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Some Tips on Basic Numeracy/Quantitative Aptitude to help the students solve such questions are presented below:

In this type of questions, the candidate is provided with substitutes for various mathematical symbols. It is followed by a question involving calculation of an expression or selecting the correct/incorrect equation. The real signs are to be put in the given equation to arrive at the right choices.

Fundamental Operations:

Fundamental operations are mathematical expressions which include division, multiplication, addition and subtraction. In complex expressions there are other operations namely 'of' and 'brackets'.

Brackets are of different types. Some of the most commonly used brackets are:

Bracket symbol | Name |

() | Paranthesis or common brackets |

{} | Braces or curly brackets |

[] | Brackets or square brackets or box brackets |

- | Vinculum |

The order in which various mathematical operations are to be performed is the rule of 'BODMAS'. where

B -> Brackets : - (), {}, []

O -> Of

D -> Division : ÷

M -> Multiplication : x

A -> Addition : +

S -> Subtraction : -

According to the rule of BODMAS, first of all brackets should be removed followed by Of, Division, Multiplication, Addition and Subtraction.

+ x + = + e.g. 4 x 4 = 16

+ x - = - e.g. 4 x (-4) = -16

- x + = - e.g. (-4) x 4 = -16

- x - = + e.g. (-4) x (-4) = 16

+ ÷ + = + e.g. 4 ÷ 4 = 1

+ ÷ - = - e.g. 4 ÷ (-4) = -1

- ÷ + = - eg. (-4) ÷ 4 = -1

- ÷ - = + (-4) ÷ (-4) = 1

In order to simplify complex expressions involving more than one brackets, the following steps are taken:

The following important topics are described under the heading arithmetical reasoning:

A shopkeeper buys goods either directly from a manufacturer or through a wholeseller.

If (S.P.) > (C.P.), then there is a gain or profit. And gain = (S.P.) - (C.P.)

If (S.P.) < (C.P.), then there is a loss. And loss = (C.P.) - (S.P.)

Gain as well as loss is always calculated on C.P.

(i) Gain % = {Gain/C.P.x100}

(ii) Loss % {Loss/C.P.x100}

Generally a shopkeeper has to bear additional expenses such as freight charges, labour charges and maintenance charges for the goods, before they are sold. Such charges are known as overhead charges.

The real cost price = total investment = (payment made for buying goods + overhead charges)

The method of finding first the value of one (unit) article from the value of the given number of articles and then the value of the required number of articles is known as the unitary method. The general rules used in this method are:

(i)

If 12 mangoes cost Rs. 60, what is the cost of 29 mangoes?

Cost of 12 mangoes = Rs. 60

Cost of 1 mango = Rs. 60/12 [less mangoes, less cost]

Cost of 29 mangoes = Rs. 60/12 x 29 = Rs. 145 [more mangoes, more cost]

Similarly a speed of '20 m per second' or 20 m/sec' means that a distance of 20 m is covered in 1 second.

The u + v = down rate and u - v is up rate.

∴ u = 1/2[down rate + up rate]

and v = 1/2[down rate - up rate].

Area of Rectangle = length x breadth = l x b

Perimeter of rectangle = 2 (length + breadth) = 2 (l + b)

Diagonal of the rectangle = √ {(lenght)

Area of square = (side)

Side of square = √area

Perimeter of square = 4 x sides = 4a

Diagonal of square = √2 x side = √2a

Side of square = Perimeter/4

Opposite sides of parallelogram are equal and parallel. Diagonals of a parallelogram bisect each other.

Area of parallelogram = base x height

Perimeter of parallelogram = 2(length + breadth)

Area of triangle = 1/2 x base x altitude

Circumference of circle = 2Πr

Area of circle = Πr

A rectangular parallelopiped whose faces be squares, and length, breadth and height be equal is called a cube.

Volume of a cube = (side)

Curved surface area of cube = 4a

Total surface area of cube = 6a

Side of cube =∛(Volume)

Side of cube = √(total surface area)/6

A parallelopiped whose faces are rectangle is called a rectangular parallelopiped or a cuboid.

