Aptitude

Formulae

Train

Time taken by a train of length l metres to pass a pole or standing man or a signal post is equal to the time taken by the train to cover l metres. Time taken by a train of length l metres to pass a stationery object of length bmetres is the time taken by the train to cover (l + b) metres. Suppose two trains or two objects bodies are moving in the same direction at um/s and v m/s, where u > v, then their relative speed is = (u - v) m/s. Suppose two trains or two objects bodies are moving in opposite directions at um/s and v m/s, then their relative speed is = (u + v) m/s. If two trains of length a metres and b metres are moving in opposite directions atu m/s and v m/s, then: The time taken by the trains to cross each other = (a + b) sec. (u + v) If two trains of length a metres and b metres are moving in the same direction atu m/s and v m/s, then: The time taken by the faster train to cross the slower train = (a + b) sec. (u - v) If two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then: (A's speed) : (B's speed) = (b : a)

Time & Distance

Speed, Time and Distance: Speed = Distance , Time = Distance , Distance = (Speed x Time). Time Speed km/hr to m/sec conversion: x km/hr = x x 5 m/sec. 18 m/sec to km/hr conversion: x m/sec = x x 18 km/hr. 5 If the ratio of the speeds of A and B is a : b, then the ratio of the the times taken by then to cover the same distance is 1 : 1 or b : a. a b Suppose a man covers a certain distance at x km/hr and an equal distance at ykm/hr. Then, the average speed during the whole journey is 2xy km/hr. x + y

Height & Distance

Trigonometry: In a right angled  OAB, where BOA = , i.   sin  = Perpendicular = AB ; Hypotenuse OB ii.   cos  = Base = OA ; Hypotenuse OB iii.  tan  = Perpendicular = AB ; Base OA iv.  cosec  = 1 = OB ; sin  AB v.   sec  = 1 = OB ; cos  OA vi.  cot  = 1 = OA ; tan  AB Trigonometrical Identities: sin2  + cos2  = 1. 1 + tan2  = sec2 . 1 + cot2  = cosec2 . Values of T-ratios: 0° (/6) 30° (/4) 45° (/3) 60° (/2) 90° sin  0 1 2 3 2 1 cos  1 3 2 1 2 0 tan  0 1 3 1 3 not defined Angle of Elevation: Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, the angle which the line of sight makes with the horizontal through O, is called the angle of elevation of P as seen from O.  Angle of elevation of P from O = AOP. Angle of Depression: Suppose a man from a point O looks down at an object P, placed below the level of his eye, then the angle which the line of sight makes with the horizontal through O, is called the angle of depression of P as seen from O

Time & Work

Work from Days: If A can do a piece of work in n days, then A's 1 day's work = 1 . n Days from Work: If A's 1 day's work = 1 , then A can finish the work in n days. n Ratio: If A is thrice as good a workman as B, then: Ratio of work done by A and B = 3 : 1. Ratio of times taken by A and B to finish a work = 1 : 3.

Simple Interest

Principal: The money borrowed or lent out for a certain period is called the principal or thesum. Interest: Extra money paid for using other's money is called interest. Simple Interest (S.I.): If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest. Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then (i). Simple Intereest = P x R x T 100 (ii). P = 100 x S.I. ; R = 100 x S.I. and T = 100 x S.I. . R x T P x T P x R

Compound Interest

Let Principal = P, Rate = R% per annum, Time = n years. When interest is compound Annually:    Amount = P 1 + R n 100 When interest is compounded Half-yearly:     Amount = P 1 + (R/2) 2n 100 When interest is compounded Quarterly:     Amount = P 1 + (R/4) 4n 100 When interest is compounded Annually but time is in fraction, say 3 years.     Amount = P 1 + R 3 x 1 + R 100 100 When Rates are different for different years, say R1%, R2%, R3% for 1st, 2ndand 3rd year respectively.     Then, Amount = P 1 + R1 1 + R2 1 + R3 . 100 100 100 Present worth of Rs. x due n years hence is given by:     Present Worth = x . 1 + R 100

Profit & Loss

Cost Price:

The price, at which an article is purchased, is called its cost price, abbreviated as C.P.

Selling Price:

The price, at which an article is sold, is called its selling prices, abbreviated as S.P.

Profit or Gain:

If S.P. is greater than C.P., the seller is said to have a profit or gain.

Loss:

If S.P. is less than C.P., the seller is said to have incurred a loss.

IMPORTANT FORMULAE

1. Gain = (S.P.) - (C.P.)
2. Loss = (C.P.) - (S.P.)
3. Loss or gain is always reckoned on C.P.
4. Gain Percentage: (Gain %)

   Gain % =		Gain x 100
   		C.P.

5. Loss Percentage: (Loss %)

   Loss % =		Loss x 100
   		C.P.

