages were in the ratio 5 : 2 : 3. Total daily wages of all amount to Rs. 318. Find the total daily wages paid to each category.

1. Rs. 240, 60, 18
2. Rs. 210, 70, 38
3. Rs. 190, 95, 33
4. Rs. 240,45,15
5. None of these

3. Ram, Shiv and Ganesh assemble for a contributory party. Ram brings 3 apples while Shiv brings 5. Since Ganesh did not have any apples, he contributed Rs. 8. How many rupees should Ram and Shiv respectively get, assuming each of the three consumes an equal portion of the apples?

1. 1, 7
2. 2, 5
3. 5, 3
4. 2, 6
5. 2,7

4. A factory employs skilled workers, unskilled workers and clerks in the proportion 8 : 5 : 1 and the wages of a skilled worker as unskilled worker and a clerk are in the ratio 5 : 2 : 3. When 20 unskilled workers are employed, the total daily wages of all, amount to Rs. 318. Find the daily wages paid to each category of employees (Rs.):

1. 240,57,19
2. 210,70,13
3. 230,65,12
4. 210,70,15
5. 240,60,18

5. The price of a diamond is proportional to the square of its weight. The diamond accidentally fell and broke into four pieces whose weights are in the ratio of 1 : 2 : 3 : 4. If the price fetched is Rs. 70,000 less than the original price, find the original price?

1. Rs. 100,000
2. Rs. 70,000
3. Rs. 160,000
4. Rs. 10800
5. Rs. 150,000

6. The speeds of three cars are in the ratio 2 : 3 : 4. The ratio between the times taken by these cars to travel the same distance is

1. 1 : 2 : 3
2. 2 : 3 : 4
3. 8 : 5 : 4
4. 4 : 3 : 6
5. 6 : 4 : 3

7. If a, b, c and d are proportional then the mean proportion between a2 + c2 and  b2 + d2 is

1. ad/dc
2. ab + cd
3. a/b + d/c
4. b2/a2 + d2/c2
5. a/b + d2/c2

8. A cat takes 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog?

1. 13 : 14
2. 15 : 11
3. 17 : 15
4. 15 : 16
5. 15 : 17

9. A bag contains one rupee, 50-paise and 25-paise coins in the ratio 2 : 3 5. Their total value is Rs. 114. The value of 50 paise coins is:

1. Rs. 28
2. Rs. 36
3. Rs. 49
4. Rs. 72
5. Rs. 50

Average(MCQ)

1. A batsman has a certain average of runs for 16 innings. In the 17th innings, he makes a score of 85 runs, thereby increasing his average by 3. What is the average after the 17th innings?
1. 60
2. 37
3. 38
4. 27
5. 68

2. In Aran’s opinion his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Arun and he thinks that Aran’s weight is greater than 60 kg but less than 70 kg. His mother’s view is that his weight cannot be greater than 68 kg. If all of them are correct in their estimation, what is the average of different probable weights of Arun?
1. 71 kg
2. 67 kg
3. 73 kg
4. 58 kg
5. 52 kg

3. The ratio between the present ages of M and N is 5 : 3 respectively. The ratio between M’s age 4 years ago and N’s age after 4 years is 1 : 1. What is the ratio between M’s age after 4 years and N’s age 4 years ago?
1. 6 : 1
2. 3 : 1
3. 4 : 1
4. 4 : 5
5. 2 : 3

4. In 1919, W. Rhodes, the Yorkshire cricketer, scored 891 runs for his country at an average of 34.27; in 1920, he scored 949 runs at an average of 28.75; in 1921, 1329 runs at an average of 42.87 and in 1922, 1101 runs at an average of 36.70. What was his batting average for the four years?
1. 37.25
2. 36.92
3. 35.58
4. 28.72
5. 25.67