Volume of cuboid = length x breadth x height = lbh

Curved surface area = 2h (l+b)

Total surface area of cuboid = 2 (lb + bh + hl)

Length of the diagonal = √(l

Length of cuboid = volume/(breadth x height)

Breadth of cuboid = volume/(length x height)

Height of cuboid = volume/(length x breadth)

Area of four walls = 2 (lenght + breadth) x height

We know that 24 hours make a day. We divide each day into two halves; for this reason we use two Latin terms a.m. and p.m.

a.m. indicates the times between 12 midnight and 12 noon (i.e, before noon or midday)

p.m. indicates the time between 12 noon and 12 midnight (i.e, after noon or midday)

Generally in railway stations 24 - hour clock timing are used. To escape from the confusion of this time table, we should remember that:

(a) 12 noon is expressed as 12:00;

(b) 12 midnight is expressed as 24:00 or 00:00;

Just watch any clock and observe the angle which is formed by the hour hand and the minute hand. Measure the degree by which a minute hand moves every 5 minutes or the hour hand moves every 15 minutes.

A- Movement of hour hand | 1. 360 degree in 12 hour |

2. 30 degree in 1 hour | |

3. 15 degree in 1/2 hour | |

B- Movement of Minute hand | 4. 360 degree in 60 minute |

5. 6 degree in 1 minute |

When you have to calculate the acute angle between a time say 3:20PM then the movement of hour hand should also be taken into account. The calculation is like this:

If hour hand is exactly at 3 hr and minute hand is at 20 minute then the angle between them will be 30 degree. The movement of hour hand in 20 minutes will be (30 degree in 1 hour or 60 minutes i.e. (30/60)*20 = 10 degrees). So the acute angle is 30 - 10 =20 degree.

Some questions asked on clock sequencing are as follows:

It is 0 hr 0 min at present. How many degrees will the minute hand move by the time it is 2 hr 30 min

a. 720

b. 900

c. 880

d. 450

A clock shows the time as 4:00. After hour hand has mover 150 degrees, the time would be

a. 8:00

b. 10:00

c. 7:00

d. 9:00

The present ages of the father and son are respectively 50 years and 5 years. After how many years will the age of the father become 6 times that of his som?

(a) 2

(b) 4

(c) 6

(d) 8

Let the required number of years be x.

The present age of father = 50 years.

The present age of son = 5 years.

∴ After x years

The age of father will be (x + 50) years.

and the age of son will be (x + 5) years.

Therefore, as per the condition given;

x + 50 = 6(x + 5)

x + 50 = 6x + 30

- 5x = - 20

x = 4 years

∴ the required number of years is 4 years.

[1]

2. A number such as four or five, not attached to any particular things or units is called an abstract number.

3. A number of particular units, such as four boys or five men, is called a concrete number.

4. All numbers are written by the means of the symbols 0, 1, 2, 3, 4, 4, 6, 7, 8, 9 which are called digits.

5. 2, 4, 6, 8 are called

6. 1, 3, 5, 7, 9 are called

7. The number to be divided is called the

8. The number by which it is divided is called the

9. The number which tells how many times the divisor is contained in the dividend is called the

Dividend ÷ Divisor = Quotient

or Quotient x Divisor = Dividend

In inexact division:

Quotient x Divisor + Remainder = Dividend

12. For the number of coins in the successive groups, each of which is formed by putting together ten of the next smaller groups, we have the following names:

a unit,

ten,

one hundred,

one thousand,

ten thousand,

one hundred thousand,

one million,

ten million,

one hundred million,

one thousand million,

ten thousand million,

one hundred thousand million,

one billion, etc.

e.g. we may express the number 9373 in words as nine thousand, three hundred and seventy three.