6. Selling Price: (S.P.)

   SP =		(100 + Gain %)	x C.P
   		100

7. Selling Price: (S.P.)

   SP =		(100 - Loss %)	x C.P.
   		100

8. Cost Price: (C.P.)

   C.P. =		100	x S.P.
   		(100 + Gain %)

9. Cost Price: (C.P.)

   C.P. =		100	x S.P.
   		(100 - Loss %)

10. If an article is sold at a gain of say 35%, then S.P. = 135% of C.P.
11. If an article is sold at a loss of say, 35% then S.P. = 65% of C.P.
12. When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by:

    Loss % =		Common Loss and Gain %		2	=		x		2	.
    		10					10
13. If a trader professes to sell his goods at cost price, but uses false weights, then

    Gain % =		Error	x 100	%.
    		(True Value) - (Error)

Patnership

Partnership: When two or more than two persons run a business jointly, they are calledpartners and the deal is known as partnership. Ratio of Divisions of Gains: When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments. Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year: (A's share of profit) : (B's share of profit) = x : y. When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now gain or loss is divided in the ratio of these capitals. Suppose A invests Rs. x for p months and B invests Rs. y for q months then, (A's share of profit) : (B's share of profit)= xp : yq. Working and Sleeping Partners: A partner who manages the the business is known as a working partner and the one who simply invests the money is a sleeping partner.

Percentage

Concept of Percentage: By a certain percent, we mean that many hundredths. Thus, x percent means x hundredths, written as x%. To express x% as a fraction: We have, x% = x . 100     Thus, 20% = 20 = 1 . 100 5 To express a as a percent: We have, a = a x 100 %. b b b     Thus, 1 = 1 x 100 % = 25%. 4 4 Percentage Increase/Decrease: If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is: R x 100 % (100 + R) If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is: R x 100 % (100 - R) Results on Population: Let the population of a town be P now and suppose it increases at the rate of R% per annum, then: 1. Population after n years = P 1 + R n 100 2. Population n years ago = P 1 + R n 100 Results on Depreciation: Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then: 1. Value of the machine after n years = P 1 - R n 100 2. Value of the machine n years ago = P 1 - R n 100 3. If A is R% more than B, then B is less than A by R x 100 %. (100 + R) 4. If A is R% less than B, then B is more than A by R x 100 %. (100 - R)

Ages

1. If the current age is x, then n times the age is nx.

2. If the current age is x, then age n years later/hence = x + n.

3. If the current age is x, then age n years ago = x - n.

4. The ages in a ratio a : b will be ax and bx.

5. If the current age is x, then	1	of the age is	x	.
	n		n

Calender

Odd Days: We are supposed to find the day of the week on a given date. For this, we use the concept of 'odd days'. In a given period, the number of days more than the complete weeks are calledodd days. Leap Year: (i). Every year divisible by 4 is a leap year, if it is not a century. (ii). Every 4th century is a leap year and no other century is a leap year. Note: A leap year has 366 days. Examples: Each of the years 1948, 2004, 1676 etc. is a leap year. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year. None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year. Ordinary Year: The year which is not a leap year is called an ordinary years. An ordinary year has 365 days. Counting of Odd Days: 1 ordinary year = 365 days = (52 weeks + 1 day.)  1 ordinary year has 1 odd day. 1 leap year = 366 days = (52 weeks + 2 days)  1 leap year has 2 odd days. 100 years = 76 ordinary years + 24 leap years   = (76 x 1 + 24 x 2) odd days = 124 odd days.   = (17 weeks + days)  5 odd days.  Number of odd days in 100 years = 5. Number of odd days in 200 years = (5 x 2)  3 odd days. Number of odd days in 300 years = (5 x 3)  1 odd day. Number of odd days in 400 years = (5 x 4 + 1)  0 odd day. Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days. Day of the Week Related to Odd Days: No. of days: 0 1 2 3 4 5 6 Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

Clock

Minute Spaces: The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces. Hour Hand and Minute Hand: A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand. In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand. In every hour, both the hands coincide once. The hands are in the same straight line when they are coincident or opposite to each other. When the two hands are at right angles, they are 15 minute spaces apart. When the hands are in opposite directions, they are 30 minute spaces apart. Angle traced by hour hand in 12 hrs = 360° Angle traced by minute hand in 60 min. = 360°. If a watch or a clock indicates 8.15, when the correct time is 8, it is said to be 15 minutes too fast. On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.

Averages

Average: Average = Sum of observations Number of observations Average Speed: Suppose a man covers a certain distance at x kmph and an equal distance at ykmph. Then, the average speed druing the whole journey is 2xy kmph. x + y

