Compound Interest - Aptitude Concepts (Important Facts And Formulae)

Compound Interest - Aptitude Concepts (Important Facts And Formulae)

Monday, March 17, 2014 /

IMPORTANT FACTS AND FORMULAE

Compound Interest: Sometimes it so happens that the borrower and the lender agree to fix up a certain unit of time, let say it yearly or half-yearly or may be quarterly to settle the previous account.

In such cases, amount after the first unit of time becomes the principal for second unit, amount after second unit becomes the principal for the third unit and so on.

After a specified period of time, the difference between the amount and the money borrowed is called the Compound Interest (abbreviated as C.I.) for that particular period.

Let Principal = P, Rate = R% per annum, Time = n years.

I. When interest is compound Annually:
   

Amount = P(1+R/100)^n

II. When interest is compounded Half-yearly:

     Amount = P[1+(R/2)/100]^2n

III. When interest is compounded Quarterly:

      Amount = P[ 1+(R/4)/100]^4n

IV. When interest is compounded AnnuaI1y but time is in fraction, say 3(2/5) years.

     Amount = P(1+R/100)^3 x (1+(2R/5)/100)

V. When Rates are different for different years, say Rl%, R2%, R3% for 1st, 2nd and 3rd year respectively.
     Then,

Amount = P(1+R1/100)(1+R2/100)(1+R3/100)

VI. Present worth of Rs.x due n years hence is given by :

Present Worth = x/(1+(R/100))^n

Volume And Surface Area - Aptitude Concepts (Important Facts And Formulae)

Volume And Surface Area - Aptitude Concepts (Important Facts And Formulae)

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IMPORTANT FORMULAE

I. CUBOID

Let length = 1, breadth = b and height = h units. Then,

1. Volume = (1 x b x h) cubic units.

2. Surface area= 2(lb + bh + lh) sq.units.

3. Diagonal.=l^2 +b^2 +h^2 units

II. CUBE

Let each edge of a cube be of length a. Then,

1. Volume = a3 cubic units.

2. Surface area = 6a2 sq. units.

3. Diagonal = 3 a units.

III. CYLINDER

Let radius of base = r and Height (or length) = h. Then,

1. Volume = (pi r^2h) cubic units.

2. Curved surface area = (2pi rh). units.

3. Total surface area =2pi*r (h+r) sq. units

IV. CONE

Let radius of base = r and Height = h. Then,

1. Slant height, l = h^2+r^2

2. Volume = (1/3) pi*r2h cubic units.

3. Curved surface area = (pi*rl) sq. units.

4. Total surface area = (pi*rl + pi*r2 ) sq. units.

V. SPHERE

Let the radius of the sphere be r. Then,

1. Volume = (4/3)pi*r3 cubic units.

2. Surface area = (4pi*r2) sq. units.


VI. HEMISPHERE

Let the radius of a hemisphere be r. Then,

1. Volume = (2/3)pi*r3 cubic units.

2. Curved surface area = (2pi*r2) sq. units.

3. Total surface area = (3pi*r2) units.

    Remember: 1 litre = 1000 cm3.

Percentage - Aptitude Concepts (Important Facts And Formulae)

Percentage - Aptitude Concepts (Important Facts And Formulae)

Wednesday, March 12, 2014 /

IMPORTANT FACTS AND FORMULAE

1. Concept of Percentage : By a certain percent ,we mean that many hundredths. Thus the x percent means x hundredths and can be written as x%.

To express x% as a fraction : We have , x% = x/100.

Thus, 

        20% =20/100 =1/5; 

        48% =48/100 =12/25, etc.

To express a/b as a percent : We have, a/b =((a/b)*100)%.

Thus, 

        ¼ =[(1/4)*100] = 25%; 

        0.6 =6/10 =3/5 =[(3/5)*100]% =60%.

2. If the price of a particular commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is 

[R/(100+R))*100]%.

If the price of a particular commodity decreases by R%, then the increase in consumption so as to decrease the expenditure is 

[(R/(100-R)*100]%.