13. Local value of the number is the face value of the number e.g. the local value of 9 in 9373 is nine.

14. Place value of the number is the value in whose unit's place it occupies e.g. the place value of 9 in 9373 is nine thousand.

15. If two or more numbers be multiplied together, the result is called the

Thus 6, being the result of multiplying 2 by 3, is called the product of 2 and 3; 2 and 3 are called the factors of 6.

All numbers are multiples of unity, and each contains itself once. Any number can therefore be divided by itself and unity, but if it cannot be divided without remainder by any other number, it is said to be

16. Numbers which are not prime are called

17. A series of numbers in which each is greater by 1 than that which precedes it, such as 5, 6, 7, 8 are called consecutive numbers.

18. An even number is of the form 2n, and odd number is of the form 2n + 1 or 2n - 1 where n is a whole number or any integer.

19.

(i) 2 is a factor of all numbers whose digit can be divided by 2.

(ii) 4 is a factor if the number composed of the last two digits can be divided by 4.

(iii) 8 is a factor if the number composed of the last three digits can be divided by 8.

(iV) 5 is a factor if the last digit be either 0 or 5.

(v) 3 is a factor if the sum of the digits can be divided by 3.

(vi) 9 is a factor if the sum of the digits can be divided by 9.

(vii) 6 is a factor if both 2 and 3 are factors.

(viii) 11 is a factor if, when 1

(ix) 12 is a factor if both 3 and 4 are factors.

(x) 25 is a factor if last two digits are both zero or 25, 50 or 75.

(xi) 125 is a factor if last three digits are either zeros or any number divisible by 125 namely 125, 250, 375, 500, 625, 750 or 875.

20. The following are the prime numbers between 1 and 1000.

2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 |

73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 |

179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 | 233 | 239 | 241 | 251 | 257 | 263 | 269 | 271 | 277 | 281 |

283 | 293 | 307 | 311 | 313 | 317 | 331 | 337 | 347 | 349 | 353 | 359 | 367 | 373 | 379 | 383 | 389 | 397 | 401 | 409 |

419 | 421 | 431 | 433 | 439 | 443 | 449 | 457 | 461 | 463 | 467 | 479 | 487 | 491 | 499 | 503 | 509 | 521 | 523 | 541 |

547 | 557 | 563 | 569 | 571 | 577 | 587 | 593 | 599 | 601 | 607 | 613 | 617 | 619 | 631 | 641 | 643 | 647 | 653 | 659 |

661 | 673 | 677 | 683 | 691 | 701 | 709 | 719 | 727 | 733 | 739 | 743 | 751 | 757 | 761 | 769 | 773 | 787 | 797 | 809 |

811 | 821 | 823 | 827 | 829 | 839 | 853 | 857 | 859 | 863 | 877 | 881 | 883 | 887 | 907 | 911 | 919 | 929 | 937 | 941 |

947 | 953 | 967 | 971 | 977 | 983 | 991 | 997 | 1009 |

21. If a and b represent any two numbers, the following general principles are frequently useful:

(i) (a + b) x (a - b) = a

(ii) (a

(iii) (a

(iv) (a + b)

(v) (a - b)

(vi) (a + b)

(vii) (a - b)

(viii) (a

(ix) (a

(x) (a

(xi) {(a + b)

(xii) (a + b)

22. A number which is a factor of two or more numbers is said to be a

For example, it is required to find the H.C.F. of 8 and 12.

Factors of 8 are 1, 2, 4, 8 and

Factors of 12 are 1, 2, 3, 4, 6, 12

The common factors are 1, 2, 4; but the highest of these is 4, hence 4 is the H.C.F.

23. A number which can be divided without remainder by two others, or more, is a

(i) If A and B are two numbers, then the product of their L.C.M. and H.C.F. is equal to the product of the two numbers i.e. L>C.M. x H.C.F. = A x B.

Reasoning: Let 6 and 8 be two numbers

H.C.F. = 2; L.C.M. = 24

2 x 24 = 6 x 8

hence the above property is established.