Areas

FUNDAMENTAL CONCEPTS Results on Triangles: Sum of the angles of a triangle is 180°. The sum of any two sides of a triangle is greater than the third side. Pythagoras Theorem: In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2. The line joining the mid-point of a side of a triangle to the positive vertex is called the median. The point where the three medians of a triangle meet, is called centroid.The centroid divided each of the medians in the ratio 2 : 1. In an isosceles triangle, the altitude from the vertex bisects the base. The median of a triangle divides it into two triangles of the same area. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle. Results on Quadrilaterals: The diagonals of a parallelogram bisect each other. Each diagonal of a parallelogram divides it into triangles of the same area. The diagonals of a rectangle are equal and bisect each other. The diagonals of a square are equal and bisect each other at right angles. The diagonals of a rhombus are unequal and bisect each other at right angles. A parallelogram and a rectangle on the same base and between the same parallels are equal in area. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area. IMPORTANT FORMULAE 1.   Area of a rectangle = (Length x Breadth).        Length = Area and Breadth = Area . Breadth Length 2.   Perimeter of a rectangle = 2(Length + Breadth). Area of a square = (side)2 = (diagonal)2. Area of 4 walls of a room = 2 (Length + Breadth) x Height. 1.   Area of a triangle =  x Base x Height. 2.   Area of a triangle = s(s-a)(s-b)(s-c)       where a, b, c are the sides of the triangle and s = (a + b + c). 3.   Area of an equilateral triangle = 3 x (side)2. 4 4.   Radius of incircle of an equilateral triangle of side a = a . 23 5.   Radius of circumcircle of an equilateral triangle of side a = a . 3 6.   Radius of incircle of a triangle of area  and semi-perimeter s = s 1.   Area of parallelogram = (Base x Height). 2.   Area of a rhombus =  x (Product of diagonals). 3.   Area of a trapezium =  x (sum of parallel sides) x distance between them. 1.   Area of a circle = R2, where R is the radius. 2.   Circumference of a circle = 2R. 3.   Length of an arc = 2R , where  is the central angle. 360 4.   Area of a sector = 1 (arc x R) = R2 . 2 360 1.   Circumference of a semi-circle = R. 2.   Area of semi-circle = R2 . 2

Volume & Surface Area

CUBOID Let length = l, breadth = b and height = h units. Then Volume = (l x b x h) cubic units. Surface area = 2(lb + bh + lh) sq. units. Diagonal = l2 + b2 + h2 units. CUBE Let each edge of a cube be of length a. Then, Volume = a3 cubic units. Surface area = 6a2 sq. units. Diagonal = 3a units. CYLINDER Let radius of base = r and Height (or length) = h. Then, Volume = (r2h) cubic units. Curved surface area = (2rh) sq. units. Total surface area = 2r(h + r) sq. units. CONE Let radius of base = r and Height = h. Then, Slant height, l = h2 + r2 units. Volume = r2h cubic units. Curved surface area = (rl) sq. units. Total surface area = (rl + r2) sq. units. SPHERE Let the radius of the sphere be r. Then, Volume = r3 cubic units. Surface area = (4r2) sq. units. HEMISPHERE Let the radius of a hemisphere be r. Then, Volume = r3 cubic units. Curved surface area = (2r2) sq. units. Total surface area = (3r2) sq. units. Note: 1 litre = 1000 cm3.

Permutations & Combinations

Factorial Notation: Let n be a positive integer. Then, factorial n, denoted n! is defined as: n! = n(n - 1)(n - 2) ... 3.2.1. Examples: We define 0! = 1. 4! = (4 x 3 x 2 x 1) = 24. 5! = (5 x 4 x 3 x 2 x 1) = 120. Permutations: The different arrangements of a given number of things by taking some or all at a time, are called permutations. Examples: All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb). All permutations made with the letters a, b, c taking all at a time are: ( abc, acb, bac, bca, cab, cba) Number of Permutations: Number of all permutations of n things, taken r at a time, is given by: nPr = n(n - 1)(n - 2) ... (n - r + 1) = n! (n - r)! Examples: 6P2 = (6 x 5) = 30. 7P3 = (7 x 6 x 5) = 210. Cor. number of all permutations of n things, taken all at a time = n!. An Important Result: If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind, such that (p1 + p2 + ... pr) = n. Then, number of permutations of these n objects is = n! (p1!).(p2)!.....(pr!) Combinations: Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination. Examples: Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA. Note: AB and BA represent the same selection. All the combinations formed by a, b, c taking ab, bc, ca. The only combination that can be formed of three letters a, b, c taken all at a time is abc. Various groups of 2 out of four persons A, B, C, D are: AB, AC, AD, BC, BD, CD. Note that ab ba are two different permutations but they represent the same combination. Number of Combinations: The number of all combinations of n things, taken r at a time is: nCr = n! = n(n - 1)(n - 2) ... to r factors . (r!)(n - r)! r! Note: nCn = 1 and nC0 = 1. nCr = nC(n - r) Examples: i.   11C4 = (11 x 10 x 9 x 8) = 330. (4 x 3 x 2 x 1) ii.   16C13 = 16C(16 - 13) = 16C3 = 16 x 15 x 14 = 16 x 15 x 14 = 560.

Numbers

1. (a + b)(a - b) = (a² - b²)
2. (a + b)² = (a² + b² + 2ab)
3. (a - b)² = (a² + b² - 2ab)
4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
5. (a³ + b³) = (a + b)(a² - ab + b²)
6. (a³ - b³) = (a - b)(a² + ab + b²)
7. (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac)
8. When a + b + c = 0, then a³ + b³ + c³ = 3abc.

