Math tips, Math formulas

Algebra Tips


Algebra is a branch of pure mathematics and forms an interesting curriculum of education in almost all schools, all over the world. In this article,


Area of rectangle (A)= Length (l) X Breadth (b)

Now, if you know the value of either of the two quantities, among the three variables, we can substitute the value of the two known variables and get the third one! One of the simple algebra tips! Isn't it? We need algebra because it drastically reduces the space for problem solving and lengthy problems can be solved in a few lines. Algebra finds many applications in fields of engineering, finance, economics and architecture!

The mathematical problems in algebra are expressed through algebraic equations. Algebraic equations are a combination of variables and constants. For instance, consider the equation, 3X= 9. This is a simple algebraic equation in one variable. Dividing both sides by 3, we get the solution of this equation as, X=3. Similarly, 3X + 2Y = 8, is an algebraic equation in two variables. Now, since this equation has two variables, we will require another equation with the same variables, to find the solution of this equation. Now consider,

3X + 1Y - 7Z= 8
This is an example of a linear equation in three variables. X,Y and Z are the three variables in this equation. Let's check out some other simple algebra tips! Here in this equation. X, Y and Z are the three variables, whose values can change with the constraint that the left hand side (L.H.S) of the equation must always be equal to the right hand side (R.H.S) of the equation. The numbers 3, 1 and -7 are called the coefficients of X, Y and Z respectively. The respective values of X, Y and Z, which make the L.H.S=R.H.S are called solutions of the given algebraic expression. This is so, because if we take X=3, Y= -1 and Z=0 and put in the above equation, we would get L.H.S=R.H.S.! Try it! You will get 8=8. Its as simple as that! Here, we got the solution with the help of hit and trial method. Suppose, we had to find the solution of this equation algebraically, then we would require two other linear equations in three variables.

Some important algebra tips that are the basics of algebra are as follows

1. We can add the same number on both sides of an equation. e.g., 5X - 8 = 7 is the same as 5X - 8 + 8 = 7 + 8.
2. We can subtract the same number on both sides of an equation. e.g. 2a + 4 - 5 = 6 – 5.
3. We can multiply both sides of an equation by the same (non-zero) number. e.g. (5/6)x = 7 is the same as 6 X (5/6)x = 7 X 6.
4. We can divide both sides of an equation by the same (non-zero) number. e.g. 2r = 10 is the same as (2r/2) = (10/2).

Set Properties and Operations

Several properties and operations have been defined for sets. For the purpose of this section, sets are assumed to be collections of numbers. Set is defined as the set .


 Properties

An object is an element of a set when it is contained in the set. For example, 1 is an element of . This is written as . Similarly, the fact that 11 is not an element of is written as . The universe (usually represented as ) is a set containing all possible elements, while the empty set or null set (represented as or ) is a set containing no elements.

The cardinality of a set is the number of elements in the set. The cardinality of (written as or ) is 10.

The complement of a set is the set containing all elements of the universe which are not elements of the original set. For example, if the universe is defined as , then the complement of with respect to (written as ) is . The cardinalities of a set and its complement together equal the cardinality of the universe. Thus, the universe and the null set are complements of each other.

A set is a subset of another when all the elements in the first set are contained in the second set. Given sets and , is a subset of , notated as , if and only if for all , implies . All sets are subsets of the universe. By definition, all sets are subsets of themselves and by convention, the null set is a subset of all sets. For example, . Any given set has subsets.

Two sets are equal if they are subsets of each other.

A set's proper subsets are all subsets except the set itself. This relationship is notated by

Operations

1. The union of two sets is the set containing all elements of either or , including elements of both and . This operation is written as . For example, .
2. The intersection of two sets is the set containing all elements of both and . This is written as . For example, .
3. The sum of the cardinalities of the intersection and union of two sets is equal to the sum of the cardinalities of the two sets.

