INTRODUCTION
1. PERCENTAGE
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1. PERCENTAGE
Hundredth Part:
Out of 100 equal parts, each part is known as its hundredth part.
Thus 6 hundredths = 6/100, 17 hundredths = 17/100 etc.
Percentage:
By a certain percentage, we mean that many hundredths.
We denote x percent by x% .
Thus x% = x hundredths = x/100
54% = 54 hundredths = 54/100
TO CONVERT A PERCENTAGE INTO A FRACTION
We have x% = x/100
Rule: For converting a percentage into a fraction, divide if by 100 and remove the sign %
TO CONVERT A FRACTION INTO A PERCENTAGE
Rule: For converting a fraction into a percentage multiply the fraction by 100 and put the sign %.
PERCENTAGE AS A RATIO
Rule: A percentage can be expressed as ratio with first term equal to given percentage and second term 100.
PERCENTAGE AS A DECIMAL
Rule: First convert the given percentage as a fraction and then convert this fraction into decimal form.
RATIO AS A PERCENTAGE
Rule: First write the ratio as a fraction and then multiply with 100 to get percentage.
DECIMAL AS A PERCENTAGE
Rule: Convert the given decimal into a fraction and then multiply by 100 to get percentage.
TWO IMPORTANT RESULTS
2. PROFIT AND LOSS​
COST PRICE : The price at which an article is purchased is called cost price (C.P.).
SELLING PRICE : The price at which an article is sold is called selling price (S.P.).
PROFIT : If the selling price of an article is more than the cost price then the seller is said to have made a profit, where Profit = S.P. – C.P.
LOSS : If the selling price of an article is less than the cost price then the seller is said to have incurred a loss, where Loss = C.P. – S.P.
PERCENTAGE PROFIT : It is always reckoned on C.P.
PERCENTAGE LOSS : It is always reckoned on C.P
. 
COST PRICE IN TERMS OF PROFIT :
COST PRICE IN TERMS OF LOSS :
SELLING PRICE IN TERMS OF PROFIT :
SELLING PRICE IN TERMS OF LOSS :
SALE at equal gain and loss % : When a person sells two identical 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:
3. TIME AND WORK​
- If A can do a piece of work in x days, then A's 1 day's work = 1/x.
- If A's work efficiency is k times as compared to B’s then Ratio of work done by A and B is k :1.
- If A can do a piece of work in x days and B can do the same piece of work in y days, then in 1 day together A and B can do part of work.
do part of work.- If A, B and C can do a work in x, y and z days respectively then all of them working together can do the work in days.
days.4. RATIO AND PROPORTION​
RATIO : Ratio of two quantities a and b is given by a:b. Here a and b are called antecedent and consequent respectively.
PROPORTION : When two ratio are equal to each other then they are said to be proportional.
Here a and d are called extreme terms while b and c are called mean terms.
Numerically, Product of Means = Product of Extremes
Properties and different types of Ratio and Proportion:
- If each term of the ratio is multiplied by a non-zero quantity then the ratio does not change.
- If each term of the ratio is divided by a non-zero quantity then the ratio does not change.
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Compounded Ratio : The compounded ratio of a:b, c:d and e:f is ace:bdf.
- Duplicate Ratio: Duplicate ratio of a:b is a^2:b^2.
- Sub-duplicate Ratio: Sub-duplicate ratio of a:b is a^1/2:b^1/2.
- Triplicate Ratio: Triplicate ratio of a:b is a^3:b^3.
- Sub-triplicate Ratio: Sub triplicate ratio of a:b is a^1/3:b^1/3.
- Componendo and Dividendo :
| if |  |
- For two or more equal ration, each ratio is equal to Sum of antecedents: Sum of consequents.
- Third Proportional: If a:b::c:d then cis called the third proportional.
- Fourth Proportional: If a:b::c:d then dis called the fourth proportional.
- Mean Proportional: If a:x::x:b then x is called the mean proportional where x = ab.
- Direct Variation: If the variations in x is directly proportional to the variations in y then where k is called the constant of proportionality.
where k is called the constant of proportionality.- Indirect Variation: If the variations in x is indirectly proportional to the variations in y then , where k is called the constant of proportionality.
, where k is called the constant of proportionality.5. MIXTURE AND ALLIGATION
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: It is the cost of a unit quantity of the mixture.