5. In hotel Jaysarmin, the rooms are numbered from 101 to 130 on-the first floor, 221 to 260 on the second floor and 306 to 345 on the third floor. In the month of June 2002, the room occupancy was 60% on the first floor, 40% on the second floor and 75% on the third floor. If it is also known that the room charges are Rs. 200, Rs. 100 and Rs. 150 on each of the floors, then find the average income per room for the month of June 2002:
1. Rs. 150.3
2. Rs. 88.18
3. Rs. 78.3
4. Rs. 70.9
5. Rs. 75.3

6. A salesman gets a bonus according to the following structure:
If he sells articles worth Rs. x then he gets bonus of Rs. (x/100 – 1). In the month of January, his sales value was Rs. 100, in February it was Rs. 200, from March to November it was Rs. 300 for every month and in December it was Rs. 1200. Apart from this, he also receives a basic salary of Rs. 30 per month from his employer. Find his average income per month during the year.
1. Rs. 35.78
2. Rs. 30.26
3. Rs. 32.58
4. Rs. 38.88
5. Rs. 30.50

7. The weight of a body as calculated by the average of 7 different experiments is 53.735 g. The average of the first three experiments is 54.005 g, of the fourth is 0.004 g greater than the fifth, while the average of the sixth and seventh experiment was 0.010 g less than the average of the first three. Find the weight of the body obtained by the fourth experiment.
1. 49.353 g
2. 51.718 g
3. 53.342 g
4. 54.512 g
5. 52.242 g

8. Find the average weight of four containers, if it is known that the weight of the first container is 100 kg and the total of the second, third and fourth containers’ weight is defined by f(x) = x2 – 3/4 (x2), where x = 100.
1. 650 kg
2. 880 kg
3. 760 kg
4. 460 kg
5. 780 kg

9. There are five boxes in a cargo hold. The weight of the first box is 200 kg and the weight of the second box is 20% higher than the weight of the third box, whose weight is 25% higher than the first box’s weight. The fourth box at 350 kg is 30% lighter than the fifth box. Find the difference in the average weight of the four heaviest boxes and the four lightest boxes.
1. 80 kg
2. 75 kg
3. 37.5 kg
4. 116.8 kg
5. 65 kg

Average

Average
The average of a given number of quantities of the same kind is expressed as
Average = Sum of the quantities/ Number of the quantities
Average is also called the Arithmetic Mean.
Also, Sum of the quantities = Average × Number of the quantities
Number of quantities = Sum of the quantities/ Average
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Simplication

Simplification

In simplification of an expression there are certain laws which should be strictly adhered to. These laws are as follows:

‘VBODMAS’ Rule

This rule gives the correct sequence in which the mathematical operation are to be executed so as to find out the value of a given expression. Here, ‘V’ stands for Vinculum (or Bar), ‘B’ stands for ‘Bracket’, ‘O’ stands for ‘Of’, ‘D’ stands for ‘Division’, ‘M’ stands for ‘Multiplication’, ‘A’ stands for ‘Addition’ and ‘S’ stands for ‘Subtraction’.
(1) Here, ‘VBODMAS’ gives the order of simplification. Thus, the order of performing the mathematical operations in a given expression are

1. First : Vinculum or line bracket or bar
2. Second: Bracket
3. Third: Of
4. Fourth: Division
5. Fifth: Multiplication
6. Sixth: Addition &
7. Seventh: Subtraction

The above order should strictly be followed.
(2) There are four types of brackets.

1. Square brackets [ ]
2. Curly brackets { }
3. Circular brackets ( )
4. Bar or Vinculum –

Thus, in simplifying an expression all the brackets must be removed in the order ‘–’, ‘( )’, ‘{ }’ and ‘[ ]’.