LCM & HCF

Factors and Multiples: If number a divided another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.): The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly. There are two methods of finding the H.C.F. of a given set of numbers: Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F. Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number. Similarly, the H.C.F. of more than three numbers may be obtained. Least Common Multiple (L.C.M.): The least number which is exactly divisible by each one of the given numbers is called their L.C.M. There are two methods of finding the L.C.M. of a given set of numbers: Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors. Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers. Product of two numbers = Product of their H.C.F. and L.C.M. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1. H.C.F. and L.C.M. of Fractions:     1. H.C.F. = H.C.F. of Numerators L.C.M. of Denominators     2. L.C.M. = L.C.M. of Numerators H.C.F. of Denominators H.C.F. and L.C.M. of Decimal Fractions: In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers. Comparison of Fractions: Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

Simplification

'BODMAS' Rule: This rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of given expression. Here B - Bracket, O - of, D - Division, M - Multiplication, A - Addition and S - Subtraction Thus, in simplifying an expression, first of all the brackets must be removed, strictly in the order (), {} and ||. After removing the brackets, we must use the following operations strictly in the order: (i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction. Modulus of a Real Number: Modulus of a real number a is defined as |a| = a, if a > 0 -a, if a < 0 Thus, |5| = 5 and |-5| = -(-5) = 5. Virnaculum (or Bar): When an expression contains Virnaculum, before applying the 'BODMAS' rule, we simplify the expression under the Virnaculum.

Surds & Indices

Laws of Indices: am x an = am + n am = am - n an (am)n = amn (ab)n = anbn a n = an b bn a0 = 1 Surds: Let a be rational number and n be a positive integer such that a(1/n) = a Then, a is called a surd of order n. Laws of Surds: a = a(1/n) ab = a x b = a b (a)n = a (a)m = am

Ratio & Proportion

Ratio: The ratio of two quantities a and b in the same units, is the fraction  and we write it as a : b. In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent. Eg. The ratio 5 : 9 represents 5 with antecedent = 5, consequent = 9. 9 Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3. Proportion: The equality of two ratios is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion. Here a and d are called extremes, while b and c are called mean terms. Product of means = Product of extremes. Thus, a : b :: c : d  (b x c) = (a x d). Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c. Third Proportional: a : b = c : d, then c is called the third proportion to a and b. Mean Proportional: Mean proportional between a and b is ab. Comparison of Ratios: We say that (a : b) > (c : d)      a > c . b d Compounded Ratio: The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf). Duplicate Ratios: Duplicate ratio of (a : b) is (a2 : b2). Sub-duplicate ratio of (a : b) is (a : b). Triplicate ratio of (a : b) is (a3 : b3). Sub-triplicate ratio of (a : b) is (a1/3 : b1/3). If a = c , then a + b = c + d .     [componendo and dividendo] b d a - b c - d Variations: We say that x is directly proportional to y, if x = ky for some constant k and we write, x  y. We say that x is inversely proportional to y, if xy = k for some constant k and we write, x  1 . y

Pipes & Cistern

Inlet: A pipe connected with a tank or a cistern or a reservoir, that fills it, is known as an inlet. Outlet: A pipe connected with a tank or cistern or reservoir, emptying it, is known as an outlet. If a pipe can fill a tank in x hours, then: part filled in 1 hour = 1 . x If a pipe can empty a tank in y hours, then: part emptied in 1 hour = 1 . y If a pipe can fill a tank in x hours and another pipe can empty the full tank in yhours (where y > x), then on opening both the pipes, then the net part filled in 1 hour = 1 - 1 . x y If a pipe can fill a tank in x hours and another pipe can empty the full tank in yhours (where x > y), then on opening both the pipes, then the net part emptied in 1 hour = 1 - 1 . y x

Boats & Streams

Downstream/Upstream: In water, the direction along the stream is called downstream. And, the direction against the stream is called upstream. If the speed of a boat in still water is u km/hr and the speed of the stream is vkm/hr, then: Speed downstream = (u + v) km/hr. Speed upstream = (u - v) km/hr. If the speed downstream is a km/hr and the speed upstream is b km/hr, then: Speed in still water = 1 (a + b) km/hr. 2 Rate of stream = 1 (a - b) km/hr.

Allegations & Mixtures

Alligation: It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price. Mean Price: The cost of a unit quantity of the mixture is called the mean price. Rule of Alligation: If two ingredients are mixed, then     Quantity of cheaper = C.P. of dearer - Mean Price Quantity of dearer Mean price - C.P. of cheaper We present as under: C.P. of a unit quantity of cheaperC.P. of a unit quantity of dearer (c) Mean Price (m) (d) (d - m) (m - c)  (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c). Suppose a container contains x of liquid from which y units are taken out and replaced by water. After n operations, the quantity of pure liquid = x 1 - y n units.

Logerithms

Logarithm: If a is a positive real number, other than 1 and am = x, then we write: m = logax and we say that the value of log x to the base a is m. Examples: (i). 103 1000      log10 1000 = 3. (ii). 34 = 81      log3 81 = 4. (iii). 2-3 = 1      log2 1 = -3. 8 8 (iv). (.1)2 = .01      log(.1) .01 = 2. Properties of Logarithms: 1. loga (xy) = loga x + loga y 2. loga x = loga x - loga y y 3. logx x = 1 4. loga 1 = 0 5. loga (xn) = n(loga x) 6. loga x = 1 logx a 7. loga x = logb x = log x . logb a log a Common Logarithms: Logarithms to the base 10 are known as common logarithms. The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'. Characteristic: The internal part of the logarithm of a number is called itscharacteristic. Case I: When the number is greater than 1. In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number. Case II: When the number is less than 1. In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative. Instead of -1, -2 etc. we write 1 (one bar), 2 (two bar), etc. Examples:- Number Characteristic Number Characteristic 654.24 2 0.6453 1 26.649 1 0.06134 2 8.3547 0 0.00123 3 Mantissa: The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.