- RULE OF ALLIGATION: If two ingredients are mixed, then
6. PERMUTATION AND COMBINATION
1. FUNDAMENTAL PRINCIPLE OF COUNTING
1.1 MULTIPLICATION PRINCIPLE
If an operation can be performed in ‘m’ different ways; following which a second operation can be performed in ‘n’ different ways, then the two operations in succession can be performed in m × n ways. This can be extended to any finite number of operations.
1.2 ADDITION PRINCIPLE
If an operation can be performed in ‘m’ different ways and another operation, which is independent of the first operation, can be performed in ‘n’ different ways. Then either of the two operations can be performed in (m + n) ways. This can be extended to any finite number of mutually exlusive operations.
2. FACTORIALS
If n is a natural number then the product of all natural numbers upto n is called factorial n and it is denoted by
n ! or .
Thus n ! = n (n – 1) (n – 2) ...... 3.2.1.
It is obvious to note that
= n
= n (n – 1)
= n (n – 1) (n – 2) etc.
Note : 0 ! = 1, 1 ! = 1, 2 ! = 2, 3 ! = 6, 4 ! = 2, 5 ! = 120, 6 ! = 720
3. PERMUTATION
Each of the different arrangements which can be made by taking some or all of a number of things is called a permutation.
4. COUNTING FORMULAS FOR PERMUTATIONS
4.1 WITHOUT REPETITION
(i) The number of permutations of n different things, taking r at a time is denoted by nPr or P(n, r)
then (0 < r < n)
= n(n – 1) (n – 2) ..... (n – r + 1), n ϵ N and r ϵ W
(ii) The number of arrangements of n different objects taken all at a time is nPn = n !
Note : nP1 = n, nPr = n. n–1Pr–1, nPr = (n–r+1). nPr–1, nPn = nPn–1
4.2 WITH REPETITION
(i) The number of permutations of n things taken all at a time, p are alike of one kind, q are alike of second kind and r are alike of a third kind and the rest n – (p + q + r) are all different is
(ii) The number of permutations of n different things taken r at a time when each thing may be repeated any number of times is nr.
4.3. NUMBER OF PERMUTATIONS UNDER CERTAIN CONDITIONS​
- The number of permutation of n different things taken all together when r particular things are to be place at some r given places
= n–rPn–r =
- The number of permutations of n different things taken r at a time when m particular things are to be placed at m given places = n–mPr–m.
- Number of permutations of n different things, taken r at a time, when a particular thing is to be always included in each arrangement, is.
r . n – 1Pr – 1
- Number of permutation of n different things, taken r at a time, when a particular thing is never taken in each arrangement is
n – 1Pr
- Number of permutations of n different things, taken all at a time, when m specified things always come together is
m ! × (n – m + 1)!
- Number of permutations of n different things, taken all at a time, when m specified things never come together is
n ! – m ! × (n – m + 1) !
(i) Arrangement round a circular table: The number of circular permutations of n different things taken all at a time is (n – 1) !, if clockwise and anticlockwise orders are taken as different.
(ii) Arrangement of beads or flowers (all different) around a circular necklace or garland: The number of circular permutations of n different things taken all at a time is (n –1) !, if clockwise and anticlockwise orders are taken as not different.
(iii) Number of circular permutations of n different things taken r at a time:
- Case I : If clockwise and anticlockwise orders are taken as different, then the required number of circular permutations = (nPr)/r.
- Case II : If clockwise and anticlockwise orders are taken as not different, then the required number of circular permutations = (nPr)/(2r).
(iv) Restricted Circular Permutations: When there is a restriction in a circular permutation then first of all we shall perform the restricted part of the operation and then perform the remaining part treating it similar to a linear permutation.
5. COMBINATION
Each of the different groups or selections which can be made by some or all of a number of given things without reference to the order of the things in each group is called a combination.
6. DIFFERENCE BETWEEN PERMUTATION AND COMBINATION​
| Problems of permutations | Problems of combinations |
| 1. Arrangements | Selections, choose |
| 2. Standing in a line, seated in a row | Distributed group is formed |
| 3. Problems on digits | Committee |
| 4. Problems on letters from a word | Geometrical problems |
7. COUNTING FORMULAS FOR COMBINATION
7.1 SELECTION OF OBJECTS WITHOUT REPETITION
The number of combinations of n different things taken r at a time is denoted by nCr or C (n, r) or
Then ; (0 < r < n)
; n ϵ N and r ϵ W
If r > n, then nCr = 0.