Modulus of a Real Number

The modulus of a real number x is defined as = x, if a > 0
|x| = x, if a < 0

Basic Formulae

(i) (a + b)² = a² + 2ab + b²
(ii) (a – b)² = a² – 2ab + b²
(iii) (a + b)² – (a – b)² = 4ab
(iv) (a + b)² + (a – b)² = 2(a² + b²)
(v) (a² – b²) = (a + b) (a – b)
(vi) (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(vii) (a³ + b³) = (a + b) (a² – ab + b²)

HCF and LCM (MCQ)

1. The least number which when divided by 4, 6, 8, 12 and 16 leaves a remainder of 2 in each case is:
1. 20
2. 43
3. 50
4. 59
5. 60

2. The HCF of two numbers is 8. Which one of the following can never be their LCM?
1. 32
2. 24
3. 48
4. 56
5. 60

3. The product of the LCM and HCF of two numbers is 24. The difference of the two numbers is 2. Find the numbers.

1. 2 and 6
2. 9 and 14
3. 2 and 4 4. 6 and 4
5. 6 and 10

4. Find the greatest number that will divide 640, 710 and 1526 so as to leave 11, 7, 9 as remainders respectively.

1. 36
2. 37
3. 42
4. 29
5. 47

5. If LCM of two natural numbers is 36 and their sum is 30. Find the two numbers.

1. 15, 4
2. 12, 18
3. 9, 21
4. 4, 26
5. 20, 26

6. There are 5 consecutive natural numbers with LCM as 60. The product of the first two numbers is equal to the 5th number. What is the sum of the 5 numbers?
1. 0
2. 19
3. 20
4. 25
5. 24

7. Find the least number which when divided by 16, 18 and 20 leaves a remainder 4 in each case, but is completely divisible by 7.

1. 465
2. 3234
3. 2884
4. 3234
5. 464

8. HCF of two numbers is 15 and their LCM is 180. If their sum is 105, then the numbers are:
1. 30 and 70
2. 35 and 70
3. 50 and 75
 4. 45 and 60
5. 55 and 60

9. A certain type of board is sold in lengths of multiples of 2 feet. The shortest board sold is 6 feet, and the longest is 24 feet. A builder needs a large quantity of this type of board in feet lengths. For minimum waste the lengths to be ordered should be:
1. 23 ft
2. 28 ft
3. 22 ft
4. 52 ft
5. 25 ft

HCF and LCM

Highest Common Factor
The highest common factor of two or more given numbers is the largest of their common factors. It is known as GCD also.
eg, Factors of 20 are 1, 2, 4, 5, 10, 20
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36
Here greatest and common factor of 20 and 36 is 4.
:-> HCF of 20 and 36 is 4.
Least Common Multiple
The least common multiple of two or more given numbers is the least of their common multiples.
eg, Multiple of 25 are 25, 50, 75, 100, 125, 150, 175, ..
Multiple of 30 are 30, 60, 90, 120, 150, 180, 210, ..
Here 150 is least common multiple of 25 and 30
:-> LCM of 25 and 30 is 150.

Square Root & Cube Root

Square Root & Cube Root

Square Root

The square root of a number is that number the product of which itself gives the given number, ie, the square root of 400 is 20, the square root of 625 is 25.
The process of finding the square root is called evaluation. The square root of a number is denoted by the symbol called the radical sign.

How to Find the Square Root of an Integer?

(i) By the method of Prime Factors: When a given number is a perfect square, we resolve it into prime factors and take the product of prime factors, choosing one out of every two.
(ii) By the method of Long Division: This method can be used when the number is large and the factors cannot be determined easily. This method can also be used when we want to add a least number or to subtract a least number from a given number so that the resulting number may give a perfect square of some number.

Surds

Indices and Surds

In the expression xⁿ, n is called the exponent or index and x is called the base and xⁿ is read as ‘x to the power of n’ or ‘x raised to the power n’.
eg,
1. 3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729
4³ = 4 × 4 × 4 = 64
The expression (x^m)ⁿ is read as ‘x raised to the power m whole raised to the power n’.
2. (2⁴)³ = (16)³ = 2¹²

Laws of Indices

Surds

Decimal fraction(MCQ)