(ii) Lowest Common Divisor (L.C.D.) = 1 ÷ (G.C.D.)

24. If a/b, c/d, e/f be the proper fractions, then their L.C.M. is given by

(L.C.M. of numberators a, c, e, ----)/(H.C.F. of denominator b, d, f, ...)

where as their H.C.F. is equal to (H.C.F. of numberators a, c, e, ----)/(L.C.M. of denominator b, d, f, ...)

e.g. H.C.F. of 1/2, 3/5, 4/7 and 5/21 is equal to

(H.C.F. of 1, 3, 4, 5)/(L.C.M. of 2, 5, 7, 21) = 1/210

L.C.M. of 1/2, 3/5, 4/7 and 5/21 is equal to

(L.C.M. of 1, 3, 4, 5)/(H.C.F. of 2, 5, 7, 21) = 60/1 = 60.

25. For all types of arithmetical simplification, the rule of BODMAS is very useful. The letters, B, O, D, M, A, S in order of preference, are explained as follows:

B stands for brackets

O stands for of (means multiplication)

D stands for division

M stands for multiplication

A stands for addition

S stands for subtraction

Caution: The above order of preference is to be strictly maintained, any carelessness shown, results in wrong conclusion.

26. Law of indices:

(i) a

(iii)(a

(iv) If a

(v) If a

(vi) If a

27. Metric Measures:

(a) Measures of weight is in GRAMS (g)

(b) Measures of length is in METRES (m)

(c) Measures of capacity is in LITRES (l)

TABLE (Metric system)

10 milli (of a unit) = centi (of a unit)

10 centi (of a unit) = 1 deci (of a unit)

10 deci (of a unit) = unit

10 units = 1 deca unit

10 deca units = 1 hecta unit

10 hecta units = kilo unit

units may be grams, metres or litres.

One quintal = 100 kilograms

One tonne = 10 quintals = 1000 kilograms

One tonne = 2204 pounds

Land Measure:

100 centiares = 1 are = 100 sq. metres

100 ares = 1 hectare = 10000 sq. metres

100 hectares = 1 sq. kilometre

A decimal in which a digit or a set of digits is repeated continually is called a Recurring decimal. Recurring decimals are written in a shortened form, the digits which are repeated being marked by dots placed over the first and the last of them, thus

8/3 = 2.666. ..... = 2.6

1/7 = .142857142857142857.... = .142857

21/22 = .9545454.... = .954

Such a decimal as .142857, in which all the digits recur, is called a

SQUARE ROOT of a number is that number which when multiplied by itself is equal to the given number. Thus 2 is the square root of 4 and 9 is the square root of 81.

SURDS are those numbers whose square roots cannot be exactly found. Thus √2, √3, √5 cannot be exactly found and hence they are known as surds. The square roots of the numbers can be found by means of prime factors as well as by general method of long division. If we are asked a question of the type:

"By what least number, the number 3888 be multiplied or divided by so that the resulting number is perfect square"

We use the method of prime factor.

Ans if the problem is put as"what least number should be subtracted from or added to 15410 to make the result a perfect square",

We use the general method of long division.

Let us find the square root of 1296 by prime factors

1296 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3.

= 2

Square root = 2 x 2 x 3 x 3 = 36.

1. Ratio:

The relation which one quantity bears to another of the same kind with respect to magnitude is called the ratio of the one to the other. A ratio may be expressed in the form of a vulgar fraction. A ratio, for example, of 3 kg. of tea to 5 kg. of tea may be expressed either in the form 3 : 5 or in the form 3/5.

A ratio is said to be in its

2. Proportion is an equality of ratios, for example 12 : 20 = 3 : 5 or 12/20 = 3/5. Each of these expressions indicates a

∴ When four numbers are in proportions, the rule is that

"product of the extremes = product of the means"

Now if a : b = c : d or a/b = c/d, then a, b, c, d are called proportionals and are said to be in proportion. ∴ a x d = b x c.

