Transcript Amity School of Business

Amity School of Business
Amity School of Business
BBA Semester IV
ANALYTICAL SKILL BUILDING
1
Ratio
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Ratio is simply the quotient of two numbers.
This is where we get the word "rational
numbers." A rational number is any number
that can be expressed as the ratio or quotient of
two integers (denominators cannot equal
zero). Every time you write a fraction, you have
written a ratio. A proportion is simply the
equating of two ratios. Whenever one ratio
(or fraction) equals another ratio (or fraction),
this is a proportion
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Ratio
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To compare ratios, write them as fractions. The
ratios are equal if they are equal when written as
fractions.
Example: Are the ratios 3 to 4 and 6:8 equal?
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are
equal; that is, if 3 x 8 = 4 x 6.
Since both of these products equal 24, the
answer is yes, the ratios are equal.
3
Ratio
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A ratio is a comparison of two numbers.
We generally separate the two numbers in
the ratio with a colon ":".
4
Ratio
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A comparison of numbers with the same
units so units are not required.
3:9
4/12
5 to 20
Ratio1:3 <1/3
5
Rate
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A comparison of 2 measurements with
different units.
Example Unit
10 km per 2 h
$ 6 per 3 h
Rate 5 km per h <$ 2/h
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Few facts
Duplicate ratio of a : b
a2 : b2
Triplicate ratio of a : b
a3 : b3
Subduplicate ratio
√a : √b
Subtriplicate ratio
3√a
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: 3√b
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Properties of ratio
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Invertendo
a
c

b
d
b
d

a
c
Alternendo
a
c

b
d
a
b

c
d
componendo
a
c

b
d
Dividendo
a
c

b
d
Componendo
- Dividendo
a
c

b
d
Equal ratio
a
c e
 
b
d f
ab
cd

b
d
a-b
c-d

b
d
ab
cd

a-b
c-d
ace
bdf
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Ratio
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Let x : y be a ratio, which can also be written as x/y. We
will try to find out what will happen when a constant a is
added both to the numerator and denominator
CASE I (x/y < 1 )
If x/y<1, then addition of a constant positive number to
numerator and denominator leads to a bigger ratio than
the ratio itself, i.e.
x/y < (a+x)/(a+y) for x/y<1
where a is a constant positive number. e.g. 1/2 is less
than 1 and when we add 2 to both numerator and
denominator we get 3/4 and 3/4 is greater than 1/2
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Ratio
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CASE I (x/y < 1 )
If x/y<1, then addition of a constant positive number to
numerator and denominator leads to a bigger ratio than
the ratio itself, i.e.
x/y < (a+x)/(a+y) for x/y<1
where a is a constant positive number. e.g. 1/2 is less
than 1 and when we add 2 to both numerator and
denominator we get 3/4 and 3/4 is grater than 1/2 .
Similarly, Subtraction leads to a similiar ratio, i.e., x/y >
(x-a)/(y-a) for x/y<1 Lets consider a fraction 5/10, if 5 is
subtracted from numerator as well as denominator, we
get 0 and it is less than 5/10(i.e. 1/2) Thus the rule for
the case of subtraction is the reverse of the case of
addition, as can be easily seen by the given example
10
Ratio
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CASE II (x/y > 1 )
The above rule gets totally and exactly
reversed for x/y >1. Therefore,
x/y > (x+a)/(y+b) and x/y < (x-a)/(y-b).
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Proportion
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Four (non-zero) quantities of the same kind a, b,
c, d are in proportion,
written as a : b :: c : d if a/b = c/d
The non-zero quantities of the same kind a, b, c,
d, ... are in continued proportion if
a/b = b/c = c/d = ...
In particular, a, b, c are in continued proportion if
a/b = c/d. In this case b is called the mean
proportion; b = ac; c is called third proportional.
If a, b, c, d are in proportion, then d is called
fourth proportional.
12
Proportion
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The four parts of the proportion are separated into two
groups, the means and the extremes, based on their
arrangement in the proportion. Reading from left-to-right
and top-to-bottom, the extremes are the very first
number, and the very last number. This can be
remembered because they are at the extreme beginning
and the extreme end. Reading from left-to-right and topto-bottom, the means are the second and third numbers.
Remembering that "mean" is a type of average may help
you remember that the means of a proportion are "in the
middle" when reading left-to-right, top-to-bottom. Both
the means and the extremes are illustrated below.
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Proportion
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Algebra properties tell us that the products of the
means is equal to the product of the extremes
fraction one-half is equal to two-fourths.
This is shown as a proportion below.
1/2 = 2/4
2*2=1*4
4=4
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Proportion
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A proportion is an equation with a ratio on each
side. It is a statement that two ratios are equal.
3/4 = 6/8 is an example of a proportion. When
one of the four numbers in a proportion is
unknown, cross products may be used to find
the unknown number. This is called solving the
proportion. Question marks or letters are
frequently used in place of the unknown number.
Example: Solve for n: 1/2 = n/4.
Using cross products we see that 2 x n = 1 x 4
=4, so 2n = 4. Dividing both sides by 2, n = 4 / 2
so that n = 2.
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Proportion
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Used to compare two ratios or make
equivalent fractions. To solve one can:
1.use equivalent fractions.
2.cross multiply.
1/2 = 3/4
1x4=2x3
Note: The product of the outside (first and
last) numbers equals the product of the
two middle numbers.
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Mixtures
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• Simple mixture – When two different
ingredients are mixed together, it is known
as simple mixture.
• Compound Mixture – When two mixtures
are mixed together to form another
mixture, it is known as compound mixture
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Alligation
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When different quantities of the same or
different ingredients of different costs (one
cheap and other dear) are mixed together to
produce a mixture of a mean cost, the ratio of
their quantities is inversely proportional to the
difference in their cost from the mean cost.
Quantity _ Of _ Cheap Pr ice _ Of _ Dear  mean Pr ice