1. If 2805 ¸ 2.55 = 1100, then 280.5 ¸ 25.5 is:
1. 111
2. 1.1
3. 0.11
4. 11
5. 1.11

2. The value of 213 + 2.013 + 0.213 + 2.0013 is:
1. 217.2273
2. 21.8893
3. 217.32
4. 3.217.32
5. None of these

3. The numerator of a non-zero rational number is five less than the denominator. If the denominator is increased by eight and the numerator is doubled, then again we get the same rational number. The required rational number is:
1. 1/8
2. 4/9
3. 2/8
4. 3/8
5. 3/8

4. What will be the approximate value of 779.5 × 5 – 46.5 × 19 – 9?
1. 10800
2. 18008
3. 10080
4. 10008
5. 18080

5. A fraction becomes 2 when 1 is added to both the numerator and the denominator and it becomes 3 when 1 is subtracted from both numerator and denominator. The numerator of the given fraction is:
1. 7
2. 5
3. 3
4. 8
5. 6

6. What approximate value should come in place of z in the following question? 242 + 2.42 + 0.242 + 0.0242 = z
1. 670
2. 575
3. 580
4. 585
5. 680

7. What should come in place of the question mark (?) in the following question?
248.8 – 190.48 + 68.08 = ?
1. 126.4
2. 127.4
3. 128.4
4. 124.6
5. 126.4

8. Find a fraction such that if 1 is added to the denominator it reduces to ½ and reduces to 3/5 on adding 2 to the numerator.
1. 12/25
2. 13/25
3. 11/13
4. 11/25
5. 14/25

9. xy is a number that is divided by ab where xy < ab and gives a result 0. xyxyxy... then ab equals:
1. 77
2. 33
3. 99
4. 66
5. 88

Decimal fractions

A fraction is a part of the whole (object, thing, region). It forms the part of basic aptitude of a person to have and idea of the parts of a population, group or territory. Candidate must have a feel of ‘fractional’ thinking. eg, , here ‘12’ is the number of equal part into which the whole has been divided, is called denominator and ‘5’ is the number of equal parts which have been taken out, is called numerator.

Numerical ability(MCQ)

1. Five-eighth of three-tenth of four-ninth of a number is 45. What is the number?
1. 470
2. 550
3. 560
4. 540
5. None of these

2. Which of the following numbers should be added to 11158 to make it exactly divisible by 77?
1. 9
2. 8
3. 6
4. 5
5. 7

3. If n is odd, (11)ⁿ + 1 is divisible by:
1. 11 + 1
2. 11 – 1
3. 11
4. 10 + 1
5. 10 – 1

4. Find unit digit in (515)³¹ + (515)⁹⁰:
1. 0
2. 5
3. 1
4. 4
5. None of these

5. The number 899 is:
1. a number with 5 factors
2. a number with 4 factors
3. a number with more than 4 factors
4. a perfect cube
5. a number with 3 factors

6. The numerator of a fraction is multiple of two numbers. One of the numbers is greater than the other by 2. The greater number is smaller than the denominator by 1. If the denominator is given as 5 + c(c is a constant), then the minimum value of the fraction is:
1. 2/3
2. –2
3. –1/2
4. 1/2
5. 1/3

7. Find the number which when multiplied by 13 is increased by 180:
1. 20
2. 15
3. 12
4. 5
5. 14

8. Find the number of divisors of 10800.
1. 57
2. 60
3. 72
4. 62
5. 70

9. The sum of the digits in a two-digit number is 5. If 9 is subtracted from the number, the result is the number with the digits reversed. The number is:
1. 23
2. 24
3. 41
4. 14
5. 32

10. Three consecutive numbers such that twice the first, 3 times the second and 4 times the third together make 182. The numbers in question are:
1. 18, 22 and 23
2. 18, 19 and 20
3. 19, 20 and 21
4. 20, 21 and 22
5. 21, 22 and 23

Numerical ability

Place Value (Indian)

Face Value and Place Value of a Digit

Face Value: It is the value of the digit itself eg, in 3452, face value of 4 is ‘four’, face value of 2 is ‘two’.
Place Value: It is the face value of the digit multiplied by the place value at which it is situated eg, in 2586, place value of 5 is 5 × 102 = 500.