3. Compound Ratio:

When the antecedents (Numerators) and the consequents (denominators) of two or more ratios are multiplied to get a new antecedent and the new consequent, the new ratio is called their compound ratio.

e.g. the compound ratio of 2 : 3, 3 : 4, 5 : 6 = 2/3 x 3/4 x 5/6 = 5/12 or 5 : 12

4. Direct Ratio:

If the increase or decrease in the value of x is the increase or decrease in the value of y, then x and y are said to be in the direct ratio and is written as x : y.

5. Indirect Ratio:

If the increase or decrease in the value of x is the decrease or increase in the value of y, then x and y are said to be in indirect ration and is written as y : x.

e.g. "If the cost of 7 shirts is Rs. 350, what is the cost of 9 shirts."

Here increase in the number of shirts results in increase in the value of money.

∴ direct ratio of shirts is also the direct ratio of money

-> 7 shirts : 9 shirts : : Rs. 350 : Rs. x

∴ 7x = Rs. 350 x 9

x = (350 x 9)/7 = Rs. 450.

The method by which we first find the value of a unit is called

If 17/18 of an estate be worth Rs. 850. What is the value of 3/5 of the estate?

Let x be the value of the estate, so that 17/18x = Rs. 850.

x = Rs. 850 x 18/17, then

3/5x = Rs. 850 x 18/17 X 3/5 = Rs. 540.

When a series of quantities are so connected with one another, that we know how much of the first kind is equivalent to a given quantiry of second, how much of the second is equivalent to a given quantity of the third and son on, the rule by which we find how much of the last kind is equivalent to a given quantity of the first kind is called the "chain rule".

Six men earn as much as 8 women, 2 women earn as much as 3 boys and 4 boys earn as much as 5 girls. If a girl earns 50 p. a day, what does a man earn a day?

Let Rs. x = Earning of 1 man

6 men = 8 women

2 women = 3 boys

4 boys = 5 girls

1 girl = 1/2 rupee

x X 6 X 2 X 4 X 1 = 1 X 8 X 3 X 5 X 1/2

x = (8 X3 X X 5 X 1/2)/(6 X 2 X X 4 X 1)

= Re. 1.25 P

(1)

(2)

(3) A sleeping partner only provides capital for the business whereas a working partner provides capital for the business but also takes part in the management of the business.

A and B are partners in a business. A invests Rs. 1000 for 8 months and B Rs. 1500 for 6 months. They gain Rs. 340, what is B's share?

A puts Rs. 1000 for 8 months. If he wants to have same amount of profit as before in one month, he has to increase his investment 8 times.

∴ He must invest Rs. 8000 for 1 month.

Similarly B must also invest Rs. 6 x 1500 i.e. Rs. 9000 for 1 month.

Hence, Rs. 340 must be divided in the ratio of

8000 : 9000 or 8 : 9

∴ B's share is Rs. 8/17 x 340 = Rs. 160.

If A invests Rs. 3000 for 1 year in the business, how much B should invest so that profits after 1 year may be distributed in the ration of 3 : 2.

As the profits are always divided in proportion to the investments of the partners.

∴ B's investment is Rs. 2/3 x 3000 = Rs. 2000.

(1) A fraction with 100 as its denominator is called a

(2) Remember the following points. Their direct use helps in solving objective type problems on percentages.

100% =1; 50% = 1/2; 25% = 1/4

12 1/2% 1/8; 6 1/4% = 1/16; 37 1/2% = 3/8

62 1/2% = 5/8; 87 1/2% = 7/8; 75% = 3/4

20% = 1/5; 30% = 3/10; 70% = 7/10

(3) If we are given two numbers x and y, one of the numbers can be expressed in term of the percentage of the other.

(ii) 'y' as a percentage of 'x' = y/x x 100

What rate percent is 3 minutes 36 seconds to an hour?