Quantity _ Of _ Dear
Mean Pr ice  Pr ice _ Of _ Cheap
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Alligation
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If a vessel contains ‘a’ litres of liquid A, and if ‘b’
liters be withdrawn and replaced by liquid B,
then if ‘b’ litres of mixture be withdrawn and
replaced by liquid B, and the operation repeated
n
‘n’ times in all. Then :
 a b


Liquid A left after nth operation
 a 
Liquid B left after nth operation
 a b
 1 

 a 
n
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When X1 quantity of A of cost C1 and
X2 of Business
quantity of B of cost C2 are mixed, then the cost
of the mixture is
Cm
C1 X 1  C 2 X 2

X1  X 2
When two mixtures M1 and M2, each containing
ingredient A and B in the ratio a:b and x:y are
mixed in the roportion of the ingredients A and B
i.e. Qa:Qb in the mixture is
qa
qb
 x 
 a 

M1
  M2


x

y
 ab




y 
 b 

M1
  M2


x

y
 ab


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Alligation
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 x   qa 


  
quantiy _ M 1  x  y   qa  qb 

quantity _ M 2  qa   a 

  

 qa  qb   a  b 
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Problem
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A milkman with draws 1lit of milk from a vessel containing 10 lit of pure milk and
replaces it with water. Lets see how the concentration changes after each operation.
Initial amount of milk = 10 lit
After first replacement, amount of milk = 9 lit and water = 1lit
As per the formula, Amount of milk left / Initial amount of milk = 1 – 1/ 10 =
9/10 (same as above)
Second operation:
Now the vessel contains 9lt milk and 1lt water
The1 lt mixture that ‘ll taken out of the container in second operation ‘ll contain 0.1 lt
of water and 0.9lt of milk and that ‘ll be replaced by 1 lt of water
So amount of milk in the mixture after second operation = 9 – 0.9 = 8.1
and amount of water in the mixture after second operation = 1 – 0.1 + 1 = 1.9 (=1081.)
Now as per formula
Amount of milk left / Initial amount of milk
= (1 – 1/10)^2
= 81/100 (same as above)
Notes:
1. Some time the amount of liquid taken out and replaced by are not same. In that
case don’t use the formula. Calculate the final composition by the method explained
above
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Intersection of a Set and its
Complement
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The intersection of a set and its complement is the
empty set.
For example, let A consist of all the unbroken
plates in a set of plates, and let A' consist of all
the broken plates. Then (A . A‘) is empty
because there are no plates that are unbroken
and broken at the same time.
We summarize this law in symbols as follows:
A . A' = 0.
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Venn diagrams
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Venn diagrams are illustrations used in
the branch of mathematics known as set
theory. They are used to show the
mathematical or logical relationship
between different groups of things (sets).
A Venn diagram shows all the logical
relations between the sets
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Venn diagrams
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25
Complement of an Intersection
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Since the intersection of two sets A and B is given
by
A . B = {x: x is in A and x is in B},
the complement of this intersection is given by
(A . B)' = {x: not-(x is in A and x is in B)}.
