Job Alert ExamMission: October 2017

Clock - Important Formulas |Aptitude

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.
1. In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.
2. In every hour, both the hands coincide once.
3. The hands are in the same straight line when they are coincident or opposite to each other.
4. When the two hands are at right angles, they are 15 minute spaces apart.
5. When the hands are in opposite directions, they are 30 minute spaces apart.
6. Angle traced by hour hand in 12 hrs = 360°
7. Angle traced by minute hand in 60 min. = 360°.
8. 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.

Problems on Ages - Important Formulas | Aptitude

Important Formulas on "Problems on 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

Partnership - Important Formulas |Aptitude

Partnership:
When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.
Ratio of Divisions of Gains:
1. 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.
2. 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.

Permutation and Combination - Important Formulas |Aptitude

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:
1. We define 0! = 1.
2. 4! = (4 x 3 x 2 x 1) = 24.
3. 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:
1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
2. 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:
ⁿP_r = n(n - 1)(n - 2) ... (n - r + 1) = n!
(n - r)!
Examples:
1. ⁶P₂ = (6 x 5) = 30.
2. ⁷P₃ = (7 x 6 x 5) = 210.
3. 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 p_r are alike of r^th kind,
such that (p₁ + p₂ + ... p_r) = n.
Then, number of permutations of these n objects is = n!
(p₁!).(p₂)!.....(p_r!)
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:
1. 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.
2. All the combinations formed by a, b, c taking ab, bc, ca.
3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
4. Various groups of 2 out of four persons A, B, C, D are:
  AB, AC, AD, BC, BD, CD.
5. 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:
ⁿC_r = n! = n(n - 1)(n - 2) ... to r factors .
(r!)(n - r)! r!
Note:
1. ⁿC_n = 1 and ⁿC₀ = 1.
2. ⁿC_r = 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.
3! 3 x 2 x 1

Area - Important Formulas |Aptitude

FUNDAMENTAL CONCEPTS

Results on Triangles:
1. Sum of the angles of a triangle is 180°.
2. The sum of any two sides of a triangle is greater than the third side.
3. Pythagoras Theorem:
  In a right-angled triangle, (Hypotenuse)² = (Base)² + (Height)².
4. The line joining the mid-point of a side of a triangle to the positive vertex is called the median.
5. 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.
6. In an isosceles triangle, the altitude from the vertex bisects the base.
7. The median of a triangle divides it into two triangles of the same area.
8. 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:
1. The diagonals of a parallelogram bisect each other.
2. Each diagonal of a parallelogram divides it into triangles of the same area.
3. The diagonals of a rectangle are equal and bisect each other.
4. The diagonals of a square are equal and bisect each other at right angles.
5. The diagonals of a rhombus are unequal and bisect each other at right angles.
6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
7. 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 r = .
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

Square Root and Cube Root - Important Formulas |Aptitude

Square Root:
If x² = y, we say that the square root of y is x and we write y = x.
Thus, 4 = 2, 9 = 3, 196 = 14.
Cube Root:
The cube root of a given number x is the number whose cube is x.
We, denote the cube root of x by x.
Thus, 8 = 2 x 2 x 2 = 2, 343 = 7 x 7 x 7 = 7 etc.
Note:
1. xy = x x y
2.
x
y
= x = x x y = xy .
y y y y

Decimal Fraction - Important Formulas | Aptitude

Decimal Fractions:
Fractions in which denominators are powers of 10 are known as decimal fractions.
Thus, 1 = 1 tenth = .1; 1 = 1 hundredth = .01;
10 100
99 = 99 hundredths = .99; 7 = 7 thousandths = .007, etc.;
100 1000
Conversion of a Decimal into Vulgar Fraction:
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
Thus, 0.25 = 25 = 1 ; 2.008 = 2008 = 251 .
100 4 1000 125
Annexing Zeros and Removing Decimal Signs:
Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
Thus, 1.84 = 184 = 8 .
2.99 299 13
Operations on Decimal Fractions:
1. Addition and Subtraction of Decimal Fractions: The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
2. Multiplication of a Decimal Fraction By a Power of 10: Shift the decimal point to the right by as many places as is the power of 10.
  Thus, 5.9632 x 100 = 596.32; 0.073 x 10000 = 730.
3. Multiplication of Decimal Fractions: Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.
  Suppose we have to find the product (.2 x 0.02 x .002).
  Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.
  .2 x .02 x .002 = .000008
4. Dividing a Decimal Fraction By a Counting Number: Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend.
  Suppose we have to find the quotient (0.0204 Ã· 17). Now, 204 Ã· 17 = 12.
  Dividend contains 4 places of decimal. So, 0.0204 Ã· 17 = 0.0012
5. Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
  Now, proceed as above.
  Thus, 0.00066 = 0.00066 x 100 = 0.066 = .006
  0.11 0.11 x 100 11
Comparison of Fractions:
Suppose some fractions are to be arranged in ascending or descending order of magnitude, then convert each one of the given fractions in the decimal form, and arrange them accordingly.
Let us to arrange the fractions 3 , 6 and 7 in descending order.
5 7 9
Now, 3 = 0.6, 6 = 0.857, 7 = 0.777...
5 7 9
Since, 0.857 > 0.777... > 0.6. So, 6 > 7 > 3 .
7 9 5
Recurring Decimal:
If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.
n a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set.
Thus, 1 = 0.333... = 0.3; 22 = 3.142857142857.... = 3.142857.
3 7
Pure Recurring Decimal: A decimal fraction, in which all the figures after the decimal point are repeated, is called a pure recurring decimal.
Converting a Pure Recurring Decimal into Vulgar Fraction: Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.
Thus, 0.5 = 5 ; 0.53 = 53 ; 0.067 = 67 , etc.
9 99 999
Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
Eg. 0.1733333.. = 0.173.
Converting a Mixed Recurring Decimal Into Vulgar Fraction: In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated. In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.
Thus, 0.16 = 16 - 1 = 15 = 1 ; 0.2273 = 2273 - 22 = 2251 .
90 90 6 9900 9900
Some Basic Formulae:
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.

Job Alert ExamMission

Sunday, October 1, 2017