3 minutes 36 seconds = (3 x 60 + 36) seconds = 216 seconds

Number of seconds in an hour = 3600 seconds

∴ Rate percent = (216 x 100)/3600 % = 6 %

A man spent Rs. 229.50 P which is 85% of what he earned. How much does he earn?

85% of earning = Rs. 229.50 P.

His earning = Rs. 229.50 x 100/85 = Rs. 270.

Find the ratio in which rice at Rs. 7.20 P a kg. be mixed with rice at Rs. 5.70 P a kg. to produce a mixture worth Rs. 6.30 P a kg.

Difference of the 1

Difference of the mean price and the 2

Now mixture must be in the inverse ratio i.e. 60 : 90 or 2 : 3.

How much water must be added to 14 litres of milk worth Rs. 5.40 P a litre so that the value of the mixture may be Rs. 4.20 P a litre?

The mean value is Rs. 4.20 P a litre and the price of water is zero paisa (i.e. free of cost) per litre

By allegation method:

Milk/water = (Rs. 4.20 - 0)/(Rs. 5.40 - Rs. 4.20) = 420/120 = 7/2

∴ Milk and water must be mixed in the ratio 7 : 2.

Since the milk is 14 litres, so water to be mixed is 4 litres.

(1) If a person reckons his expenditure during an ordinary year, and divides the amount by 365, the quotient gives what is called his average daily expenditure.

If a man buys 3 horses for Rs. 300, Rs. 400 and Rs. 500 respectively, he buys 3 horses for Rs. 1200, at an average price of Rs. (1200 ÷ 3) i.e. Rs. 400 each.

If a train travels 200 km. in 5 hours, the speed may vary from time to time, but the average rate is 40 km per hour.

∴ the average of any number of quantities of the same kind is the result of dividing the the sum of the quantities by the number, average is also called the mean of the quantities.

In general the average of n quantities of the same kind, denoted by a, b, c, d, ..... is 1/n (a + b + c + d...); and conversely, if A be the average of n quantities, the sum of the quantities will be nA.

(2)

A man's daily expenditure is Rs. 10 during May, Rs. 14 during June and Rs. 15 durinng July. Find the average daily expenditure for the three months.

The total expenditure = (10 x 31 + 14 x 30 + 15 x 31) rupees

= (310 + 420 + 465) = Rs. 1195

The number of days = 31 +30 + 31 = 92

∴ the average daily expenditure = Rs. 1195/92 = Rs. 13 approx.

thus "amount = Principal + Interest."

If A stands for amount; P stands for principal; I stands for Interest; T stands for time (in years); R stands for rate per cent per annum (per year); the following relations may be remembered.

I = P x R X T/100

P = 100 x I/R x T

R = 100 x I/P x T

T = 100 x I/P x R

P = 100A/(100 + RT)

[

Rule: (i) A = P(1 + r/100)

where n is the number of times the interest is compounded.

At what rate per cent per annum will be a sum of money treble itself in 25 years?

Rs. 100 becomes Rs. 300 in 25 years i.e, interest earned on Rs. 100 is Rs. 200 over a period of 25 years.

rate % = 200/25 = 8%.

If a sum of money doubles itself in 7 years, it becomes 5 times in how many years?

Rs. 100 amounts to Rs. 200 in 7 years i.e. Rs. 100 is earned after every 7 years on Rs. 100.

Rs. 100 becoming Rs. 500 means, interest earned is Rs. 400 which shall take time 7 x 4 = 28 years.

A sum of money placed at compound interest doubles itself in 4 years. In how many years will it amount to eight times itself?

If x be the sum it becomes 2x in four years at a certain rate say r%

2x = x(1 + r/100)

or 2 = (1 + r/100)

cube both sides

2

8 = (1 + r/100)

or 8x = x(1 + r/100)

Hence the required number of years = 12 years.

(ii) If A can do a work in 3 days and B in 8 days, the ratio of the work done by A and B in the same time is 8 : 3.

A's 1 day's work is 1/3 amd B's 1 day work is 1/8

(1 + r/100)

Ratio of the work is 1/3 : 1/8 or 8 : 3.

