RATE

In general, Interest is defined as the rent or charge which we pay for utilizing a particular thing or money (Principal). Most often it represents the payment for giving or using money (principal) for a period of time. Interest cannot be charged randomly, they are charged on some rates. The rate is decided by the owner of the property and the owner only can enjoy the benefits of the money acquired as interest. The formula and example problems for finding the rate of interest when the principal and time duration is given is seen in the following sections.

Formula to Calculate Rate

The General formula to calculate the simple interest is as follows,

Simple Interest, (S.I)=

-->P*N*R/100

Where,

P = Principal amount

N = Number of years

R = Rate of Interest.

From the above formula, the formula for rate is derived as,

Rate of Interest =

--> {(S.I)*100/P.N}

Polynomial Expressions

In Polynomial Expression, we use letters like a, b, x and y to denote numbers. Performing addition, subtraction, multiplication, division or extraction of roots on these symbols and real numbers, we obtain what are called algebraic expressions.

The word algebra is derived from the Arabic word al–jab. In Arabic language, ‘al’ means ‘the’ and ‘jabr’ means ‘reunion of broken parts’. The usage of the word can be understood by a simple example. In the equation x + 5 = 9, the left hand side is the addition (sum) of two parts x and 5. If we add (unite) (–5) to each side of the equation, we get

(x + 5) + (–5) = 9 + (–5) or x + [5 + (–5)] = 9 – 5 or x + 0 = 4 or x = 4.

Here 9 and −5 are reunited to get 4. This type of mathematics is called algebra. Indian mathematicians like Aryabhatta, Brahmagupta, Mahavir, Sridhara, Bhaskara II have developed this subject very much. The Greek mathematician Diophantus has developed this subject to a great extent and hence we call him the father of Algebra.

Symbols in an algebraic expression are called variables of the expression. For example, in ax + b, if a and b are specific numbers and x is not specified, then x is the variable of ax + b. In 2x2 + 3xy + y2, x and y are variables. If the variables in an algebraic expression are replaced with specific numbers, then the expression yields a number and this number is called a value of the expression. For example, 2x2 + y is an algebraic expression and x and y are variables of the expression.

Direct Variation

The mathematical relationship between two variables, expressed in the equation in which one of the variable is equal to the constant multiple of the other of an equation or function expressing direct variation. The relationship between the two variables remains in a constant ratio. It is called as direct variation.

If two variables A and B are so related that when A increases ( or decreases) in a given ratio, B also increases ( or decreases) in the same ratio, then A is said to vary directly as B ( or A is said to vary as B). This is symbolically written as,

-->A ∝ B ( A varies as B)
Suppose a train moving at a uniform speed travels D km in T minutes.

Now, consider the following table:


-->

D(km)	24	12	48	36
T(km)	30	15	60	45

The table shows that T is increased or decreased in the same ratio as the distance D. Hence, the variables D and T are in Direct Variation (i.e., D ∝ T)

We also see that , (24 / 30) = 4/5, (12 / 15) = 4/5, (48 / 60) = 4/5, (36 / 45) = 4/5

The ratio of the corresponding values of D and T is always same. So, we can say that, the value of D/T is constant . If this constant be k, then D/T = K or, D= KT. This constant is called the Constant of Variation.

Hence, if D ∝ T then D = KT where K = Constant of Variation.
Thus, if A ∝ B then A=mB where m is the constant of variation and is independent of A and B.
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Sequence and its types

A sequence is a list of items. We can specify any item in the list by its place in the list: first, second, third, fourth, and so on. Many useful lists have patterns so we know what items occur in each place in the list.

There are 2 kinds of sequences:

Finite Sequence: A finite sequence is a list made up of a finite number of items.

Infinite Sequence: An infinite sequence is a list that continues without end.

Examples of Finite Sequences:

Listed below are some examples of Finite Sequences.

Example 1) The sequence 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the first 10 odd numbers.

Example 2) The sequence a, e, i, o, u, is the sequence of vowels in the alphabet.

Examples of Infinite Sequences:

Listed below are some examples of Infinite Sequences.

Example 1) The sequence 2, 4, 6, 8, 10, 12, 14, 16, ... is the sequence of even whole numbers. The 100th place in this sequence is the number 200.

Example 2) The sequence a, b, c, a, b, c, a, b, c, a, b, ... is the sequence of the letters a, b, c, repeating in this pattern forever.

Inductive Reasoning

Inductive reasoning is the type of reasoning which involves moving from a set of specific facts to general conclusion. It uses the premises from the objects that have been examined to establish a conclusion of an object which has not been examined. The mathematical induction is the form of deductive reasoning. It is a kind of reasoning that allows for the possibility that that is false even where all of the premises are true.

Types of inductive reasoning:

Generalization:

It proceeds from the premise about sample to a conclusion. The conclusion about the population. The proportion Q of the sample has attribute A. Therefore the proportion Q of the population has attribute A.

Statistical syllogism:

It proceeds from a generalization to the conclusion. This conclusion about an individual. Let us consider an example. The proportion Q of the population has attribute A. Individual I is a member of p. Therefore there is a probability which corresponds to Q that I has A.

Simple induction:

It proceeds from a premise about a sample group to the conclusion. This conclusion about another individual. Example:Proportion Q of the known instances of population has attribute A. An Individual I is another member of p.Therefore there is a probability corresponding to Q that I has A.(source: Wikipedia)

Causal inference:

Based on conditions of the occurrence of an effect it draws the conclusion about a causal connection.

Prediction:

From the past sample a prediction draws the conclusion about a future individual.

• These types are very important for inductive reasoning, which is derived from the definition of inductive reasoning.

For example on Inductive reasoning 

Axiom

Axiom is a statement that is accepted as true. It does not require any proof. Geometry axioms is a statement that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for published, and serves as a starting point for reducing and inferring other truths. In mathematics, axiom refers to two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that works as a starting point from which other statements are logically derived.

Example of an Axiom

• A line contains infinitely many points.
• Things which are equal to the same things are equal to each other.

Let us see below some of Geometry Axioms

Straight Angles: All the straight angles are congruent in nature.

Linear Pair: If two angles are linear, than they are supplementary.
Vertical Angles: All the vertical angles are congruent in nature.

Triangle Sum: The sum of all the three angles in a triangle is 180º.

Rhombus: If 4 sides of a rhombus are congruent, then it is a parallelogram.

Congruent Supplements: If an angle is supplement, then it is congruent.

Construction: Two or more points form a line.

Sum of two sides: The sum of two sides of a triangle is greater than the third side.

Longest side: The largest side of a triangle is from its largest angle.