Probebility

Experiment: An operation which can produce some well-defined outcomes is called an experiment. Random Experiment: An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment. Examples: Rolling an unbiased dice. Tossing a fair coin. Drawing a card from a pack of well-shuffled cards. Picking up a ball of certain colour from a bag containing balls of different colours. Details: When we throw a coin, then either a Head (H) or a Tail (T) appears. A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face. A pack of cards has 52 cards. It has 13 cards of each suit, name Spades, Clubs, Hearts and Diamonds. Cards of spades and clubs are black cards. Cards of hearts and diamonds are red cards. There are 4 honours of each unit. There are Kings, Queens and Jacks. These are all called face cards. Sample Space: When we perform an experiment, then the set S of all possible outcomes is called the sample space. Examples: In tossing a coin, S = {H, T} If two coins are tossed, the S = {HH, HT, TH, TT}. In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}. Event: Any subset of a sample space is called an event. Probability of Occurrence of an Event: Let S be the sample and let E be an event. Then, E  S.  P(E) = n(E) . n(S) Results on Probability: P(S) = 1 0  P (E)  1 P() = 0 For any events A and B we have : P(A  B) = P(A) + P(B) - P(A  B) If A denotes (not-A), then P(A) = 1 - P(A).

True Discount

IMPORTANT CONCEPTS

Suppose a man has to pay Rs. 156 after 4 years and the rate of interest is 14% per annum. Clearly, Rs. 100 at 14% will amount to R. 156 in 4 years. So, the payment of Rs. now will clear off the debt of Rs. 156 due 4 years hence. We say that:

Sum due = Rs. 156 due 4 years hence;

Present Worth (P.W.) = Rs. 100;

True Discount (T.D.) = Rs. (156 - 100) = Rs. 56 = (Sum due) - (P.W.)

We define: T.D. = Interest on P.W.;     Amount = (P.W.) + (T.D.)

Interest is reckoned on P.W. and true discount is reckoned on the amount.

IMPORTANT FORMULAE

Let rate = R% per annum and Time = T years. Then,

1.   P.W. =	100 x Amount	=	100 x T.D.
	100 + (R x T)		R x T

2.   T.D. =	(P.W.) x R x T	=	Amount x R x T
	100		100 + (R x T)

3.   Sum =	(S.I.) x (T.D.)
	(S.I.) - (T.D.)

4.   (S.I.) - (T.D.) = S.I. on T.D.

5.   When the sum is put at compound interest, then P.W. =	Amount
	1 + R T 100

QUANTS – 1 (Number system)