(iii) If A is thrice as good a workman as B, then ratio of the work done by A and B is 3 : 1.

A can do piece of work in 6 days, and B can do it in 12 days. What time will they require to do it working together?

Part of the work done by A in one day = 1/6

Part of the work done by B in one day = 1/12

Part of the work done by A and B in one day = 1/6 + 1/12 = 3/12 = 1/4

Time required by A and B working together to finish the work = 4 days.

A and B undertake to do a piece of work for Rs. 200. A can do it in 6 days and B in 8 days. With the help of C they finish it in 3 days. How much is paid to C?

Each of three is working for three days.

A's work for 3 days = 3 x 1/6 = 1/2

∴ A gets 1/2 x Rs. 200 = Rs. 100

B's work for 3 days = 3 x 1/8 = 3/8

∴ B gets 3/8 x Rs. 200 = Rs. 75

∴ C gets the rest i.e. Rs. 200 - (100 + 75) = Rs. 25

Three pipes A, B and C can fill a cistern in 12, 15 and 20 minutes respectively. Pipes A and C work for one minute and Pipes B and C work in next minute. In how many minutes the cistern will be full if theis proceedure is continued?

Work done by pipes A and C in one minute = 1/12 + 1/20 = (5 +3)/60 = 8/60 of the cistern.

Work done by pipes B and C in the next minute = 1/15 + 1/20 = (4 + 3)/60 = 7/60 of the cistern

Part of the cistern filled in 2 minute = 8/60 + 7/60 = 15/60 = 1/4 ∴ The cistern shall be full in 8 minutes.

1.

2.

3.

4.

5.

6.

7.

1. Stock = (Investment x 100)/M.V. Also

2. Stock = (Income x 100)/Rate

3. Investment = (Stock x M.V.)/100 Also

4. Investment = (Income x M.V.)/Rate

5. Income = (Stock x Rate)/100 Also

6. Income = (Investment x Rate)/M.V.

How much of 5% stock at 5 above par can be purchased by investing Rs. 7980?

Investment = Rs. 7980;

M.V. = 100 + Rs. 5 = Rs. 105

Stock = (Rs. 7980 x 100)/105 = 7600.

A man's net income after paying income tax of 8 paise in the rupee is Rs. 146.50. Find his total income?

Suppose his gross income= Re. 1 or 100 paise

His net income = 100 P - 8 P = 92 P

If net income is Rs. 92/100, the gross income = Re. 1

If net income is Rs. 1460.50, the gross income = Rs. 1 x 100/92 x 1460.50

= Rs. 1587.50

The income of a man increases by Rs. 3000 but the rate of income tax decreases from 10% to 7%. He pays the same amount of income tax as before. Find his income?

If the income of the man be Rs. x, then the tax in the first case @ 10% is 10%x.

In the second case, his income becomes (x + 3000) rupees and he now pays the tax at the rate of 7%.

∴ tax in this case = 7%(x + 3000).

Thus 10%x = 7%(x + 3000)

-> 10x = 7x + 21000

-> 3x = 21000

x = 7000

Thus his income is Rs. 7000.

For what sum should goods worth Rs. 1150 be insured at 8% so that in case of loss, the owner may recover the premium as well as the goods?

If the owner insures his goods for Rs. 100, then in case of loss he would recover Rs. 8 premium and Rs. (100 - 8) or Rs. 92 as the value of goods.

Now Rs. 92 must be insured for Rs. 100

Rs. 1150 must be insured for Rs. 100/92 x 1150 = Rs. 1250

[Multiply the value of the goods by 100/(100 - rate)]

1. A land measure unit is are. One are measures an area equal to 100m

2.

3.

4.

A (surface area or area) = l x b

D (diagonal) = √(l

5.

P = 4x or x = 1/4 P

A = x

D (diagonal) = √2x or D

6.

P = 4x or x = 1/4 P

A = 1/2 x Product of the diagonals = 1/2 x AC x BD.