Some important results :
| (i) nCn = 1, nC0 = 1, | (ii) nCr = |
| (iii) nCr = nCn–r, | iv) nCx = nCy => x + y = n, |
| (v) nCr + nCr–1 = n+1Cr | (vii) nCr = (n – r + 1) nCr–1, |
| (vi) nCr = . n–1Cr–1, | |
| (viii) nC1 = nCn–1 = n |
7.2 SELECTION OF OBJECTS WITH REPETITION​
The total number of selections of r things from n differents things when each thing may be repeated any number of times is
n + r–1Cr
7.3 RESTRICTED SELECTION / ARRANGEMENT
(i) The number of combinations of n different things taken r at a time,
- when k particular objects occur is n – kCr – k.
- If k particular objects never occur is n – kCr.
(ii) The number of arrangements of n distinct objects taken r at a time so that k particular object are
- always included = n–kCr–k . r !
- never included = n–kCr .r !
(iii) The number of combinations of n objects, of which p are identical, taken r at a time is
- n–pCr + n–pCr–1 + n–pCr–2 + ......... + n–pC0 if r £ p.
- n–pCr + n–pCr–1 + n–pCr–2 + ......... + n–pCr–p if r > p.
7.4 SELECTION FROM DISTINCT OBJECTS
The number of ways (or combinations) of n different things selecting at least one of them is
nC1 + nC2 + nC3 +......+ nCn = 2n – 1. This can also be stated as the total number of combination of n different things.
7.5 SELECTION FROM IDENTICAL OBJECTS
- The number of combination of n identical things taking r(r < n) at a time is 1.
- The number of ways of selecting r things out of n alike things is n +1 (where r = 0, 1, 2,..., n).
- The number of ways to select some or all out of (p + q + r) things where p are alike of first kind, q are alike of second kind and r are alike of third kind is
= (p + 1) (q + 1) (r + 1) – 1
7.6 SELECTION WHEN BOTH IDENTICAL AND DISTINCT OBJECTS ARE PRESENT
If out of (p + q + r + t) things, p are alike one kind, q are alike of second kind, r are alike of third kind and t are different, then the total number of combinations is
(p + 1) (q + 1) (r + 1) 2t – 1
7.7 Divisors of a given natural number :
Let , where p1, p2, p3, .... pk are different primes and α1, α2, α3, ........, αk are natural numbers then :
- The total number of divisors of N including 1 and N is
= (α1 + 1) (α2 + 1) (α3 + 1) .... (αk + 1)
- The total number of divisors of N excluding 1 and N is
= (α1 + 1) (α2 + 1) (α3 + 1) .... (αk + 1) – 2
- The total number of divisors of N excluding 1 or N is
= (α1 + 1) (α2 + 1) (α3 + 1) .... (αk + 1) – 1
- The sum of these divisors is
= (p10 + p11 + p11 + ....+ ) (p20 + p21 + p22 + ....+ )...... (pk0 + pk1 + pk2 + ....+ )
- The number of ways in which N can be resolved as a product of two factors is
(α1 + 1) (α2 + 1) .... (αk + 1), If N is not a perfect square
[(α1 + 1) (α2 + 1) .... (αk + 1) + 1], If N is a perfect square
- The number of ways in which a composite number N can be resolved into two factors which are relatively prime (or co-prime) to each other is equal to 2n – 1 where n is the number of different factors in N.
8. DERANGEMENT THEOREM
Any change in the given order of the thing is called a Derangement.
(i) If n items are arranged in a row, then the number of ways in which they can be rearranged so that no one of them occupies the place assigned to it is
(ii) If n things are arranged at n places then the number of ways to rearrange exactly r things at right places is
9. DIVISIBILITY OF NUMBERS
The following table shows the conditions of divisibility of some numbers
| Divisible by | Condition |
| 2 | whose last digit is even |
| 3 | sum of whose digits is divisible by 3 |
| 4 | whose last two digits number is divisible by 4 |
| 5 | whose last digit is either 0 or 5 |
| 6 | which is divisible by both 2 and 3 |
| 8 | whose last three digits number is divisible by 8 |
| 9 | sum of whose digits is divisible by 9 |
| 25 | whose last two digits are divisible by 25 |
11. SUM OF NUMBERS
(i) For given n different digits a1, a2, a3 ......., an the sum of the digits in the unit place of all numbers fromed (if numbers are not repeated) is
(a1 + a2 + a3 + ...... + an) (n – 1)!
i.e. (sum of the digits) (n – 1) !