1. Find the ratio between the LCM and HCF of 5, 15 and 20?
(a) 8 : 1 (b) 14 : 3 (c) 12 : 2 (d) 12 : 1
2. Find the units digit of the expression 256^251 + 36^528 + 73^54
(a)4 (b) 0 (c) 6 (d) 5
3. If 146! is divisible by 5^n, then find the maximum value of n.
(a)34 (b) 35 (c) 36 (d) 37
4. The number of divisors of 1420.
(a)14 (b) 15 (c) 13 (d) 12
5. There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys
or girls alone. Find the total number of sections thus formed?
(a) 24 (b) 32 (c)16 (d) None of these
6. Find the remainder when the number 9^100 is divided by 8?
(a)1 (b) 2 (c) 0 (d) 4
7. Find the number of zeroes at the end of 50!
(a)13 (b) 11 (c) 5 (d) 12
8. The square of a number greater than 1000 that is not divisible by three, when divided by three,
leaves a remainder of
(a)1 always (b) 2 always (c) either 1 or 2 (d) Cannot be said
9. The product of a natural number by the number ten by the same digits in the reverse order is 2430
Find the numbers.
(a) 54 and 45 (b) 56 and 65 (c) 53 and 35 (d) None of these
10. Find the pairs of natural numbers the difference of whose squares is 55.
(a) 28 and 27 or 8 and 3
(b) 18 and 17 or 18 and 13
(c) 8 and 27 or 8 and 33
(d) None of these
11. The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM
of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.
(a) 6 and 54 (b) 8 and 56 (c) 12 and 60 (d) 7 and 55
12. |x-3| + 2|x+1| = 4 find the possible integral value of x
(a) 1 (b) -1 (c)3 (d) There are many solutions
13. The sum of two numbers is 20 and their geometric mean is 20% lower than their arithmetic mean.
Find the ratio of the numbers.
(a) 4 : 1 (b) 9 : 1 (c) 1: 1 (d) 17 : 3
14. the expression 333^555+ 555^333 is divisible by
(a)2 (b)3 (c)37 (d) All of these
EEE/CPC/R/A1/2
15. [x] denotes the greatest integer value just below x and {x} its fractional value. The sum of [x]^2 and
{x}^1 is 25.16. Find x.
(a) 5.16 (b) —4.84 (c) Both a and b (d) Cannot be determined
16. Find the pairs of a natural number whose greatest common divisor is 5 and the least common
multiple is 105.
(a) 5 and105 or 15 and 35
(b) 6 and105 or 16 and 35
(c) 5 and15 or 15 and 135
(d) None of these
17. A two digit number exceeds by 19 the sum of the squares of its digits and by 44 the double product
of its digits. Find the number.
(a)72 (b) 62 (c) 22 (d) 12
18. The power of 45 that will exactly divide 123! is
(a)29 (b) 30 (c) 31 (d) 59
19. The smallest natural number n such that n! is divisible by 990.
(a) 3 (b) 5 (c) 11 (d) None of these
20. √x * √y = √(xy) is true only when
(a) x > 0, y > 0 (b) x <0 and y > 0 (c) x > 0 and y < 0 (d) All of these
21. Find the last two digits of: 15 x 37 x 63 x 51 x 97 x 17
(a) 35 (b) 45 (c) 55 (d) 85
22. Find the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7.
(a) 686 (b) 676 (c) 666 (d) 656
23. Find the lowest of three numbers as described: If the cube of the first number exceeds their product
by 2, the cube of the second number is smaller than their product by 3, and the cube of the third
number exceeds their product by 3.
(a)3^(1/3) (b) 9^(1/3) (c) 2 (d) Any of these
24. In what base (121)x is a perfect square no
(a)2 (b)3 (c) 4 (d) all possible base greater than or equal to 3
25. How many pairs of natural numbers are there the difference of whose squares is 45.
(a) 1 (b) 2 (c) 3 (d) 4
26. Find the two-digit number if it is known that the ratio of the required number and the sum of its
digits is 8 as also the quotient of the product of its digits and that of the sum is 1419.
(a) 54 (b) 72 (c) 27 (d) None of these
27. The last three-digits of the multiplication 12345 *54321 will be
(a) 865 (b) 745 (c)845 (d) 945
28. Find the gcd (111... 11 hundred ones ; 11... 11 sixty ones).
(a) 111 forty ones (b) 111... twenty five ones (c) 111.. .twenty ones (d) None of these
29. Q=(361)x400x441x484x529. The number of factors for Q =
(a) 5103 (b) 1701 (c) 32 (d) None of these
30. What is the remainder when 222….upto 1000 digits is divided by 3125.
(a) 222 (b) 347 (c) 472 (d) 722

Key

1. d 2. b 3. b 4. d 5. c 6. a 7. d 8. a 9. a 10. a 11. a 12. b 13. a 14. d 15. c 16 a 17. a 18. a 19. c 20. a
21. a 22. b 23. a 24. d 25. c 26. b 27. b 28. c 29. a 30. b

QUANTS – 2 (Sequence and Series)