But not-(x is in A and x is in B) has the same
meaning as x is not in A or x is not in B, and
{x: x is not in A or x is not in B} = A' + B'.
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Complement of an Intersection
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we have the following law for the complement of
an intersection:
(A n B)' = A' u B'.
This is another of De Morgan's laws for sets.
For example, let the universal set be the whole
numbers from 1 to 10. Let A = {1, 2, 3, 4, 7, 8, 9,
10} and let B = {2, 4, 6, 8, 10}. Then A . B = {2,
4, 8, 10}, and so (A . B)' = {1, 3, 5, 6, 7, 9}. Also
A' = {5, 6}, B' = {1, 3, 5, 7, 9} and A' + B' = {1, 3,
5, 6, 7, 9}. Therefore, in this example, (A . B)' =
A' + B'.
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Complement of an Union
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Since the union of two sets A and B is given
by
A + B = {x: x is in A or x is in B},
the complement of the union is given by
(A + B)' = {x: not-(x is in A or x is in B)}
But not-(x is in A or x is in B) has the same
meaning as x is not in A and x is not in B,
and
{x: x is not in A and x is not in B} = A'.B'.
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Complement of an Union
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We have the following law for the complement of a union:
(A + B)' = A' . B'.
This is one of De Morgan's laws for sets.
For example, let the universal set be the set of all
substances. Let A be the set of all solids, such as stone
and iron. Let B be the set of all liquids, such as water
and oil. Then A + B is the set all substances that are
solid or liquid. The complement (A + B)' is the set of all
substances that are not solid or liquid, in other words the
set A' . B' of all substances that are not solid and not
liquid, such as oxygen and nitrogen (which are gases).
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Complement of a complement
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The complement of a complement is written in
symbols as follows: (A')' = {x: not-(x is not in A)}.
Since not-(x is not in A) has the same meaning as
x is in A, it follows that (A')' = {x: x is in A}. But {x:
x is in A} = A. Therefore we have the
complement law:
(A')' = A.
For example, let the universal set be the set of
whole numbers from 1 to 10, and let A be the set
of even numbers from 2 to 10. Then A' is the set
of odd numbers from 1 to 9, and (A')' is the
original set A of even numbers from 2 to
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Union of a set and its
complement
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The union of a set and its complement is the
universal set.
For example, let x stand for an animal, let A
= {x: x is male}, and let A' = {x: x is
female}. Then A + A' = {x: x is male or x is
female} = the set of all animals.
We write this law briefly as follows:
A u A' = U.
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Coordinate geometry
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• Distance between two points P(x1,y1) and
Q(x2,y2)
• Coordinates of a mid point
• The coordinate of the point R(x,y) which divides
a straight line joining two points (x1,y1) and
(x2,y2) internally in a given ratio m1:m2
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Coordinate geometry
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• Area of triangle whose vertices are
(x1,y1), (x2,y2) and (x3,y3) is
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