Concept of Sets

The word "set" implies a collection or grouping of similar objects or symbols. The objects in a set have at lea& one characteristic in common, such as similarity of appearance or purpose. A set of tools would be an example of a group of objects not necessarily similar in appearance but similar in purpose. The objects or symbols in a set are called members or



Element of Set



The elements of a mathematical ret are usually symbols, ouch as numerals , lines, or points. For example, the integer6 greater than zero and less than 6 form a set, as follows:



{1, 2, 3, 4}



Notice that braces are used to indicate sets. This is often done where the elements of the set are not too numerous, Since the elements of the set (2, 4, 6) are the same as the elements of (4, 2, 6}, there two seta are said to be equal. In other words, equality between sets has nothing to do with the order in which the elements are arranged. Further- more, repeated elements are not necessary. That is, the elements of (2, 2, 3, 4) are simply 2, 3, and 4. Therefore the sets (2, 3, 4) and (2, 2, 3,4) are equal. Practice problems:



1. Use the correct symbols to designate the set of odd positive integers greater than 0 and less than 10.

of names of days of the week which do not contain the letter "s".

3. List the elements of the set of natural numbers greater than 15 and less than 20.

4. Suppose that we have sets as follows:



A = (1, 2, 3) C = (1, 2, 3, 4)



B = (1, 2, 2, 3) D = (1, 1, 2, 3)



Which of these sets are equal?



Answers:



1. 1, 3, 5, 7, 9

2. {Monday, Friday}

3. 16, 17, 18, and 19

4. A=B=D



SUBSETS



Since it is inconvenient to enumerate all of the elements of a set each time the set is mentioned, sets are often designated by a letter. For example, we could let S represent the set of all integers greater than 0 and less than 10. In symbols, this relationship could be stated as follows:



s = (1, 2, 3, 4, 5, 6, 7, 8,9)



Now suppose that we have another set, T, which comprises all positive even integers less than 10. This set is then defined as follows:



T - (2, 4, 6, 8)



Notice that every element of T is also an element of S. This establishes the SUBSET relationship; T is said to be a subset of 9.

Multiply By 4




To quickly multiply by four, double the number and then double it again.

Often this can be done in your head.
 
 
Multiply By 5


To quickly multiply by 5, divide the number in two and then multiply it by 10. Often this can be done quickly in your head.
 
 
Finger Math:
 
9X Rule




To multiply by 9,try this:

(1) Spread your two hands out and place them on a desk or table in front of you.

(2) To multiply by 3, fold down the 3rd finger from the left. To multiply by 4, it would be the 4th finger and so on.

(3) the answer is 27 ... READ it from the two fingers on the left of the folded down finger and the 7 fingers on the right of it.



This works for anything up to 9x10!

The fundamental concept to remember when performing calculations with percentages is that the percent symbol can be treated as being equivalent to the pure number constant 1 / 100 = 0.01. , for an example 35% of 300 can be written as (35 / 100) × 300 = 105.




To find the percentage that a single unit represents out of a whole of N units, divide 100% by N. For instance, if you have 1250 apples, and you want to find out what percentage of these 1250 apples a single apple represents, 100% / 1250 = (100 / 1250)% provides the answer of 0.08%. So, if you give away one apple, you have given away 0.08% of the apples you had. Then, if instead you give away 100 apples, you have given away 100 × 0.08% = 8% of your 1250 apples.



To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:



(50 / 100) × (40 / 100) = 0.50 × 0.40 = 0.20 = 20 / 100 = 20%.

It is not correct to divide by 100 and use the percent sign at the same time. (E.g. 25% = 25 / 100 = 0.25, not 25% / 100, which actually is (25 / 100) / 100 = 0.0025.)



The easy way to calculate Addition in percentage (discount 10% + 5%):



y = [(x1+x2) - (x1*x2)/100]



for example: Dept Store promotion: discount 10%+5%, the total discount is not 15%, but:



y = [(10% + 5%) − (10% * 5%) / 100] = [15% − 0.5%] = 14.5%

The first tip for solving addition problems is to teasch learners to "count on" counting on works by having the learnercount on from the biggest number or the largest added in an addition problem. For example, in the 6+4 the learner counts up from six(6) aloud by saying sssixxx..... and then "7, 8, 9, 10" The learner can use their figers, toes, or ?

          Touch points to help see that they have counted up free places. Touch points are basically points on a number that learner can use to "count on " with, for example, the number 4 has four easy to distinguish touch points. So for a problem like 6+4 they say six aloud then counts on by touching the four points of the number 4while seven, eight, night, ten.