1. If log (0.5), log (2x — 1) and log (2x + 5) are in arithmetic progression then x =
(1) 3/2 (2) log23 (3) log32 (4) - log32
2. The sum of the squares of three terms of a geometric progression, equals the square of the sum of
those three terms. Find the sum of the three terms.
(1) 0 (2) 1 (3) 2 (4) 3
3. The number of factors of 32, 24,405 are in
(1) arithmetic progression (2) geometric progression
(3) harmonic progression (4) None of the above.
4. If 1/(1+x),1/(1-x), 4/(1-x^2) are in arithmetic progression, then
(1) x lies between 0 and 2. (2) x lies between —2 and 0.
(3) x lies between -2 and 2. (4) There are no values of x.
5. A quadratic equation is such that the arithmetic mean of its roots equals their geometric mean. One
of the roots is a prime number divisible by 2. What is the equation?
(1) x^2-3x + 2 = 0 (2) x^2-4x = 4 (3) x^2-4x = -4 (4) x^2-x -2 = 0
6. If a+ b, b + c and c + a are in harmonic progression, then
(1) a2, b2 and c2 will be in arithmetic progression.
(2) a2, bc and c2 will harmonic progression.
(3) 2ab, a2 and b2 will in arithmetic progression.
(4) b2, a2 and c2 will in arithmetic progression.
7. Given
   1          1         1          1
  —   +    —   +   —   +    —   +….∞ terms = k
 1.3        2.5      3.7       4.9
where k is a side of a rectangle. If 1 - (1/2) + (1/3) – (1/4) is another side of the rectangle, what
would be the cost of fencing (in Rs.) the rectangle, at Rs.2 per unit?
(1) k+2 (2) 2(k+2) (3) k+1 (4) 2(k+l)
8. Each of the quantities |a|, |b| and |c| is less than 1, and a, b, c are in arithmetic progression. If
x =1+a+ a2+….∞ terms, y =1+b+ b2+….∞ terms and z =1+c+ c2+….∞ terms,
then 1/x, 1/y, 1/z are in _____
(1) geometric progression (2) arithmetic progression
(3) harmonic progression (4) None of these
9. If a, b, c are in. geometric progression, what is the ratio of the squares of the sum of the roots and the
difference of the roots of the equation
a x^2+ bx + c = 0 is
(1) 1:2 (2) 1:3 (3) 1: 4 (4) None of these
10. Let a and b be two consecutive positive integers which are prime. What should be the value of the
integer c such that 1/a, 1/b, 1/c are in arithmetic progression?
(1)1 (2) 6 (3) 2 (4) Cannot be determined
11. Given that a, 5, b are in arithmetic progression and a, 4, b are in geometric progression, then what is
the difference between a and b?
(1) 8 (2) 2 (3) 6 (4) 10
12. If the sides of a right-angled triangle are in arithmetic progression, then the minimum possible
difference between the magnitudes of the area (in cm the perimeter (in cm) of such a triangle is:
(1) 0 (2) 6 (3) 12 (4) 4.5
13. If a, b, c, d are in arithmetic progression with a not equal to b. Then ( c- b) / (d –a) is
(1) 1: 2 (2) 1:3 (3) 1: 1 (4) Dependent on a, b, c, d
14. lf (a+b+c)2 = (a + c)2+ 2 (ab +bc +ca), then
(1) a, b, c are in geometric progression (2) a, 2b, c are in geometric progression
(3) a, b, 2c are in geometric progression (4) None of the above
15. Find the product of terms (1-100)*(2-100)*(3-100)…..(1000-100)
(1)999000 (2) 990000 (3) 9990000 (4) 0
16. a, b, c are the length of the sides (in cm) of a right-angled triangle where ‘a’ denotes the shortest side
and ‘c’ denotes the hypotenuse. Given that b, 2a, c are in arithmetic progression and that the
hypotenuse and the longer of the two perpendicular sides differ by 2, find the value of b.
(1) 4 (2) 15 (3) 8 (4) 24
17. The sum of first 2n natural numbers is equal to one third the sum of the squares of first n natural
numbers. Find the value of n.
(1) 16 (2) 17 (3) 18 (4) 19
18. The sum to infinity of the series
(1/2) + (3/4) + (5/8) + (7/16) + ….∞ is
(1) 3/2 (2) 3 (3) 3/4 (4) 3/8
19. lf |x|<1and1 + 4x +10x^2+20x3+ ….∞ = 16 . Find x.
(1) 1/2 (2) 1/3 (3) 1/4 (4) 1/5
20. If 3a, b, c are in geometric progression, then one root of 3ax^2 + 2bx + c=0 is
(1) c/b (2) - c/b (3) b/3a (4) None of these
21. Sum of first 9 terms of an arithmetic progression is 54. What is the sum of first 10 terms if the first
term and the common difference are positive integers?
(1) 65 (2) 66 (3) 63 (4) Data insufficient
22. If the product of the eighteenth and forty ninth terms of a geometric progression is 36, what is the
product of the first sixty six terms of the progression?
(1) 3633 (2) 3666 (3) 3622 (4) Cannot be determined
23. Let a, b be 2 geometric means between 2 positive numbers x, y. Then (a + y)/(b + x) equals
(1) the common ratio. (2) the reciprocal of the common ratio.
(3) twice the common ratio (4) the negative of the common ratio.
EEE/CPC/R/A2/3
24. Karthikeyan who was angry with Alok started pulling his hair. First he pulled out a strand of hair,
then 2, then 3 and so on upto 7 strands. He realized that every time he pulled out ‘N’ strands, ‘N’
more strands came out along with them. He decided to stop pulling his hair at that time. How many
strands did he pull out, in total?
(1) 28 (2) 56 (3) 50 (4) 35
25. Find the sum of the first 20 terms of the series
(1)(3) + (3)(5) + (5)(7) + (7)(9) +…..∞
(1) 11460 (2) 11640 (3) 11480 (4) 11860
26. If every term of an infinite geometric progression is one-third the sum of the terms that follow, then
what is the common ratio of the series?
(1) 1/2 (2) 2/3 (3) 3/4 (4) 4/5
27. If the sum of first three odd numbered terms and first three even numbered terms of an infinite
geometric series are 6 and 4 respectively, what is the sum of the series?
(1) 12 (2) 15 (3) 1458/133 (4) 486/47
28. What is the sum of the product of the all possible pairs of distinct numbers of the first ten natural
numbers?
(1) 2640 (2)1320 (3) 1705 (4) 1920
29. There are five numbers in arithmetic progression with the second and fourth terms in the ratio 2: 5. If
the sum of the squares of the five numbers is 335, what is the square of their sum?
(1) 625 (2) 900 (3) 1225 (4) 1600
30. What is the sum Of the first ten terms of the series (2)(3)(4) + (3)(4)(5) + (4)(5)(6)…
(1) 4000 (2) 4500 (3) 5000 (4) 6000

Apti Test - I

1.  2.	A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is :
	A. 1 4 B. 1 10 C. 7 15 D. 8 15
	A can lay railway track between two given stations in 16 days and B can do the same job in 12 days. With help of C, they did the job in 4 days only. Then, C alone can do the job in:
	A. 9 1 days 5 B. 9 2 Days 5 C. 9 3 days 5 D.         10
3..  A man purchased a cow for Rs. 3000 and sold it the same day for Rs. 3600, allowing the buyer a credit of 2 years. If the rate of interest be 10% per annum, then the man has a gain of: A. 0% B. 5% C. 7.5% D. 10%

4.     Kamya purchased an item of Rs. 46,000 and sold it at loss of 12 per cent. With that amount she purchased another item and sold it at a gain of 12 per cent. What was her overall gain/loss?