3. Results on Population : Let the population of the city be P now, and suppose it increases at the rate of R% per annum.  Then :

I. Population after n years = P [1+(R/100)]^n.

II. Population n years ago = P /[1+(R/100)]^n.

4. Results on Depreciation : Let the present value of a particular machine be P. Let it depreciates at the rate R% per annum. Then,

I. Value of the machine after n years = P[1-(R/100)]n.

II. Value of the machine n years ago = P/[1-(R/100)]n.

5. If A is R% more than B, then B is less than A by

[(R/(100+R))*100]%.

If A is R% less than B , then B is more than A by

[(R/(100-R))*100]%.

H.C.F. AND L.C.M. OF NUMBERS - Basic Concepts

H.C.F. AND L.C.M. OF NUMBERS - Basic Concepts

Tuesday, March 11, 2014 /

IMPORTANT FACTS AND FORMULAE

I. Factors and Multiples :- If a number A(say) divides another number B(say) exactly, in that case we say that A is a factor of B and in this case, B is called a multiple of A.

II. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.): The H.C.F.(Highest Common Factor) of two or more than two numbers is the greatest number that divides each of them exactly.

There are mainly two methods for finding the H.C.F. of a given set of numbers and these are as follows:

1. Factorization Method : Express the each one of given numbers as the product of prime factors.The product of least powers of common prime factors gives the H.C.F.

2. Division Method: Suppose we have to find the H.C.F. of two given numbers. 

1. Divide the larger number by the smaller one. 
2. Now, divide the divisor by the remainder. 
3. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. 
4. The last divisor is the required H.C.F.

Finding the H.C.F. of more than two numbers : In the case where we have to find the H.C.F. of three numbers, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers and Similarly, the H.C.F. of more than three numbers may be obtained.

III. 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(Least Common Multiple).

There are mainly two methods for finding the L.C.M. of a given set of numbers and these are as follows:

1. Factorization Method of Finding L.C.M.: 

1. Resolve the each one of given numbers into a product of the prime factors. 
2. Then, the L.C.M. is the product of highest powers of all factors.

2. Common Division Method or Short-cut Method for Finding L.C.M.: 

1. Arrange given numbers in a row in any order. 
2. Divide by a number which divides exactly at least two of the given numbers & carry forward the numbers which are not divisible. 
3. Repeat above process till no two of the numbers are divisible by the same number except 1. 
4. The product of divisors and the undivided numbers is required L.C.M. of the given numbers.

IV. Product of two numbers =Product of their H.C.F. and L.C.M.

V. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.

VI. 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 

Numbers - Important Facts And Formulae

Numbers - Important Facts And Formulae

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IMPORTANT FACTS AND FORMULAE 

I. Numeral : In Hindu Arabic system, there are ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and these symbols called digits to represent any number.

A group of digits that denoting a number is called a numeral.

We can represent a number, say 689745132 as shown below :

We read it as :- 'Sixty-eight crores, ninety-seven lacs, forty-five thousand, one hundred and thirty-two'.

II. Local Value or Place Value of a Digit in a Numeral : 

In the above numeral : 

Place value of 2 is (2 x 1) = 2; Place value of 3 is (3 x 10) = 30; 

Place value of 1 is (1 x 100) = 100 and so on. 

Place value of 6 is 6 x 108 = 600000000 

III.Face Value : It is the value of the digit itself in a numeral at whatever place it may be. In the above numeral, 

face value of 2 is 2; 

face value of 3 is 3 and so on

IV.TYPES OF NUMBERS :

1.Natural Numbers : Counting numbers 1, 2, 3, 4, 5, 6, 7..... are called natural numbers. 

2.Whole Numbers : All the counting numbers together with the zero form the set of whole numbers. Thus, 

 (i) 0 is only the whole number which is not a natural number(zero is not a natural number). 

 (ii) Every natural number is a whole number(except zero). 

3.Integers : All the natural numbers, 0 and negatives of counting numbers i.e., {…, - 3 , - 2 , - 1 , 0, 1, 2, 3,…..} together form the set of the integers. 