(ii) Sum of the total numbers which can be formed with given n different digits a1, a2, a3 ....... an is
(a1 + a2 + a3 + ..... + an) (n – 1) ! . (111 ......... n times)
10. IMPORTANT RESULTS ABOUT POINTS
If there are n points in a plane of which m(< n) are collinear, then
- Total number of different straight lines obtained by joining these n points is nC2 – mC2 + 1
- Total number of different triangles formed by joining these n points is nC3 – mC3
- Number of diagonals in polygon of n sides is nC2 – n
- If m parallel lines in a plane are intersected by a family of other n parallel lines. Then total number of parallelograms so formed is mC2 × nC2
- n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. Then the number of parts into which these lines divide the plane is equal to-
1 + Σn
7. RELATION AND FUNCTIONS
Ordered Pair: Two elements a and b listed in a specific order , form an Ordered Pair ,denoted by (a , b).
In an ordered pair (a , b), we call a as the first component and b as the second component.
By changing the position of the components , the ordered pair is changed, i.e. (1, 4) (4, 1).
Also, (a , b) = (c , d) => a = c and b = d
Thus, (x , y) = (2 , 7) => x = 2 and y = 7
Cartesian Product of two sets : Let A and B be two non-empty sets. Then, their Cartesian product A × B is the set of all ordered pairs (a , b) such that a A and b B.
i.e. A × B = {(a, b) : a A and b B}
Relation: Let A and B be two non-empty sets. Then, every subset of A × B is called a relation from A to B.
i.e. If R ⊂ A × B, then R is relation from A to B.
Also, if (a , b) ϵ R, we say that a is related to b and we write aRb.
Representation of a Relation :
(i) Roster Form: A relation represented by the set of all ordered pairs contained in it, is said to be in roster form.
Example: If A = {1, 9, 16, 25 } and B = {1, 2, 3, 4, 5}, then a relation R from A to B defined as 'is the square' of can be represented in the roster form as :
R = {(1, 1), (9, 3), (16, 4), (25, 5)}.
(ii) Arrow Diagram : Let R be a relation from A to B. We draw closed bounded figures to represent the sets A and B and write their elements in the corresponding figures. Then, we draw arrows from A to B to indicated the pairing of the corresponding elements related to each other. Thus, we can show the relation given in example 2 by the arrow diagram as shown below.
(iii) Set-Builder Form : A relation R from A to B is said to be in a Set-Builder Form, when written as:
R = {(a, b) : a ϵ A, b ϵ B and a is connected with b by a given rule}
The relation given in example 2 can be represented in the set-builder form as :
R = {(a, b) : a ϵ A, b ϵ B and a = b2}
Domain and Range of a Relation : Let R be a relation from A to B. Then
Domain (R) = Set of first component of all ordered pairs in R.
Range (R) = Set of second component of all ordered pairs in R.
Range of Relation: The property that tells us how the first component is related to the second component of each ordered pair in R, is called the rule of the relation.
Function or Mapping : Let A and B be two non-empty sets. Then, a function or a mapping f from A to B is a rule which associates to each element x ϵ A, a unique element f(x) ϵ B, called the image of x. If f is a function from A to B, then we write f : A → B.
Remarks: For f to be a function from A to B :
(i) every element in A must have its image in B.
(ii) no element in A must have more than one images.
Representation of a function:
(i) Arrow Diagram : Let f be a relation from A to B. We draw closed bounded figures to represent the sets A and B and write their elements in the corresponding figures. Then, we draw arrows from A to B joining each elements of A with its
corresponding image in B. The function given in the above example can be represented by the arrow diagram as shown above.
(ii) Roster Form : Let f be a function from A to B. We form ordered pairs of all the elements of A with their corresponding images in B. Then, the function f is represented as the set of all such ordered pairs The function given in the above example can be represented in the roster form as f = {(1, 2), (2, 4), (3, 6)}.
(iii) Equation Form : Let f be a function from A to B. If f be represented in terms of the rule of association , then it is said to be in the equation form.
In the above example, the rule of association is given by f (x) = 2x . Let y ϵ B be the image of an element x ϵ A.
Then, We have y = 2x
Then, y = 2x is the representation of function f in the equation form.
Domain, Co-Domain and Range of A Function : Let f be a function from A to B . Then, we define:
(a) Domain (f) = A (b) Co-domain (f) = B
(c) Range (f) = Set of all images of elements of A
Function as a Relation: Let A and B be two non-empty sets and R be a relation from A to B. Then R is called a function from A to B , if domain (R) = A and no two ordered pairs in R have the same first components.
Remark: Every function is a relation. But, every relation need not be a function.
REAL VALUED FUNCTIONS: A rule f which associates to each real number x, a unique real number f(x), is called a real- valued function. Here, f(x) is an expression in x.