7.

(i) Area of any triangle = 1/2 x base x height

(ii) Area of a triangle when its sides are given as a, b and c = √(s(s-a) (s-b) (s-c)) where s = 1/2 (a + b +c).

(iii) Area of an equilateral triangle = √3/4 x (side)

(iv) Area of an isosceles triangle whose base is 'b' and equal sides are 'a' each = 1/4 x b x √(4a

8. Area of four walls of a room = 2 x height (length + breadth) = 2 h (l + b)

9.

10. What is Π? This is the ratio of the circumference of the circle with its diameter and remains constant. Its approximate value is 22/7 or 3.1416. Thus Π = circumference/diameter

11. Circumference of a circle = 2Πr

Area = Πr

12. The radius R of a circle whose area is equal to the sum of the areas of series of circles with radii r

For example, the radius of a circle whose area is equal to the sum of the areas of two circles of radii 4 cms. and 8 cms. is = √(6

13.

14.

(i) Surface area = 2 x (lb + bh + lh)

(ii) Volume = l x b x h.

(iii) Diagonal = √(l

15.

(i) Perimeter = 12 x edge

(ii) Area = 6 x (edge)

(iii) Volume = (edge)

(iv) Diagonal = √3 x edge

(v) Edge 'E' of a cube whose volume, is equal to the sum of volume of cubes of edges e

16.

(i) Surface area = 4Πr

(ii) Volume = 4/3 Πr

(iii) Area of the solid hemisphere = 3Πr

(iv) Radius 'R' of the sphere whose volume is equal to the sum of the volumes of spheres with radii r

17.

(i) Curved surface of the cylinder = base circumference x height = 2Πr x h

(ii) Total surface of a closed cylinder = 2Πr (r + h)

(iii) Volume = base area x height = Πr

(iv) Height of the cylinder = Volume of the cylinder/base area of cylinder

18.

(i) Curved surface = Πrl = Πr x √(r

(ii) Total surface = Πr (l + r)

(iii) Volume = 1/3 Πr

∴ Volume of cone/Volume of cylinder = 1/3

For example, if a conical vessel can hold 20 litres of milk, then a cylindrical vessel can hold 3 x 20 i.e. 60 litres of milk.

19. If it is to find the canvas required for making the conical tent with given dimensions, formula (i) is used. For finding how much iron sheet is required to construct a conical vessel, formula (ii) is used. To find the capacity of the cone, formula (iii) is used.

20.

(i) If the sides of a rectangel are doubled, what percent of its area is increased?

Now it is very clear that the area of the corresponding rectangle is becomes four times the original and thus the increase is 3 times or 300%.

(ii) If the cost of levelling the field in the form of a square is Rs. 25, what is the cost of levelling another square field whose side is three times the side of the first.

Since the area of the other field is 9 times the original, so the cost is (25 x 9) i.e. Rs. 225

(iii) The perimeter of a regular hexagone is 48 cm, what is the area of the hexagon.

The perimeter being 48 cm, so the side of the hexagon is (48 ÷ 6) i.e. 8 cm. Hence the area of the hexagon is 6 x √3/4 x (8)

(iv) The perimeter of a cube is 36 cm. What is its volume?

Now side of the cube = 1/12 x 36 cm = 3 cm

∴ Volume = 3

(v) The weight of an iron ball of radius 2 cm is 25 grams. What is the weight of another similar sphere whose radius is 6 cm.

Since the radius of the other sphere is 3 times the first the volume is 3

(vi) If the ratio of areas of two cylindrical vessels are as 25 : 49, what will be the ratio of their volumes?

Since area ratio is 25 : 49

∴ dimension ratio = √25 : √49 = 5 : 7

or volume ratio = 5

In the following 271 pages you will find 1127 questions and answers on quantitative aptitude with detailed solution.

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1. Passage Reading 2. Verbal Logic 3. Non Verbal Logic 4. Numerical Logic

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