(i) Positive Integers : {1, 2, 3, 4, 5 …..} is the set of all positive integers. 

(ii) Negative Integers : {- 1, - 2, - 3, -4…..} is the set of all negative integers. 

(iii) Non-Positive and Non-Negative Integers : 0 is neither positive nor negative. So, {0, 1, 2, 3, 4….} represents the set of non-negative integers, while {0, - 1 , - 2 , - 3 , -4 …..} represents the set of non-positive integers. 

4. Even Numbers : A number which is divisible by 2 is called an even number, e.g., 2, 4, 6, 8, 10, 12 etc. 

5. Odd Numbers : A number which is not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc. 

6. Prime Numbers : A number that is greater than 1 is called a prime number and if it has exactly two factors, namely 1 and the number itself. 

Prime numbers up to 100 are : 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. 

7.Composite Numbers : Numbers that are greater than 1 which are not prime, are known as the composite numbers, e.g., 4, 6, 8, 9, 10, 12. 

Note : (i) 1 is neither prime nor composite. 

(ii) 2 is the only even number which is prime. 

(iii) There are 25 prime numbers between 1 and 100. 

8. Co-primes : Two numbers say a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes, 

V. TESTS OF DIVISIBILITY 

1. Divisibility By 2 :- A number is divisible by 2, if its unit's digit is any of :- 0, 2, 4, 6, 8. 

Ex. 849372 is divisible by 2, while 659315 is not. 

2. Divisibility By 3 :- A number is divisible by 3, if the sum of its digits is divisible by 3. 

Ex.5924823 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2+3) = 33, which 

is divisible by 3. 

But, 8643292 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9+2) = 34, 

which is not divisible by 3. 

3. Divisibility By 4 :- A number is divisible by 4, if the number formed by the last two digits is divisible by 4. 

Ex. 892644 is divisible by 4, since the number formed by the last two digits is 44, which is divisible by 4. 

But, 749281 is not divisible by 4, since the number formed by the last two digits is 81, which is not divisible by 4. 

4. Divisibility By 5 :- A number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 

20820 and 50345 are divisible by 5, while 30934 and 40946 are not. 

5. Divisibility By 6 :- A number is divisible by 6, if it is divisible by both 2 and 3. Ex. The number 35256 is clearly divisible by 2. 

Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6. 

6. Divisibility By 8 :- A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8. 

Ex. 953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8. 

But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8. 

7. Divisibility By 9 :- A number is divisible by 9, if the sum of its digits is divisible by 9. 

Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9. 

But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by 9. 

8. Divisibility By 10 :- A number is divisible by 10, if it ends with 0.

Ex. 96410, 10480 are divisible by 10, while 96375 is not. 

9. Divisibility By 11 :- A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11. 

Ex. The number 4832718 is divisible by 11, since : 

(sum of digits at odd places) - (sum of digits at even places) - (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11. 

10. Divisibility By 12 :- A number is divisible by 12, if it is divisible by both 4 and 3. 

Ex. Consider the number 34632. 

(i) The number formed by last two digits is 32, which is divisible by 4, 

(ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is 

divisible by 4 as well as 3. Hence, 34632 is divisible by 12. 

11. Divisibility By 14 :- A number is divisible by 14, if it is divisible by 2 as well as 7. 

12. Divisibility By 15 :- A number is divisible by 15, if it is divisible by both 3 and 5. 

13. Divisibility By 16 :- A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16. 

Ex.7957536 is divisible by 16, since the number formed by the last four digits is 7536, 

which is divisible by 16. 

14. Divisibility By 24 :- A given number is divisible by 24, if it is divisible by both3 and 8. 

15. Divisibility By 40 :- A given number is divisible by 40, if it is divisible by both 5 and 8. 

16. Divisibility By 80 :- A given number is divisible by 80, if it is divisible by both 5 and 16. 

Note : If a number is divisible by p as well as q, where p and q are co-primes, then the given number is divisible by pq. 

If p arid q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q. 

Ex. 36 is divisible by both 4 and 6, but not divisible by (4x6) = 24, since 4 and 6 are not co-primes.