 (A) Loss of Rs. 662.40                                      (B) Profit of Rs. 662.40

(C) Loss of Rs. 642.80                           (D) Profit Rs. 642

5.     Two numbers differ by 2 are co-primes then

a.     Both are primes                   b.         One of them should be prime

c.     None of them are primes     d.         We cannot say anything

6.      The ratio between the three angles of a quadrilateral is 13 : 9 : 5 respectively. The value of the fourth angle of the quadrilateral is 36°. What is the difference between the largest and the second smallest angles of the quadrilateral?

            (A) 104°                                                (B) 108°

(C) 72°                                                  (D) 96°

7.     65% of root (𝟑𝟏𝟑𝟔)  × 5 =? ÷ 154

(A) 56                           (B) 28

(C) 35                           (D) 32

8.     Five-ninths of a number is equal to twenty five percent of the second number. The second number is equal to one-fourth of the third number. The value of the third number is 2960. What is 30 per cent of the first number?

(A) 88.8                                     (B) 99.9

C) 66.6                                      (D) Cannot be determined

9.     The ratio of the adjacent angles of a parallelogram is 7 : 8. Also, the ratio of the angles of quadrilateral is 5 : 6 : 7 : 12. What is the sum of the smaller angle of the parallelogram and the second largest angle of the quadrilateral?

(A) 168°                        (B) 228°

(C) 156°                        (D) 224°

10.  In the above figure bus will move in what direction

A.	Left	B.	Right
C.	Will not move	D.	Cannot be determined

2. 

	11.  A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively?
	A. 2 : 1 B. 3 : 2 C. 8 : 3 D. Cannot be determined
	Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:
	A. 2 : 5 B. 3 : 5 C. 4 : 5 D. 6 : 7

13.  Grapes contain 80% of water and 20% of pulp and kismiss contain 20% water and 80% of pulp. To get 18kg of kismiss how many kg of grapes required

		A.	70	B.	72
		C.	74	D.	76
	14.  Present ages of Sameer and Anand are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Anand's present age in years?
	A. 24 B. 27 C. 40  D.      None of these

15.  How many squares are there in 5x7 grid

A.	60	B.	70
C.	85	D.      90

16.  What is the circumference of the circle that passes through all the corners of the hexagon of side 3.5 m

A.	22 cm	B.	11m
C.	16.5m	D.	None of the above

17.  Reminder given by 47²³⁷ + 22²³⁷when divided by 23

A.	7	B.	18
C.	0	D.	11

18.  If 145! Is divisible by 5ⁿ then the maximum value of ‘n’ possible

A.	34	B.	35
C.	36	D.	37

19.  There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys (or) girls alone. The minimum no. of sections possible

A.	24	B.	32
C.	16	D.	None of the above

20.  In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected is

A.	21 46	B.	25 117
C.	10 50	D.	7 23

21.  No of time in a day that hands of the clock have 153⁰angle between them

A.	24	B.	22
C.	48	D.	44

22.  Angle between the hands of the clock when the time is 3:40 PM

A.	120	B.	144
C.	130	D.	None of the above

23.  In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 run

A.	6.25	B.	6.5
C.	6.75	D.	7

24.  Mr. Thomas invested an amount of Rs. 13,900 divided in two different schemes A and B at the simple interest rate of 14% p.a. and 11% p.a. respectively. If the total amount of simple interest earned in 2 years be Rs. 3508, what was the amount invested in Scheme B?

A.	Rs: 6400	B.	Rs: 6.5K
C.	Rs: 7200	D.	Rs: 7500

25.  How many days are there in x weeks x days?

A.	7x2	B.	8x
C.	8x-(7-x)	D.	7x+7

26.  On 8^th Feb, 2005 it was Tuesday. What was the day of the week on 8^th Feb, 2004?

A.	Sunday	B.	Monday
C.	Tuesday	D.	Wednesday

27.  Two friends A & B have to reach a point Y staring from X which are the two end of the diagonal of a square. A walked along the diagonal and B along the sides. What the percentage of energy saved by A than B  

A.	44.14%	B.	41.4%
C.	29.3%	D.	50%

28.  Which of the fallowing is not a prime number     

A.	213-1	B.	83
C.	311-1	D.	231-1

29.  Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?

		A.	2:3:4	B.	6:7:8
		C.	6:8:9	D.	None of the above
	30.  If 6 men and 8 boys can do a piece of work in 10 days while 26 men and 48 boys can do the same in 2 days, the time taken by 15 men and 20 boys in doing the same type of work will be:
	A. 4 days B. 5 days C. 6 days D. 7 days