Number CategoriesNatural Numbers (N): If N is the set of natural numbers, then we write N = {1, 2, 3, 4, 5, 6,…}
The smallest natural number is 1.
Whole Numbers (W): If W is the set of whole numbers, then we write W = {0, 1, 2, 3, 4, 5,…}
The smallest whole number is 0.

Integers (I): If I is the set of integers, then we write I = {– 3, –2, –1, 0, 1, 2, 3, …}
Rational Numbers: Any number which can be expressed in the form of p/q, where p and q are both integers and q # 0 are called rational numbers.

e.g. 3/2,7/9,5,2

There exists infinite number of rational numbers between any two rational numbers. Irrational Numbers Non-recurring and non-terminating decimals are called irrational numbers. These numbers cannot be expressed in the form of p/q .

e.g. √3, √5,√29

Real Numbers: Real number includes both rational and irrational numbers.

Basic Rules on Natural Numbers

1. One digit numbers are from 1 to 9. There are 9 one digit numbers. ie, 9 × 10⁰.
2. Two digit numbers are from 10 to 99. There, are 90 two digit numbers. ie, 9 × 10.
3. Three digit numbers are from 100 to 199. There are 900 three digit numbers ie, 9 × 10².
In general the number of n digit numbers are 9 × 10^(n–1)

Sum of the first n, natural numbers ie, 1 + 2 + 3 + 4 + … + n = n n 1 / 2

Sum of the squares of the first n natural numbers ie. 1² + 2³ + 3² + 4² + …+ n² =  n n 1 2n 1 / 6

Different Types of Numbers

Even Numbers: Numbers which are exactly divisible by 2 are called even numbers.
eg, – 4, – 2, 0, 2, 4…
Sum of first n even numbers = n (n + 1)
Odd Numbers: Numbers which are not exactly divisible by 2 are called odd numbers.
eg, – 5, –3, –1, 0, 1, 3, 5…
Sum of first n odd numbers = n²
Prime Numbers: Numbers which are divisible by one and itself only are called prime numbers.
eg, 2, 3, 5, 7, 11…

• 2 is the only even prime number.
• 1 is not a prime number because it has two equal factors.
• Every prime number greater than 3 can be written in the form of (6K + 1) or (6K – 1) where K is an integer.
• There are 15 prime numbers between 1 and 50 and l0 prime numbers between 50 and 100.

Relative Prime Numbers: Two numbers are said to be relatively prime if they do not have any common factor other than 1.
eg, (3, 5), (4, 7), (11, 15), (15, 4)…
Twin Primes: Two prime numbers which differ by 2 are called twin primes.
eg, (3, 5), (5, 7), (11, 13),…
Composite Numbers Numbers which are not prime arc called composite numbers
eg, 4, 6, 9, 15,…
1 is neither prime nor composite.
Perfect Number: A number is said to be a perfect number, if the sum of all its factors excluding itself is
equal to the number itself. eg, Factors of 6 are 1, 2, 3 and 6.
Sum of factors excluding 6 = 1 + 2 + 3 = 6.
6 is a perfect number.
Other examples of perfect numbers are 28, 496, 8128 etc.

Rules for Divisibility

Divisibility by 2: A number is divisible by 2 when the digit at ones place is 0, 2, 4, 6 or 8.
eg, 3582, 460, 28, 352, ....

Divisibility by 3: A number is divisible by 3 when sum of all digits of a number is a multiple of 3.
eg, 453 = 4 + 5 + 3 = 12.
12 is divisible by 3 so, 453 is also divisible by 3.

Divisibility by 4: A number is divisible by 4, if the number formed with its last two digits is divisible by 4. eg, if we take the number 45024, the last two digits form 24. Since, the number 24 is divisible by 4, the number 45024 is also divisible by 4.

Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
eg, 10, 25, 60

Divisibility by 6: A number is divisible by 6, if it is divisible both by 2 and 3.
eg, 48, 24, 108

Divisibility by 7: A number is divisible by 7 when the difference between twice the digit at ones place and the number formed by other digits is either zero or a multiple of 7.
eg, 658
65 – 2 × 8 = 65 – 16 = 49
As 49 is divisible by 7 the number 658 is also divisible by 7.

Divisibility by 8: A number is divisible by 8, if the number formed by the last 3 digits of the number is divisible by 8. eg, if we take the number 57832, the last three digits form 832. Since, the number 832 is divisible
by 8, the number 57832 is also divisible by 8..

Divisibility by 9: A number is divisible by 9, if the sum of all the digits of a number is a multiple of 9.
eg, 684 = 6 + 8 + 4 = 18.
18 is divisible by 9 so, 684 is also divisible by 9.

Divisibility by 10: A number is divisible by 10, if its last digit is 0. eg, 20, 180, 350,….

Divisibility by 11: When the difference between the sum of its digits in odd places and in even places is either 0 or a multiple of 11.

eg, 30426
3 + 4 + 6 = 13
0 + 2 = 2
13 – 2 = 11

As the difference is a multiple of 11 the number 30426 is also divisible by 11.

‘Smart’ Facts

• If p and q are co-primes and both are factors of a number K, then their product p x q will also be a factor of r. eg, Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 prime factors of 24 are 2 and 3, which are co-prime also. Product of 2 × 3 = 6, 6 is also a factor of 24.
• If ‘p’ divides ‘q’ and ‘r’, then p’ also divides their sum or difference. eg, 4 divides 12 and 20. Sum of 12 and 20 is 32 which is divisible by 4. Difference of 20 and 12 is 8 which is divisible by 4.
• If a number is divisible by another number, then it must be divisible by each of the factors of that number. 48 is divisible by 12. Factors of 12 are 1, 2, 3, 4, 6, 12. So, 48 is divisible by 2, 3, 4 and 6 also.

Division on Numbers

In a sum of division, we have four quantities.
They are (i) Dividend, (ii) Divisor, (iii) Quotient and (iv) Remainder. These quantities are connected by a relation.
(a) Dividend = Divisor × Quotient + Remainder.
(b) Divisor = (Dividend – Remainder) ÷ Quotient.
(c) Quotient = (Dividend – Remainder) – Divisor.

Example 2: In a sum of division, the quotient is 110, the remainder is 250, the divisor is equal to the sum of the quotient and remainder. What is the dividend ?
Solution. Divisor = (110 + 250) = 360
Dividend = (360 × 110) + 250 = 39850
Hence, the dividend is 39850.

Example 3: Find the number of numbers upto 600 which are divisible by 14.
Solution. Divide 600 by 13, the quotient obtained is 46. Thus, there are 46 numbers less than 600 which are divisible by 14.

Factors and Multiples

Factor: A number which divides a given number exactly is called a factor of the given number,
eg, 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6

Thus, 1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.
• 1 is a factor of every number
• A number is a factor of itself
• The smallest factor of a given number is 1 and the greatest factor is the number itself.
• If a number is divided by any of its factors, the remainder is always zero.
• Every factor of a number is either less than or at the most equal to the given number.
• Number of factors of a number are finite.

Number of Factors of a Number: If N is a composite number such that N = a^m bⁿ c^o... where a, b, c ... are prime factors of N and m, n, o ... are positive integers, then the number of factors of N is given by the expression (m + 1) (n + 1) (o + 1)

Example 4: Find the number of factors that 224 has.
Solution. 224 = 25 × 71
Hence, 224 has (5 + 1) (1 + 1) = 6 × 2 = 12 factors.

Multiple: A multiple of a number is a number obtained by multiplying it by a natural number eg,
Multiples of 5 are 5, 10, 15, 20
Multiples of 12 are 12, 24, 36, 48
• Every number is a multiple of 1.
• The smallest multiple of a number is the number itself.
• We cannot find the greatest multiple of a number.
• Number of multiples of a number are infinite.

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